Mémoire de Master 2Université Paris-DiderotAnnée 2014-2015

Representations of the Yangian inDeligne’s category Rep(Glt)

Léa BittmannUnder the supervision of Pavel Etingof and David Hernandez

Ce mémoire récapitule le travail effectué lors de mon stage de master au MIT, USA, sousla direction de Pavel Etingof. Le but de ce stage était d’étudier, et si possible de classifier, lesreprésentations du Yangian Y (g), d’une algèbre g définie comme un objet de la catégorie de DeligneRep(Glt).

Les représentations du groupe linéaire GlN , où N est un entier naturel, sont assez bien connuesdes mathématiciens. En particulier, les représentations complexes de dimension finie de GlN sontcomplètement connues (voir [10], par exemple). En 1982, P. Deligne utilise ces résultats pourdéfinir une catégorie, qu’on appellera plus tard Rep(Glt), pour t un complexe quelconque, dansson article sur les catégories Tannakiennes [6] (voir exemple 1.27). En 2007, dans son articlefondamental [5], il définit la catégorie Rep(St), en interpolant la catégorie des représentations dugroupe symétrique SN , quand N est un grand entier. Dans la dernière partie de ce même article,il donne une définition plus précise de la catégorie Rep(Glt).

C’est cette définition que nous allons utiliser dans ce mémoire, plus particulièrement la con-struction détaillée par Comes et Wilson dans [3]. Un des objets de cette catégorie, noté g, nousintéressera particulièrement, il s’agit de l’équivalent de glN dans la catégorie Rep(Glt). A partirde cette "algèbre" g, on pourra définir son Yangian Y (g), en suivant la définition de P. Etingofdans [8]. Là aussi, il s’agira de mimer la définition du Yangian Y (glN ).

Les Yangians forment une famille de groupes quantiques. Le premier Yangian étudié est juste-ment Y (glN ), considéré par L. Faddeev et l’école de St Petersburg. Ils s’en servaient à l’originepour générer des solutions de l’équation de Yang-Baxter quantique. En 1985, V. Drinfeld intro-duit pour la première fois le terme "Yangian", en honneur du physicien C.N. Yang. Le YangianY (a) d’une algèbre de Lie simple de dimension finie a est alors défini comme une déformationcanonique de l’algèbre enveloppante universelle U(a[z]) de l’algèbre des lacets a[z]. Une nouvelleprésentation du Yangian est donnée par V. Drinfeld en 1988. Le Yangian de glN bénéficie d’unetroisième présentation, où les relations entre générateurs peuvent s’écrire sous la forme d’une seuleéquation matricielle. Cette structure particulière peut expliquer pourquoi Y (glN ) avait été étudiéplus précocement.

Notre but étant d’étudier les représentations du Yangian Y (g), il y a deux aspects distincts àprésenter, d’une coté la catégorie de DeligneRep(Glt), et de l’autre les Yangians. Nous allons com-mencer par rappeler les définitions et concepts généraux de théorie des catégories dont nous auronsbesoin. Ensuite, nous détaillerons la définition de Rep(Glt), et nous étudierons cette catégorie,avant d’arriver à la définition de Y (g). Pour cela, nous donnerons d’abord la définition classique deY (glN ), que nous utiliserons ensuite pour définir Y (g). Le chapitre suivant sera consacré à l’étudedes représentations du Yangian de glN , pour lesquelles il existe une classification bien connue. Etenfin le dernier chapitre sera consacré aux représentations de Y (g). Nous présenterons des pistespour l’étude des ces représentations, en se basant sur l’étude faite au chapitre précédent.

1

IntroductionThe representation theory of the general linear group GlN has been intensively studied by math-ematicians. But in 1982, in his paper [6], P. Deligne used the concept of Tannakian categoriesto define a category which will be later be called Rep(Glt), where t is not necessarily an integer(see example 1.27). In his fundamental paper [5] of 2007, he introduced the category Rep(St) byinterpolating the well-known categories of representations of the symmetric group Rep(SN ), whenN is a large integer, to any complex number t, and in the last section of this paper, he also gavea more detailed definition of the category Rep(Glt).

In this report, we are going to define the category Rep(Glt), following the definition in [3],and in particular, one of its objects g, which interpolates the algebra glN in this category. We aregoing to be able to define the Yangian Y (g) of g, using the definition introduced by Etingof in [8].

A Yangian is a type of quantum group, which was first studied by L. Faddeev, in the context ofthe quantum inverse scattering method. At the time, it was mostly seen as a tool to generate solu-tions to the quantum Yang-Baxter equation. Faddeev and the St Petersburg School were actuallystudying the Yangian of the general linear algebra glN . The term "Yangian" was introduced in1985 by V. Drinfeld, in honor of the physicist C.N. Yang, to define a Hopf algebra associated to anysimple finite-dimensional Lie algebra a over C. The Yangian Y (a) is defined as a deformation ofthe universal enveloping algebra U(a[z]) for the polynomial current Lie algebra a[z]. The Yangianof the general linear algebra Y (glN ) stays a special case, as there is a special presentation of itin which the defining relations can be written in the form of a single relation on the matrix ofgenerators.

The aim of this work is to study the representation of the Yangian Y (g) of the algebra g.Hence there are two different directions to work with : the category Rep(Glt) and the Yangianof the general linear group Y (glN ). So we are going to begin by recalling some definitions andresults of category theory which we are going to use later, and then study first Rep(Glt), with theresults we are interested in, then explain what is a Yangian. Then we are going to get interestedmore precisely on representations of the Yangian Y (glN ), in order to get some results that couldbe used in the case of Y (g). The last chapter explains the results we have and want to get onrepresentations of Y (g).

2

Contents

1 Preliminaries on category theory 51.1 Symmetric tensor category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 C-linear category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.2 Monoidal category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.3 Symmetric category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1.4 Rigid monoidal category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 Categorical dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Additive envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.1 Additive category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3.2 Construction of the additive envelope . . . . . . . . . . . . . . . . . . . . . 11

1.4 Karoubi envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4.1 Splittings of idempotents and Karoubi category . . . . . . . . . . . . . . . . 121.4.2 Construction of the Karoubi envelope . . . . . . . . . . . . . . . . . . . . . 13

1.5 Krull-Schmidt category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 The category Rep(Glt) 162.1 The category Rep0(Glt) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1.1 Words and diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1.2 The skeleton category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.1.3 Tensor structure on Rep0(Glt) . . . . . . . . . . . . . . . . . . . . . . . . . 192.1.4 Universal property of Rep0(Glt) . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2 Definition of Rep(Glt) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.1 Universal property of Rep(Glt) . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.2 Interpolation of the classical case . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3 Definition of the Lie algebra g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.4 Indecomposable objects in Rep(Glt) . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4.1 Bipartitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.4.2 Symmetric group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4.3 The primitive idempotent eλ . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4.4 Classification of idempotents . . . . . . . . . . . . . . . . . . . . . . . . . . 262.4.5 Analogy with the classical case . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.5 Generic semisimplicity of Rep(Glt) . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Yangian of g 293.1 Classical Yangian of glN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1.3 Matrix form of the defining relations . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Yangian in complex rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2.2 Hopf algebra structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3

4 Representations of the Yangian Y (glN ), where N is an integer 354.1 Highest weight representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2 Representations of Y (gl2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2.1 A word on representations of gl2 . . . . . . . . . . . . . . . . . . . . . . . . 374.2.2 Prerequisites for the classification theorem . . . . . . . . . . . . . . . . . . . 384.2.3 The classification theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.3 Irreducible representations of Y (glN ) . . . . . . . . . . . . . . . . . . . . . . . . . . 424.3.1 A criteria for finite-dimensional representations . . . . . . . . . . . . . . . . 424.3.2 Structure of finite-dimensional irreducible representations . . . . . . . . . . 43

5 Representations of Y (glt) 465.1 Evaluation representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.2 Two conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.2.1 First conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.2.2 Second conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.3 The invariant algebra Y (g)g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.3.1 Representations of the invariant algebra . . . . . . . . . . . . . . . . . . . . 485.3.2 Generators of the invariant algebra . . . . . . . . . . . . . . . . . . . . . . . 49

5.4 Action of the invariant algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.4.1 Action on an evaluation representation . . . . . . . . . . . . . . . . . . . . . 505.4.2 Action on a tensor product of evaluation representations . . . . . . . . . . . 515.4.3 Traces of permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.4.4 Action on the evaluation of an exterior product representation . . . . . . . 53

4

Chapter 1

Preliminaries on category theoryWe will assume that the reader is familiar with the basic concepts of category theory, such asfunctors, natural transformations, etc.

As we will use some more advanced concepts precisely, we will begin by defining them.

1.1 Symmetric tensor category

The term "tensor category" can refer to different definitions, in which the conditions under whichthe category is called tensor can vary. Here, we will use a weak definition of tensor category,because it is the one we are going to need.

Definition 1.1.1. A symmetric tensor category is a C-linear rigid symmetric monoidal category,in which the tensor product is biadditive and End(1) = C.

We are going to give precise definitions for these terms, but we invite the interested reader toconsult other sources, see [9] for example, for a more detailed discussion, and more examples.

1.1.1 C-linear categoryA C-linear category is a category enriched over the category of C-vector spaces. Precisely

Definition 1.1.2. A C-linear category is a category C in which, for every objects X,Y , the Hom-space HomC(X,Y ) has the structure of a C-vector space.

Example 1.1.3. The category of C-vector spaces is a C-linear category.

1.1.2 Monoidal categoryThe following definition is the usual definition of a monoidal category, as found in [12] for example.

Definition 1.1.4. A monoidal category is a category C equipped with :

• A bifunctor : ⊗ : C × C → C called the monoidal product or tensor product.

• A object 1 called the unit object or the identity object.

• A natural isomorphism α, called associativity constraint such that for all objects X,Y and Z,αX,Y,Z is an isomorphism of C from X(⊗Y ⊗Z) to (X⊗Y )⊗Z, which satisfy the pentagonaxiom :

For all objects X,Y, Z, T , the diagram

5

X ⊗ (Y ⊗ (Z ⊗ T ))

X ⊗ ((Y ⊗ Z)⊗ T ) (X ⊗ Y )⊗ (Z ⊗ T )

(X ⊗ (Y ⊗ Z))⊗ T ((X ⊗ Y )⊗ Z)⊗ T

1X ⊗ αY,Z,T αX,Y,Z⊗T

αX,Y ⊗Z,T

αX,Y,Z ⊗ 1T

αX⊗Y,Z,T

is commutative.

• Two natural isomorphisms λ and ρ respectively called left and right unitor, such that for anyobject X of C, λ and ρ induce isomorphisms λX : 1⊗X → X and ρX : X ⊗ 1→ X, whichsatisfy the following :

For all objects X,Y , the diagram :

X ⊗ (1⊗ Y )

1X⊗λY &&

αX,1,Y// (X ⊗ 1)⊗ Y

ρX⊗1Yxx

X ⊗ Y

(1.1)

is commutative.

Remark 1.1.5. Mac Lane’s Coherence Theorem for monoidal categories states that every "formal"diagram built using α, λ and ρ, and the ⊗ product commutes. We will not go into details aboutwhat formal means (see [12] for a precise definition and the proof of the Theorem). All the futuresdiagrams we will consider will be formal and as such we will not write λ and ρ anymore, and αonly sometimes.

Example 1.1.6. The category Set of sets is a monoidal category, the monoidal product being theCartesian product, and the unit object being a set with one element.

Example 1.1.7. If R is a commutative ring, the category R-Mod of R-modules, with the tensorproduct of modules

⊗R, is a monoidal category. The unit object is R itself.

We could ask that a functor between monoidal categories preserve the tensor product and theidentity object, this would give a certain definition of monoidal functor. But we are going to use amore general definition, where the monoidal structure is preserved up to a specified isomorphism,which should then satisfy some compatibility conditions. This leads to the following definition.

Definition 1.1.8. Let C = (⊗,1, α, λ, ρ) and D = (⊗′,1′, α′, λ′, ρ′) be two monoidal categories.A monoidal functor is a tuple (F, η, χ), where

• F : C → D is a functor,

• ηX,Y : F (X)⊗′ F (Y )→ F (X ⊗ Y ) is a natural isomorphism,

• ε : F (1)→ 1′ is an isomorphism,

6

such that, for every three objects X,Y, Z of C, the following diagrams commute

F (X)⊗′ (F (Y )⊗′ F (Z))α′F (X),F (Y ),F (Z)

//

1F (X)⊗′ηY,Z��

(F (X)⊗′ F (Y ))⊗′ F (Z)

ηX,Y ⊗′1F (Z)

��

F (X)⊗′ F (Y ⊗ Z)

ηX,Y⊗Z

��

F (X ⊗ Y )⊗′ F (Z)

ηX⊗Y,Z

��

F (X ⊗ (Y ⊗ Z))FαX,Y,Z

// F ((X ⊗ Y )⊗ Z)

(1.2)

F (X)⊗′ F (1)1F (X)⊗′ε

//

ηX,1

��

F (X)⊗′ 1′

ρ′F (X)

��

F (1)⊗′ F (Y )ε⊗1F (Y )

//

η1,Y

��

1′ ⊗′ F (Y )

λF (Y )

��

F (X ⊗ 1)FρX // F (X) F (1⊗ Y )

FλY // F (Y )

(1.3)

The monoidal functor is said to be strict if F (X)⊗′ F (Y ) = F (X ⊗ Y ), and ηX,Y = 1F (X)⊗′F (Y ),for all objects X,Y of C.

Example 1.1.9. The forgetful functor from the category of representation of a group G to thecategory of vector spaces, RepG →Vect, is a monoidal functor. Actually, all forgetful functorsfrom one monoidal category to another are monoidal functors.

Definition 1.1.10. A monoidal natural transformation is a natural transformation Γ : F → G,where (F, η, ε) and (G,µ, ν) are monoidal functors from the monoidal category C to the monoidalcategory D, such that the two following diagrams commute, for X,Y any objects of C.

F (X)⊗′ F (Y )ΓX⊗′ΓY//

ηX,Y

��

G(X)⊗′ G(Y )

µX,Y

��

F (1)

ε

��

Γ1

##

F (X ⊗ Y )ΓX⊗Y

// G(X ⊗ Y ) 1′ G(1)

νoo

(1.4)

Now we can define two functor categories.Let Hom⊗(C,D) be the category of monoidal functors between two monoidal categories C and

D, the morphisms being the monoidal natural transformations.Let Hom⊗−str(C,D) be the category of strict monoidal functors between two monoidal cate-

gories C and D.

1.1.3 Symmetric categoryHere again, the following definitions are the ones found in Mac Lane’s book [12].

Definition 1.1.11. A braided monoidal category is a monoidal category C, which is equipped witha natural isomorphism γ, called the commutativity constraint such that, for all objects X,Y of C,γX,Y : X ⊗ Y → Y ⊗X is a isomorphism which satisfies the two following hexagon identities.

7

For all objects X,Y, Z of C, the diagrams

(X ⊗ Y )⊗ ZγX⊗Y,Z

// Z ⊗ (X ⊗ Y )

αZ,X,Y

((

X ⊗ (Y ⊗ Z)

1X⊗γY,Z

((

αX,Y,Z66

(Z ⊗X)⊗ YγZ,X⊗1Y

vv

X ⊗ (Z ⊗ Y )αX,Z,Y

// (X ⊗ Z)⊗ Y

(1.5)

X ⊗ (Y ⊗ Z)γX,Y⊗Z

// (Y ⊗ Z)⊗X)α−1Y,Z,X

((

(X ⊗ Y )⊗ ZγX,Y ⊗1Z

((

α−1X,Y,Z

66

Y ⊗ (Z ⊗X)

1Y ⊗γZ,X

vv

(Y ⊗X)⊗ Zα−1Y,X,Z

// Y ⊗ (X ⊗ Z)

(1.6)

are commutative.

Remark 1.1.12. Using these axioms, we can prove that the braiding is also compatible with theright and left unitors.

Example 1.1.13. The category of representations of a quantized universal enveloping algebra Uq(g)is a braided monoidal category, where γ is obtained from the universal R-matrix see [2], chap. 10.,or [11], chap. VIII.

Definition 1.1.14. A symmetric monoidal category is a braided monoidal category in which thecommutativity constraint γ satisfies

γY,XγX,Y = 1X⊗Y (1.7)

for all objects X,Y of C.

Remark 1.1.15. We have seen in Example 1.1.13 that the category of representations of Uq(g) is abraided monoidal category. But it is not, in general, a symmetric monoidal category.

Example 1.1.16. The category RepG of representations of a group G is a symmetric monoidalcategory, with the braiding γV,W : v ⊗ w 7→ w ⊗ v.

Definition 1.1.17. A braided monoidal functor is a monoidal functor (F, η, ε), between twobraided categories, which respects the braiding on both sides. That can be translated as thecommutativity of the following diagram, for all objects X,Y of C.

F (X)⊗′ F (Y )ηX,Y

//

γF (X),F (Y )

��

F (X ⊗ Y )

FγX,Y

��

F (Y )⊗′ F (X)ηY,X

// F (Y ⊗X)

(1.8)

A symmetric functor is a braided monoidal functor between two symmetric categories, with noadditional condition.

8

1.1.4 Rigid monoidal categoryLet C be a monoidal category (not necessarily symmetric) and X an object of C.

Definition 1.1.18. A dual of X is a triplet (X∗, evX , evX), where X∗ is an object of C, andevX : X∗ ⊗X → 1 and coevX : 1→ X∗ ⊗X are morphisms, such that the compositions

XcoevX⊗1X−−−−−−−→ X ⊗X∗ ⊗X 1X⊗evX−−−−−→ X (1.9)

X∗1X∗⊗coevX−−−−−−−−→ X∗ ⊗X ⊗X∗ evX⊗1X∗−−−−−−→ X (1.10)

are the identity morphisms.

Remark 1.1.19. If an object X has a dual, then it is unique up to unique isomorphism.

Now let D be another monoidal category and (F, η, ε) : C → D be a monoidal functor. Thenthe functor F and the dual X∗ of the object X of C define a dual (F (X∗), evF (X), coevF (X)) ofF (X) in D, where

evF (X) := F (X∗)⊗′ F (X)ηX∗,X−−−−→ F (X∗ ⊗X)

FevX−−−−→ F (1)ε−→ 1

′ (1.11)

coevF (X) := 1′ ε−1

−−→ F (1)FcoevX−−−−−→ F (X ⊗X∗)

η−1X,X∗−−−−→ F (X)⊗′ F (X∗) (1.12)

Definition 1.1.20. A rigid monoidal category is a monoidal category in which every object has adual.

Example 1.1.21. If k is a field, the category k-Vect of finite dimensional k-vector spaces is a rigidmonoidal category. The dual of a vector space V is it usual dual V ∗ = {φ : V → k, k-linear}. Theevaluation and coevaluation maps are defined as

evV : V ∗ ⊗ V → k ,φ⊗ v 7→ φ(v)

coevV : k → V ⊗ V ∗

λ 7→ λ ·n∑i=1

ei ⊗ e∗i ,(1.13)

where (e1, . . . , en) and (e∗1, . . . , e∗n) are dual basis of V and V ∗, respectively.

Then, (1V ⊗ evV ) ◦ (coevV ⊗ 1V )v = (1V ⊗ evV )

(n∑i=1

ei ⊗ e∗i ⊗ v)

=n∑i=1

eivi = v. The other

condition can be verified the same way.

1.2 Categorical dimension

Here we are going to work on a rigid monoidal category C. On such a category we can define acategorical trace trφ, for every object X and φ ∈ EndC(X).

trφ := evX ◦ γX,X∗ ◦ (φ⊗ 1X∗) ◦ coevX (1.14)

Remark 1.2.1. One has trφ ∈ EndC(1), so trφ ∈ C if C is a tensor category.

Definition 1.2.2. For every object X, we define the categorical dimension dimX by

dimX := tr(1X) ∈ EndC(1)

9

Remark 1.2.3. With the rigid monoidal category structure on k-Vect given in Example 1.1.21,we can see that, if V is a k-finite-dimensional vector space, or dimension n ∈ N, the categoricaldimension of V is

dimV = n.

Indeed, tr(1V ) ∈ EndC(k) = k in this case, and tr(1V )1 = evV ◦γV,V ∗ ◦coevV 1 = evV ◦γV,V ∗n∑i=1

ei⊗

e∗i = evVn∑i=1

e∗i ⊗ ei =n∑i=1

e∗i (ei) = n.

Hence, the categorical dimension is a generalization of the dimension, to any rigid monoidalcategory.

1.3 Additive envelope

Constructing the additive envelope of a category is a way of enriching a category for it to becomeadditive.

Precisely, an additive envelope of C is a pair (Cadd, ι), where Cadd is a additive category, andι : C −→ Cadd is a fully-faithful preadditive functor, such that for any additive category D, therestriction functor

Hom+(Cadd,D) → Hom+(C,D)F 7→ F ◦ ι

(η : F ⇒ G) 7→ ηι

(1.15)

is an equivalence of categories. Thus a additive envelope, when it exists, is unique up to equivalenceof categories.

Let us define the terms we used in this paragraph.

1.3.1 Additive categoryIn order to be additive, a category first needs to be preadditive.

Definition 1.3.1. A category C is preadditive if every hom-set Hom(X,Y ) in C has the structureof an abelian group, if the composition of morphisms is bilinear, and if it has a zero object 0.

We call a zero object a object 0 in C which is both initial (for every object X in C there isa unique morphism 0 → X) and terminal (for every object X in C there is a unique morphismX → 0).

A preadditive functor if a functor F : C → D, between two preadditive categories C and D, suchthat for any pair of morphisms f, g ∈ HomC(X,Y ), one has that

F (f + g) = F (f) + F (g). (1.16)

Definition 1.3.2. A category C is additive if it is preadditive and admits finite coproducts.

The notion of coproduct is a categorical generalization of the notion of disjoint union in sets, itis an internal addition law.

For I a finite set and {Xi}i∈I objects in the category C, their coproduct∐i∈I Xi is a object of

C, equipped with morphisms ιXi : Xi →∐k∈I Xk, for every i ∈ I , satisfying a universal property.

For every object Z of C and set of morphisms fi : Xi → Z, there is a unique morphismF :

∐i∈I Xi → Z, such that, for every i 6= j in I we have a commuting diagram

10

Xi

fi$$

ιXi //∐k∈I Xk

F

��

Xj

ιxjoo

fjzz

Z

(1.17)

Let us denote by Hom+(C,D) the category of preadditive functors between two preadditivecategories C and D, the morphisms being any natural transformation.

Definition 1.3.3. A object X of the category C is said to be indecomposable if when writing acoproduct decomposition of X = X1

∐X2, with maps ιi : Xi → X and πi : X → Xi, there is an

i ∈ {1, 2} such that ιi ◦ πi = 0 ∈ EndC(X).

1.3.2 Construction of the additive envelope

From an existing preadditive category C, we can build Cadd, which is going to be an additivecategory. It is defined as follow

Objects X = (Xj)j∈J finite-length tuples of objects of C.

Morphisms φ : (Xj)j∈J → (Yi)i∈I , whereφ = (φi,j)i∈I,j∈J and φi,j : Xj

// Yi

Composition φ = (φi,j)i∈I,j∈J : (Xj)j∈J → (Yi)i∈I and ψ = (ψk,i)k∈K,i∈I : (Yj)j∈J → (Zk)k∈Kare composed via matrix multiplication

(ψ ◦ φ)k,j =∑i∈I

ψk,i ◦ φi,j : Xj → Zk. (1.18)

Cadd is indeed an additive category, the coproduct being the concatenation of tuples.{(Xj

i )i∈Ij}j∈J has as coproduct∐j∈J(Xj

i )i∈Ij := (Xji )j∈J,i∈Ij . The injection maps are built

using the zero object 0.Let j0 be fixed in J , φj0 : (Xj0

i )i∈Ij0 → (Xji )j∈J,i∈Ij is made of φj0i,j,k : Xj0

i → Xjk, which is

equal to 1Xj0i

if (i, j0) = (k, j) and Xj0i → 0→ Xj

k if not.It can be seen easily that the universal property of the finite coproducts is satisfied.

It is easy to see why Cadd is indeed an additive envelope of C. The functor ι : C → Cadd isdefined by sending any object X of C to the length-1 tuple (X) and any morphism φ of C to the1x1-matrix (φ). It is fully-faithful.

We have to understand why the universal property of the additive envelope holds. The restric-tion functor Hom+(Cadd,D)→ Hom+(C,D) is essentially surjective (and even surjective) becauseif we have a given preadditive functor F : C → D, we can easily build an antecedent F ′ under therestriction functor. It is enough to see that functors on Cadd are entirely defined by the images ofthe length-1 tuple, so by the images of the objects of C. Furthermore, we can map each naturaltransformation of functors from C to D to one and only one natural transformation from Cadd toD. Thus the restriction functor is also fully-faithful and is an equivalence of categories.

Hence, the additive envelope, as we built it, in the only additive envelope, up to equivalence ofcategories.

11

1.4 Karoubi envelope

As with the additive envelope, the Karoubi envelope of an existing category C is a Karoubi categorybuilt from the category C. The definition is similar.

A Karoubi envelope of C is a pair (Ckar, ι), where Ckar is a Karoubi category, and ι : C −→ Ckaris a fully-faithful preadditive functor, such that for any Karoubi category D, the restriction functor

Hom+(Ckar,D) → Hom+(C,D)F 7→ F ◦ ι

(η : F ⇒ G) 7→ ηι

(1.19)

is a equivalence of categories. Thus a Karoubi envelope, when it exists, is unique up to equivalenceof categories.

Let us explain what a Karoubi category is and how to build the Karoubi envelope.

1.4.1 Splittings of idempotents and Karoubi categoryLet C be a category and X an object in C.

Definition 1.4.1. e ∈ EndC(X) is called an idempotent of C if e2 = e. A splitting of an idempotente is a tuple (im e, ιe, πe) where im e is an object of C,

ιe : im e // X, πe : X // im e,

are morphisms of C, such that

1. e = ιe ◦ πe,

2. idim e = πe ◦ ιe,

If there exists a splitting (im e, ιe, πe) for an idempotent e, we will say that e splits, and callim e the image of e.

Definition 1.4.2. A category is said to be Karoubi if every idempotent in that category splits.

We also introduce a few more notions, for which the category C is supposed to be C-linear.

Definition 1.4.3. Two idempotents e, e′ of EndC(X) are said to be orthogonal if ee′ = e′e = 0.An idempotent e is said to be primitive if it is nonzero and it cannot be written as the sum of

two nonzero orthogonal idempotents.

Lemma 1.4.4. If C is a C-linear Karoubi category and X an object of C, then X is indecomposableif and only if idX is a primitive idempotent.

Proof. SupposeX is indecomposable and write idX = e+e′, where e, e′ are orthogonal idempotents.As the category is Karoubi, we have splittings : (im e, ιe, πe) and (im e′, ιe′, πe′). It is enough toshow that X = im e⊕ im e′, because as it is indecomposable, that would imply e = 0 or e′ = 0.

Let Y be an object of C, with morphisms fe : im e → Y , fe′ : im e′ → Y . The morphismF = fe ◦ πe + fe′ ◦ πe′ : X → Y satisfies the universal property of the coproduct.

The reverse can be shown in a similar way.

12

1.4.2 Construction of the Karoubi envelope

The Karoubi envelope Ckar of a category C is defined as follow

Objects The tuples (X, e) where X is an object of C and e an idempotent of EndC(X).

Morphisms HomCkar ((X, e), (Y, f)) = {φ ∈ HomC(X,Y )|f ◦ φ = φ = φ ◦ e}Let φ be a morphism Ckar, and φ0 denote the same morphism, considered as a morphism ofC. Then the Ckar-morphisms φ and ψ are equal if and only if they have the same source andtarget in Ckar and φ0 = ψ0.

Composition The composition of morphisms in Ckar is inherited from the composition of mor-phisms in C. ψ ◦ φ is defined by the source of φ, the target of ψ and (ψ ◦ φ)0 = ψ0 ◦ φ0. Wehave (id(X,e))0 = e.

In Ckar every idempotent has a splitting, thus it is a Karoubi category. Indeed, let φ be anidempotent of EndCkar (X, e), φ has a splitting (imφ, ιφ, πφ), where imφ = (X,φ) and (ιφ)0 =(πφ)0 = φ0.

The functor ι : C → Ckar defined by ιX = (X, idX) and (ιφ)0 = φ is fully-faithful, and is suchthat the universal property of the Karoubi envelope holds. This is pretty straightforward to check.

Remark 1.4.5. With this construction of the Karoubi envelope, we can see that any object of thecategory Ckar is isomorphic to the image of an idempotent of C.

1.5 Krull-Schmidt category

A Krull-Schmidt category is a category in which the Krull-Schmidt theorem holds, which meansthat every object can be uniquely written as a sum of indecomposable objects. Of course we haveto work on categories that are at least additive to be able to define the sum of objects. Thefollowing definition is sometime used (in [3] for example).

Definition 1.5.1. A Krull-Schmidt category is an additive, Karoubi category C with finite-dimensional Hom-spaces.

In particular, if C is a preadditive category with finite-dimensional Hom-spaces, then (Cadd)karis a Krull-Schmidt category.

Proposition 1.5.2. Let C be a C-linear category and Ckar its Karoubi envelope, as built above.Let X be an object of C, and e, e′, e′′ idempotents of EndC(X). Then

1. im e is indecomposable if and only if e is a primitive idempotent, and up to isomorphism, allindecomposables are thus obtained.

2. if e = e′ + e′′ and e′, e′′ are orthogonal, then im e ∼= im e′ ⊕ im e′′

3. Suppose further that EndC(X) is finite-dimensional, then im e ∼= im e′ if and only if e, e′ areconjugate in EndC(X).

Proof. We begin by the first part. We already now that im e is indecomposable if and only if idim e

is a primitive idempotent. So we have is to show that

idim e is a primitive idempotent ⇔ e is a primitive idempotent

13

Suppose e = e′ + e′′, where e′, e′′ are orthogonal idempotents in C, then

idim e = πe ◦ e ◦ ιe = πe ◦ e′ ◦ ιe + πe ◦ e′′ ◦ ιe.

Suppose now that idim e = f + f ′, where f, f ′ are primitive idempotents, then

e = ιe ◦ idim e ◦πe = ιe ◦ f ◦ πe + ιe ◦ f ′′ ◦ πe.

Hence, a decomposition as a sum of orthogonal idempotents in C of e gives us a decompositionin sum of orthogonal idempotents of idim e, and the other way round. Thus we have a bijectionbetween orthogonal idempotent decompositions of e in C and of idim e in Ckar. This proves the firstpart of the first assumption.

Moreover, as we noted in remark 1.4.5, every object of Ckar is isomorphic to the image of anidempotent of C. the proves the second part of the first assumption.

The proof of the second assumption is really similar to the proof of Lemma 1.4.4.For the third assumption, suppose first that e′ = φeφ−1. Then the morphism πe′ ◦ φ ◦ ιe :

im e→ im e′ is an isomorphism, with inverse πe ◦ φ−1 ◦ ιe′ . Conversely, let Φ : im e→ im e′ be anisomorphism. Then the morphisms

(EndC)e′ → (EndC)e , (EndC)e → (EndC)e

′

α 7→ α ◦ ιe′ ◦ Φ ◦ πe ◦ e β 7→ β ◦ ιe ◦ Φ−1 ◦ πe′ ◦ e′(1.20)

are mutually inverse. Indeed, if we write α ∈ (EndC)e′ as α = α′ ◦ e′, the composition of the

morphisms gives us α′ ◦ ιe′ ◦Φ ◦πe ◦ e ◦ ιe ◦Φ−1 ◦πe′ ◦ e′ = α′ ◦ ιe′ ◦Φ ◦Φ−1 ◦πe′ ◦ e′ = α′ ◦ ιe′ = α.The composition in the other way is symmetrical. Hence the modules (EndC)e and (EndC)e

′ areisomorphic. This implies that the idempotents e and e′ are conjugate, as EndC is finite-dimensional(see [1] for example, for a proof of this classical fact, in the case of finite-dimensional algebras).

Using this proposition, we can now prove the following Theorem, which explains why we use theterm "Krull-Schmidt category". Let us note that Proposition 1.5.2 is satisfied in a Krull-Schmidtcategory C, as in this C is its own Karoubi envelope.

Theorem 1.5.3. A Krull-Schmidt category C satisfies the Krull-Schmidt Theorem, in the sensethat every object in C can be written as a sum of indecomposable objects.

Proof. For every object X in C, it is clear that im idX = X, thus is it enough to shown that everyobject of the form im e can be written as a sum of indecomposable objects.

Let X be an object of C and let e be an idempotent of EndC(X). If im e is not indecomposable,then by Proposition 1.5.2 e is not primitive, and if e = e′ + e′′, with e′ and e′′ orthogonal, thenim e ∼= im e′ ⊕ im e′′. If im e′ or im e′′ is not indecomposable, we can decompose it the same way.

We iterate this process, and we obtain im e ∼=n⊕i=1

im ei, with e1, . . . , en ∈ EndC(X) idempotents

such that e = e1 + · · · + en. These idempotents also satisfy various conditions of orthogonality.The conditions can be defined recursively.

(?) : A sum of idempotents e = e1 + · · ·+ en satisfies the condition (?) if

• n = 1, or

• n = 2, and e1, e2 are orthogonal, or

• n > 2, and there is a k ∈ {2, . . . , n − 1} such that e1 + · · · + ek and ek+1 + · · · + en areorthogonal and both satisfy the condition (?).

14

Lemma 1.5.4. If e1, . . . , en are idempotents such that e1 + · · · + en satisfies the condition (?),then (e1, . . . , en) is free.

We will admit this Lemma.As EndC(X) is finite-dimensional, there can not be a sum of an arbitrary number of idempotents

which satisfies the condition (?). Hence, the process we detailed above stops at some point, and

we can write im e ∼=n⊕i=1

im ei, where each im ei, for 1 ≤ i ≤ n, is indecomposable.

15

Chapter 2

The category Rep(Glt)

We are now going to define the Deligne category, Rep(Glt), which interpolates the category offinite-dimensional representations of the general linear group Rep(GlN ), to complex values ofN = t.

We are going to follow the construction of Rep(Glt) introduced by Deligne in the end of [5],and more precisely, the more detailed version of this construction, found in [3].

For that we first define a smaller category, called the skeleton category Rep0(Glt), then weare going to take successively the additive envelope and the Karoubi envelope of this category toobtain Rep(Glt).

2.1 The category Rep0(Glt)

We start by introducing the diagrams we will use to define Rep0(Glt).

2.1.1 Words and diagramsLet w and w′ be two finite words in the two letter alphabet {◦, •}.

Definition 2.1.1. A (w,w′)-diagram is a graph such that

1. The vertices are the letters of w and w′, placed in two rows. The upper row forms w′, andthe lower w.

2. Each vertex is adjacent to exactly one edge.

3. An edge can be adjacent to two letters of different colors only if they are placed on the samerow, it can be adjacent to two letters of the same color only if they are placed on differentrows.

In a (w,w′)-diagram, an edge between a vertex of the upper row and one of the lower row iscalled a propagating edge.

Example 2.1.2. For these examples, we list all the possible diagrams. 1 denotes the empty word.

(• • ◦◦,1)-diagrams :

• ◦• ◦ • ◦• ◦

(•◦, ◦ • •◦)-diagrams :◦ • • ◦

• ◦

◦ • • ◦

• ◦

◦ • • ◦

• ◦

◦ • • ◦

• ◦

◦ • • ◦

• ◦

16

◦ • • ◦

• ◦

◦ • • ◦

• ◦

Proposition 2.1.3. Let w and w′ be two finite words in the alphabet {◦, •}. Let r (resp. r′) bethe number of black letters in w (resp. w′) and let s (resp. s′) be the number of white letters in w(resp. w′). Then

#{(w,w′)-diagrams } = (r + s′)! if r + s′ = r′ + s

= 0 if not

Proof. First of all, it is clear that the number of possible (w,w′)-diagrams depends only on thenumber of black and white letters in each word, and not on the order in which they appear withinthe word.Next, we can build a bijection from the (w,w′)-diagrams to the (r+ s′, r′ + s)-diagrams with onlypropagating edges, by putting all the blacks letters of w′ and the white letters of w on the upperrow, and the rest on the lower row. Example :

◦ • • ◦

• ◦

• • ◦

• ◦ ◦This is clearly a bijection, the inverse function being to exchange the white letters of the upper

and lower row, while keeping the edges.Hence we have

#(w,w′)-diagrams } = #{(r + s′, r′ + s)-diagrams with onlypropagating edges}= #{ bijections from (r + s′) to (r′ + s)}= (r + s′)! if r + s′ = r′ + s

= 0 if not

Now we are going to define a product operation on these diagrams.Let w, w′ and w′′ be three finite words on {◦, •}. Let X be a given (w,w′)-diagram and Y

a (w′, w′′)-diagram. We begin by defining the diagram Y ? X, which is the diagram obtained bystacking Y on top X such that the upper row of X is identified with the lower row of Y .

Next, we define the (w,w′′)-diagram Y · X whose vertices are the upper row of Y and lowerrow of X and whose edges are obtained by composing the edges of X and Y .

Finally, let `(X,Y ) denote the number of cycles appearing in Y ?X (i.e. the number of connectedcomponents of Y ? X minus the number of connected components of Y ·X).

Example 2.1.4. We follow the construction of Y ·X on an explicit example.

Let X =

• ◦ ◦ • • ◦ • • • ◦

• • ◦ • ◦ •

and Y =

◦ • • ◦ • ◦ • •

• ◦ ◦ • • ◦ • • • ◦

17

Then Y ? X =

◦ • • ◦ • ◦ • •

• ◦ ◦ • • ◦ • • • ◦

• • ◦ • ◦ •

Thus Y ·X =

◦ • • ◦ • ◦ • •

• • ◦ • ◦ •and `(X,Y ) = 1

2.1.2 The skeleton categoryWith this setup we can now define what we are going to call the skeleton category Rep0(Glt), forany t ∈ C.

Definition 2.1.5. Let t ∈ C, the category Rep0(Glt) has

Objects finite words on the alphabet {◦, •}.

Morphisms for w,w′ ∈ Obj(Rep0(Glt)), Hom(w,w′) is the C-vector space with basis{(w,w′)-diagrams }.

Composition for w,w′, w′′ ∈ Obj(Rep0(Glt)),

Hom(w′, w′′)×Hom(w,w′)→ Hom(w,w′′) (2.1)

is the C-linear application defined by

(Y,X) 7→ Y X = t`(X,Y )Y ·X (2.2)

for X any (w,w′)-diagram and Y any (w′, w′′)-diagram.

We must of course verify that this definition is correct.

Proposition 2.1.6. Rep0(Glt) as defined above is indeed a category, for every t ∈ C.

Proof. Associativity As the composition is C-linear, we only have to check the associativity onbasis vectors. Let X be a (w,w′)-diagram, Y a (w′, w′′)-diagram and Z a (w′′, w′′′)-diagram.We have to verify that the (w,w′′′)-diagrams Z(Y X) and (ZY )X are equals.

Z(Y X) = Z(t`(X,Y )Y ·X) = t`(X,Y )Z(Y ·X) composition is C-linear

= t`(X,Y )t`(Y ·X,Z)Z · (Y ·X)

= t`(X,Y )t`(Y ·X,Z)(Z · Y ) ·X as the · product is clearly associative

= t`(X,Y )+`(Y ·X,Z)−`(X,Z·Y )−`(Z,Y )(ZY )X

Thus it is enough to show that

`(X,Y ) + `(Y ·X,Z) = `(X,Z · Y ) + `(Z, Y )

But we have

`(Y ·X,Z) = `(Y ? X,Z)− `(X,Y ) =

= `(X,Z ? Y )− `(X,Y ) = `(X,Z · Y ) + `(Z, Y )− `(X,Y )

18

Identity For all w, word on {◦, •}, there is a (w,w)-diagram 1w which satisfies the followingproperty.

For all w′ ∈ Obj(Rep0(Glt)), f ∈ Hom(w,w′) and g ∈ Hom(w′, w), f = f1w and g = 1wg.

We get 1w by simply putting an edge between the corresponding letters on the upper andlower row. For example

1••◦•◦•• =

• • ◦ • ◦ • •

• • ◦ • ◦ • •

2.1.3 Tensor structure on Rep0(Glt)

Now we are going to see that the category we built can be equipped with the structure of a tensorcategory.

First we have to build the tensor product, which is a functor⊗: Rep0(Glt)×Rep0(Glt) −→ Rep0(Glt).

Here is how this functor acts on the category Rep0(Glt)×Rep0(Glt).

On objects (w1, w2) 7→ w1 ⊗ w2 := w1w2 concatenation of words.

On morphisms Let Xi be a (wi, w′i)-diagram, for i = 1, 2. Then (X1, X2) 7→ X1 ⊗ X2 = the

(w1w2, w′1w′2)-diagram obtained by placing X2 directly on the right of X1.

Proposition 2.1.7. Rep0(Glt) equipped with

•⊗

as the tensor product,

• the empty word 1 as the unit object,

• identity as the associativity constraint, and left and right unitors,

• the commutativity constraint γ defined by γw1,w2is the (w1w2, w2w1)-diagram obtained by

putting an edge between the ith letter of w1 (resp. w2) on the upper row and the ith letter ofw1 (resp. w2) on the lower row,

is a rigid symmetric category.

Proof. we are going to prove that statement by checking each term with regard to the definitionswe gave earlier.

Monoidal⊗

as defined is clearly a bifunctor. The associativity constraint αw1w2w3is equal

to 1w1w2w3 , hence the pentagon axiom is satisfied. For the left and right unitors, we haveλw : 1 ⊗ w = w → w is equal to 1w, the same goes for ρw, so the triangle diagram is alsosatisfied.

Thus Rep0(Glt) is a monoidal category.

Symmetric First, let us take a look at what the commutativity constraint does, on an example

γ••◦,•◦◦• =

• ◦ ◦ • • • ◦

• • ◦ • ◦ ◦ •

19

We can see that only the number of letters in each word have an effect on the form of thediagram, so we will not draw the letters anymore in this part.

We have to prove that the hexagon identities are satisfied, but in our case there are onlytriangular diagrams, as the associativity constraint is the identity. We are going to prove oneof these identities. For all w1, w2, w3

w1w2w3

1w1⊗γw2,w3

&&

γw1w2,w3 // w3w1w2

γw3,w1⊗1w2

xxw1w3w2 must be commutative.

Hence we have that two (w1w2w3, w1w3w2)-diagrams are equal.

1w1 ⊗ γw2w3 = ···

··· ···w1 w3 w2

w1 w2 w3

γw3,w1⊗ 1w2

=

··· ···

···

w1 w3 w2

w3 w1 w2

··· ··· ···

w1 w2 w3

γw1w2,w3 =

It is clear by looking at these figures that we have indeed

1w1⊗ γw2w3

= γw3,w1⊗ 1w2

+ γw1w2,w3

We have now shown that Rep0(Glt) is a braided monoidal category. To show that itis a symmetric category, it is enough to show that the braiding is symmetric, i.e. thatγw2,w1

γw1,w2= 1w1w2

.

This property is illustrated in the following figure

20

··· ···

··· ···

w1 w2

w2 w1

w1 w2

=

w1 w2

w1 w2

···

Rigid For any w ∈ Obj(Rep0(Glt)) we can define a dual object w∗ by replacing all the blackletters with white letters and vice versa. Now we build the morphism evw, (resp. coevw),which is a (w∗w,1)-diagram (resp. (1, ww∗)-diagram in which there is an edge between theith letter of w and the ith letter of w∗ (it is correct because they are of opposite color).

For example :

ev••◦ = ◦ ◦ • • • ◦ and coev••◦ = • • ◦ ◦ ◦ •

We have to check that evw and coevw make w∗ a dual to w. For that we have to check thatthe two following diagrams

ww∗w

1w⊗evw

��

w∗ww∗

evw⊗1w∗

��

and

w

coevw⊗1w

BB

1w // w w∗

1w∗⊗coevw

@@

1w∗ // w∗

are commutative.

Let us look at what the first diagram, for example, looks like in terms of (w,w)-diagram.

···

···

w

w

w w∗ w = ···

w

w

This figure clearly shows the commutativity of the first diagram, a similar figure would showthe same way the commutativity of the second diagram.

Therefore we have shown that Rep0(Glt) is a rigid symmetric category.

As we also have End(1) = C, (there is only one (1,1)-diagram, which is the empty diagram).Thus Rep0(Glt) is a tensor category with the definition given before.

21

2.1.4 Universal property of Rep0(Glt)

As the skeleton category Rep0(Glt) is a tensor category, we have a concept of dimension, whichis a complex number attached to each object (see 1.2).

In our situation, the dimension is quite simple.

Proposition 2.1.8. Let w be a word in the category Rep0(Glt). Then

dimw = t|w| (2.3)

Proof. Here once again the result is quite clear if we look at a diagram. Let w be a word and letus write tr(w) = evw ◦ γw,w∗ ◦ coevw ∈ C.

··· ···w∗ w

w w∗

We have tr(w) = t` where ` is the number of connected com-ponents in the diagram. Then it is obvious that the number ofconnected components in the diagram is equal to |w|, the numberof letters in w.Hence we have tr(w) = t|w|.

Remark 2.1.9. We can see that in this case we have

dim(w1 ⊗ w2) = dim(w1) · dim(w2) = t|w1|+|w2| (2.4)

This result is actually true for any tensor category C with this definition of categorical dimension.

In particular, we have dim(•) = t.In fact, Rep0(Glt) is characterized as the universal tensor category generated by an object of

dimension t and its dual. More precisely, it satisfies the following universal property

Proposition 2.1.10. Let T be any tensor category and let Tt denote the category of t-dimensionalobjects in T . Then the following functor induces an equivalence of categories

Θ : Hom⊗−str(Rep0(Glt), T ) → TtF 7→ F (•)

(η : F ⇒ G) 7→ η•

(2.5)

This property was first stated by Deligne in [5],with the idea of the proof, and a more detailedproof is given in [3].

2.2 Definition of Rep(Glt)

From Rep0(Glt) we define Rep(Glt) as we announced before, which is

Rep(Glt) := ((Rep0(Glt))add)kar (2.6)

As we have seen in 1.3, to be able to define an additive envelope over an existing category C,C has to already be preadditive. As the hom-spaces of C are vector spaces, they also are abeliangroups, and the composition is bilinear.

It is clear thatRep(Glt) is also a symmetric tensor category, the tensor structure being inheritedfrom that of Rep0(Glt).

22

2.2.1 Universal property of Rep(Glt)

For any tensor category T , let Hom′(Rep(Glt), T ) denote the full subcategory ofHom⊗−str(Rep(Glt), T ), whose objects are the functors whose restriction Rep0(Glt)→ T yieldsa strict tensor functor. The category Rep(Glt) satisfies the following universal property.

Proposition 2.2.1. Let T be any tensor category and let Tt denote the category of t−dimensionalobjects in T . Then the following functor induce an equivalence of categories.

Hom′(Rep(Glt), T ) → TtF 7→ F (•)

(η : F ⇒ G) 7→ η•

(2.7)

As for the universal property of Rep0(Glt), the property was first mentioned by Deligne in [5],and a more detailed statement, and proof, of it can be found in [3].

2.2.2 Interpolation of the classical caseLet us denote by Rep(GlN ) the classical category of complex finite-dimensional representations ofGlN , where N is an integer. We are going to see the link between this category and the categoryRep(Glt) which we just defined.

The category Rep(GlN ) contains the vector representation V = CN . It is known that all theirreducible finite-dimensional representation of GlN lies in some V ⊗r ⊗ V ∗⊗s, with r, s integers.

What we did earlier was an interpolation of this classical case, with the correspondence

V ⊗r ⊗ V ∗⊗s ←→ • • · · · •︸ ︷︷ ︸r

◦ ◦ · · · ◦︸ ︷︷ ︸s

. (2.8)

Remark 2.2.2. One can see that we have indeed

dim(• • · · · •︸ ︷︷ ︸r

◦ ◦ · · · ◦︸ ︷︷ ︸s

) = dim(V ⊗r ⊗ V ∗⊗s) = r + s

Hence, the definition of categorical dimension is coherent with the usual definition of dimension inthis case also.

We already showed that there was some correspondence between the Deligne category we definedRep(Glt) and the category Rep(GlN ), when t = N is an integer. We can wonder if these categoriesare equivalent, when t = N is a integer. The answer is negative, there are not equivalent.

But, Deligne showed that even if these categories are not exactly the same, the functor • 7→ CN ,from Rep(Glt=N ) to Rep(GlN ), still induce an equivalence of categories (see [5])

Rep(GlN )/N → Rep(GlN ), (2.9)

where N is the ideal of negligible morphisms of Rep(GlN ).

Remark 2.2.3. Let us recall that in a C-linear tensor category C a morphism f : X → Y is callednegligible if, for all morphisms u : Y → X, we have tr(f ◦ u) = 0. The trace being the categoricaltrace defined in 1.2.

In the rest of the paper, we will often compare the situation in Rep(Glt) to the correspondingsituation in Rep(GlN ), in order to better understand the mechanics in place. As we have seen,these two categories are very similar and a lot of phenomenons are going to work the same way.

23

2.3 Definition of the Lie algebra g

With this definition of Rep(Glt) it is possible to give a meaning to Glt, for any complex numbert, as the fundamental group of the category (see [6]). Then it is also possible to define Lie(Glt), asthe Lie algebra of this group scheme. Details about that can be found in [8].

The result of this process is a simple object of the category Rep(Glt), and we are going todefine it as such.

First, let us denote by V the tautological object •, equivalent of CN in the general case. It isthe basis object from which all the category is obtained. As we saw with the universal property,V holds a lot of the category’s structure.

Definition 2.3.1. Using the tautological object V we are going to define g = Lie(Glt) by

g := V ⊗ V ∗. (2.10)

In other word, it is the object •◦ of the category Rep(Glt).

Remark 2.3.2. This is actually the interpolation in complex rank of the classical case, in which wehave

Lie(GlN ) = glN = CN ⊗ C∗N . (2.11)

g as it is defined is actually an object of the category Rep0(Glt) and thus of the categoryRep(Glt). It is not necessarily a Lie algebra in a classical sense, if t is not an integer, but we aregoing to consider it as such.Remark 2.3.3. If we want to see the objects of Rep(Glt) as representations, we can look at themas g-modules. It is clear that for every object X of Rep0(Glt), the hom-space Hom(g ⊗X,X) isnonzero, with the result of Proposition 2.1.3, i.e. there is a morphism g ⊗X → X. From that isit easy to see that for every object X of Rep(Glt) there is also a morphism g⊗X → X.

2.4 Indecomposable objects in Rep(Glt)

The paper [3] gives all the details on the classification of indecomposable representations ofRep(Glt). Here we are only going to give the main results.

First of all, as we have seen in 1.5, indecomposables objects ofRep(Glt) correspond to primitiveidempotents of Rep0(Glt). So that is what we will have to study.

Some notations are going to be useful. First of all, for nonnegative integers r, s, let wr,s be theword of Rep0(Glt)

wr,s = • • · · · •︸ ︷︷ ︸r

◦ ◦ · · · ◦︸ ︷︷ ︸s

. (2.12)

It is clear that every object of the category Rep0(Glt) is isomorphic to a unique wr,s.We write CBr,s, or simply Br,s for the algebra EndRep0(Glt)(wr,s), called the wall Brauer

algebra.Our aim is to have a classification of conjugacy classes of primitive idempotents, for that we

will use bipartitions and the action of symmetric groups.

2.4.1 BipartitionsDefinition 2.4.1. A partition λ is a tuple of nonnegative integers λ = (λ1, λ2, . . .), such that for

all i ≥ 1, λi > λi+1, and λi = 0 for all but finitely many i. We will write |λ| =∞∑i=1

λi for the size

of λ, and l(λ) for the length of λ, the smallest integer n such that λn 6= 0 and ∀i > n, λi = 0.

24

Let P be the set of all partitions.

Definition 2.4.2. A bipartition is an element of P×P. For a bipartition λ, we write λ = (λ•, λ◦).The size of λ is |λ| = (|λ•|, |λ◦|), and its length is l(λ) = l(λ•) + l(λ◦).

We define a partial order on the set of bipartitions, (a, b) ≤ (c, d) if a ≤ c and b ≤ d. We alsoset (λ•, λ◦)∗ = (λ◦, λ•).

2.4.2 Symmetric groupLet Sr denote the symmetric group on r elements, and CSr its group algebra. Then CSr isnaturally isomorphic to Br,0 and to B0,r, via

Br,0 ← Sr → B0,r

σ• ←[ σ 7→ σ◦(2.13)

where σ• is the (wr,0, wr,0)-diagram in which the ith letter of the bottom row is adjacent to theσ(i)th letter of the upper row. The definition of σ◦ is exactly the same, with white letters insteadof black ones. For example, if σ = (123)(45) ∈ S5, then

σ• =

• • • • •

• • • • •

More generally, we have an inclusion

C[Sr ×Ss] ↪→ Br,s

(σ, τ) 7→ σ• ⊗ τ◦

For now on, we will consider C[Sr ×Ss] as a subalgebra of Br,s.We will admit the following result (see [4], Proposition 2.3, (2) for the proof).

Lemma 2.4.3. There is also a surjection of algebra π : Br,s � C[Sr ×Ss], such that

π(a) = a for all a ∈ C[Sr ×Ss] ⊂ Br,s.

2.4.3 The primitive idempotent eλ

It is a well known result that partitions of r are in bijection with primitive idempotents of CSr,up to conjugation. Let us write zλ for the primitive idempotent of CSr corresponding to thepartition λ. If λ is a bipartition λ = (λ•, λ◦), with |λ•| = r and |λ◦| = s, then the idempotentof C[Sr ×Ss] corresponding to λ will be written zλ = z•λ• ⊗ z◦λ◦ . zλ is not necessarily primitivein the larger algebra Br,s. Write a decomposition of zλ is sum of primitive idempotents of Br,smutually orthogonal zλ = e1 +· · ·+ek. Then π(e1), . . . , π(ek) are mutually orthogonal idempotentsof C[Sr ×Ss] such that zλ = π(e1) + · · ·+ π(ek). As zλ is a primitive idempotent of C[Sr ×Ss],there a unique 1 ≤ i ≤ k such that π(ei) 6= 0. Let eλ = ei. It is a primitive idempotent of Br,s,defined up to conjugation.

Proposition 2.4.4. If λ 6= (∅,∅), or t 6= 0, we can define, up to conjugation, a primitiveidempotent e(i)

λ of Br+i,s+i, for i > 0, such that im eλ and im e(i)λ are isomorphic in Rep(Glt).

This is the result of Section 4.4 of [3]. The proof is lengthy so we are not going to write is here,will we admit this result.

25

2.4.4 Classification of idempotentsWe have a theorem of classification of conjugacy classes of idempotents in walled Brauer algebras,using the notations of Proposition 2.4.4. As stated in [3], the following result is merely a translationof the classification of simple modules for walled Brauer algebras (see [4], Theorem 2.7) to thelanguage of primitive idempotents.

Theorem 2.4.5. 1. If r 6= s or t 6= 0 then

{e(i)λ | |λ| = (r − i, s− i), 0 ≤ i ≤ min(r, s)}

is a complete set of pairwise non-conjugate primitive idempotents of Br,s.

2. If t = 0 and r > 0 then{e(i)λ | |λ| = (r − i, r − i), 0 ≤ i < r}

is a complete set of pairwise non-conjugate primitive idempotents of Br,r.

Now we come back to the case of indecomposable objects of Rep(Glt). Let λ be a bipartitionof size (r, s), and eλ its corresponding primitive idempotent in Br,s. Let L(λ) = im eλ, the imageof eλ in Rep(Glt). Using Proposition 1.5.2, as eλ is a primitive idempotent of Rep0(Glt) definedup to conjugation, then L(λ) is a indecomposable object, defined up to isomorphism.

Theorem 2.4.6. The map λ 7→ L(λ) defines a bijection{bipartitions ofarbitrary size

}bij.−−→

{nonzero indecomposable objects in

Rep(Glt), up to isomorphism

}(2.14)

Proof. Let X be an indecomposable object in Rep(Glt), by Proposition 1.5.2, it is isomorphicto the image im e of a primitive idempotent e ∈ End(w) of Rep0(Glt). But w is isomorphic tosome wr,s, with r, s ≥ 0. Hence, without loss of generality, we can assume that e ∈ Br,s, for somer, s ≥ 0. Thus by Theorem 2.4.5, e = e

(i)λ , for a certain bipartition λ and i an integer. Finally X

is isomorphic to im e = im e(i)λ = im eλ = L(λ). We have proven that λ 7→ L(λ) is surjective.

Let λ, µ be bipartitions. We write |λ| = (r, s) and |µ| = (r′, s′). Suppose L(λ) ∼= L(µ). Wehave Hom(L(λ), L(µ)) = Hom(im eλ, im eµ) ⊂ Hom(wr,s, wr′,s′). By Proposition 2.1.3, this hom-space is nonzero if and only if r + s′ = r′ + s. Suppose that r ≥ r′, then there is an integeri ≥ 0 such that (r, s) = (r′ + i, s′ + i). Let us put aside the case when µ = (∅,∅) and t = 0,for which it can be shown that λ = (∅,∅) also. If µ 6= (∅,∅) or t 6= 0 then e(i)

µ is defined, andim e

(i)µ∼= im eµ ∼= im eλ. By Proposition 1.5.2, this implies that e(i)

µ and eλ are conjugate in Br,s.By Theorem 2.4.5, e(i)

µ = eλ and thus µ = λ.

2.4.5 Analogy with the classical caseIt is well known that in the classical case of finite-dimensional representations of the general lineargroup GlN (C), irreducible representations are in correspondence with dominant weights of GlN (C).There is a link between this classic correspondence and the study we just did.

As in 2.2.2, let Rep(GlN ) denote the category of complex finite-dimensional representation ofGlN . For 1 ≤ i ≤ N , let εi denote the function that maps any matrix to its (i, i) entry, and let Γdenote the set of dominant weights of GlN (C).

Γ = {γ =

N∑i=1

γiεi | γi ∈ Z, γ1 ≥ · · · ≥ γN}. (2.15)

26

The weights of Γ are in bijection with the bipartitions λ of size l(λ) ≤ N , via

{bipartitions of length ≤ N} ∼−→ Γ

λ = (λ•, λ◦) 7→ wt(λ) =N∑i=1

λ•εi −N∑j=1

λ◦εN−j+1(2.16)

When N ≥ r + s, the bipartition λ = (λ•, λ◦), where λ• = λ•1 ≥ · · · ≥ λ•r and λ◦ = λ◦1 ≥· · · ≥ λ◦s, can be written in terms of a weight (λ•1, . . . , λ

•r , 0 . . . , 0︸ ︷︷ ︸N−r−s

,−λ◦s, . . . ,−λ◦1). Reciprocally, if

γ =N∑i=1

γiεi ∈ Γ, let λ• = {γ1 ≥ γ2 ≥ · · · ≥ 0} and λ◦ = {−γN ≥ −γN−1 ≥ · · · ≥ 0}, and

λ = (λ•, λ◦), then clearly wt(λ) = γ.As every finite-dimensional indecomposable representation L of GlN is isomorphic to a highest

weight module of GlN , Vγ , for some γ ∈ Γ, hence if we write Vλ for Vwt(λ), then L ∼= Vλ forsome bipartition λ. Hence, isomorphism classes of indecomposable object of Rep(GlN ), up toisomorphism, are in bijection with bipartitions of size l(λ) ≤ N . The situation is the same as forRep(Glt).

But in the case when N is an integer, we can give explicit expressions of representations offundamental weights. If V = CN (the tautological object for Rep(GlN )) and k ≤ N , then ΛkV isan irreducible highest weight representation of weight (1, . . . , 1︸ ︷︷ ︸

k

, 0, . . . , 0︸ ︷︷ ︸N−k

), and ΛkV ∗ is an irreducible

highest weight representation of GlN of weight (0, . . . , 0︸ ︷︷ ︸N−k

,−1, . . . ,−1︸ ︷︷ ︸k

). With the bijection (2.16),

ΛkV corresponds to the bipartition (λ•, λ◦), with λ• = 1 + 1 + · · ·+ 1︸ ︷︷ ︸k

and λ◦ = 0, and ΛkV ∗

corresponds to its dual (λ◦, λ•).

More generally, if N is large enough, the representationm⊗i=1

ΛiV ⊗ri ⊗n⊗j=1

ΛjV ∗⊗sj is of highest

weight

(r1 + · · ·+ rm, r2 + · · ·+ rm, . . . , rm, 0, . . . , 0,−sn,−sn − sn−1, . . . ,−sn − · · · − s1). (2.17)

Hence, if wr denotes the partition (1, . . . , 1︸ ︷︷ ︸r

, 0, . . . , 0), and

λ• =

m∑i=1

riwi, λ◦ =

n∑j=1

sjwj , (2.18)

then the irreducible representation of GlN corresponding to this bipartition (λ•, λ◦) is the highest

weight representation of weight (2.17). Thus it is a quotient ofm⊗i=1

ΛiV ⊗ri ⊗n⊗j=1

ΛjV ∗⊗sj .

As every bipartition can be written in the form (2.18), every irreducible finite-dimensionalrepresentation of GlN is a quotient of such a tensor product.

2.5 Generic semisimplicity of Rep(Glt)

In this section we are going to see for which t the category Rep(Glt) is a semisimple category. Wehave the following theorem.

27

Theorem 2.5.1. The category Rep(Glt) is semisimple if and only if t /∈ Z.

Remark 2.5.2. If C is a C-linear Krull-Schmidt category, it is semisimple if and only if EndC(L) ∼= Cfor any indecomposable object L of C, and HomC(L,L

′) = 0 if L,L′ are non isomorphic indecom-posable objects.

Proof. We are going to use the criteria in Remark 2.5.2 to prove the result. First we have tolook at the walled Brauer algebras Br,s = End(wr,s) , as in 2.4. Let us admit that if t ∈ Zthen Br,s is not semisimple for some r, s > 0 (see [4] for a complete proof). Then, in that case,with Theorem 2.4.5, there exists two different bipartitions λ, µ, with l(λ) = (r − i, s − i), l(µ) =

(r − j, s − j), with i, j ≥ 0, such that e(i)λ Br,se

(j)µ 6= 0. But e(i)

λ Br,se(j)µ∼= Hom(im e

(j)µ , im e

(i)λ ),

by e(i)λ fe

(j)µ = ι

e(i)λ

πe(i)λ

fιe(j)µπe(j)µ7→ π

e(i)λ

fιe(j)µ∈ Hom(im e

(j)µ , im e

(i)λ ), and g 7→ ι

e(i)λ

gπe(j)µ; by the

splitting relations, these two maps are reciprocally inverse. Hence, if t ∈ Z, Hom(L(µ), L(λ)) =

Hom(im e(j)µ , im e

(i)λ ) 6= 0, and Rep(Glt) is not a semisimple category.

Let us suppose now that t /∈ Z, then for all r, s, Br,s is a semisimple algebra. Let λ, µ bebipartitions such that Hom(L(µ), L(λ)) 6= 0. If we write l(λ) = (r, s) and l(µ) = (r′, s′), then as inproof of Theorem 2.4.6, this implies that (r′, s′) = (r + i, s+ i). Let us suppose that i ≥ 0. Thentake e(i)

λ ∈ Br+i,s+i = Br′,s′ . We have Hom(L(µ), L(λ)) = Hom(im eµ, im e(i)λ ) 6= 0. But as Br′,s′

is semisimple, this imply that e(i)λ and eµ are conjugate, and by Theorem 2.4.5, that λ = µ. The

semisimplicity of Br,s also implies that End(L(λ)) ∼= eλBr,seλ = C.

28

Chapter 3

Yangian of gAs we said in the introduction, a Yangian is a type of quantum group. For any finite-dimensionalsemisimple Lie algebra a, the Yangian Y (a) is a Hopf algebra deformation of U(a[z]). See [7] fora construction of the Yangian of a simple Lie algebra with an invariant scalar product, a. TheYangian is there defined by taking h = 1 in a quantization of the polynomial current Lie algebrag = a[u].

3.1 Classical Yangian of glNThere are different ways to define to define the Yangian of a semisimple Lie algebra. In [2], thereare two different presentations of the Yangian of a simple finite-dimensional Lie algebra, and theisomorphisms between these presentations.

In the case of glN there exists a third presentation of Y (g), called the Faddeev-Reshetikhin-Takhtajan presentation, or RTT presentation, and it the one we are going to use.

3.1.1 MotivationThe definition of the Yangian Y (glN ) is modeled on the structure of the universal envelopingalgebra U(glN ).

Consider the general linear glN , and its basis vectors Eij , for 1 ≤ i, j ≤ N . We have thefollowing relations.

[Eij , Ekl] = δkjEil − δilEkj , ∀ 1 ≤ i, j, k, l ≤ N (3.1)

Now, introduce the matrix E of size N×N , with coefficients in glN whose ij-th entry is Eij . Then,for p ≥ 1, and 1 ≤ i, j ≤ N , (Ep)ij =

∑Eik1Ek1k2 · · ·Eks−1l ∈ U(glN ). Then, we can prove the

following result by induction, using (3.1).

[Eij , (Ep)kl] = δkj(E

p)il − δil(Ep)kj , ∀ 1 ≤ i, j, k, l ≤ N. (3.2)

With (3.2), and another induction, we get the following equation, for p, q ≥ 0 (E0 = 1 by conven-tion).

[(Ep+1)ij , (Eq)kl]− [(Ep)ij , (E

q+1)kl] = (Ep)kj(Eq)il − (Eq)kj(E

p)il (3.3)

The algebra Y (glN ) is the one obtained by replacing (Ep)ij by an abstract generator t(p)ij .

3.1.2 DefinitionDefinition 3.1.1. The Yangian Y (glN ) of the general linear algebra glN , for N an integer, is aunitary associative algebra over C with generators t(m)

ij , where i, j = 1, . . . , N and m ∈ N∗, andrelations

[t(p+1)ij , t

(q)kl ]− [t

(p)ij , t

(q+1)kl ] = t

(q)kj t

(q)il − t

(p)kj t

(q)il (3.4)

where p, q ∈ N, and t(0)ij = δij by convention.

29

Remark 3.1.2. The relation we use to define Y (glN ) in (3.4) is actually not the same as (3.3), theleft-hand side is multiplied by −1. We use here this this other definition of Y (glN ) because it willbe convenient afterwards, but the two definitions give isomorphic algebras, the isomorphism beingdefined by t(p)ij 7→ t

(p)ji .

Proposition 3.1.3. The system of the defining relations (3.4) of the Yangian is equivalent to thefollowing system, where i, j = 1, . . . , N

[t(p)ij , t

(q)kl ] =

min{p,q}∑a=1

(t(p+q−a)kj t

(a−1)il − t(a−1)

kj t(p+q−a)il

)(3.5)

Proof. Let us suppose that p < q. We are going to prove the result by induction on p.

[t(1)ij , t

(q)kl ] = [t

(0)ij , t

(q+1)kl ] + t

(q)kj t

(0)il − t

(0)kj t

(q)il

= δilt(q)kj − δkjt

(q)il

So (3.5) is satisfied for p = 1. Let us suppose that it is satisfied up to a certain rank p. Then

[t(p+1)ij , t

(q)kl ] = [t

(p)ij , t

(q+1)kl ] + t

(q)kj t

(p)il − t

(p)kj t

(q)il

=

p∑a=1

(t(p+q−a)kj t

(a−1)il − t(a−1)

kj t(p+q−a)il

)+ t

(q)kj t

(p)il − t

(p)kj t

(q)il

=

p+1∑a=1

(t(p+q−a)kj t

(a−1)il − t(a−1)

kj t(p+q−a)il

)

Let us define the formal generating series

tij(u) := δij +∑k∈N∗

t(k)ij u

−k ∈ Y (glN )[[u−1]]. (3.6)

Then the relation (3.4) can be written in the form

(u− v)[tij(u), tkl(v)] = tkj(v)til(u)− tkj(u)til(v) (3.7)

where the indeterminates u and v are considered to be commuting with each other and with theelements of the Yangian.

3.1.3 Matrix form of the defining relations

Let us define T (u), an element of glN ⊗ Y (glN )[[u−1]].

T (u) :=

N∑i,j=1

eij ⊗ tij(u) (3.8)

where eij ∈ glN are the standard basis matrices, with 1 in position (i, j) and 0 elsewhere.We can translate the defining relations (3.4) of the Yangian with only one equation on T (u).

First we define the permutation

σ : CN ⊗ CN → CN ⊗ CN , x⊗ y 7→ y ⊗ x. (3.9)

30

As an element of End(CN ) = glN ⊗ glN , it can be written σ =N∑

i,j=1

eij ⊗ eji.

Then we introduceR(u) := 1 +

σ

u∈ gl⊗2

N

We write 1 instead of 1⊗ 1 for brevity.

Remark 3.1.4. R is called the Yang R-matrix and it satisfies the Yang-Baxter equation.

R12(u)R13(u+ v)R23(v) = R23(v)R13(u+ v)R12(u) (3.10)

where R12(u) denotes 1 + 1u

N∑i,j=1

eij ⊗ eji⊗ 1 = 1 + (12)u , and the same goes for R13(u) and R23(u).

All are elements of gl⊗3N .

Proof.

R12(u)R13(u+ v)R23(v)−R23(v)R13(u+ v)R12(u) =(1 +

(12)

u

)(1 +

(13)

u+ v

)(1 +

(23)

v

)−(

1 +(23)

v

)(1 +

(13)

u+ v

)(1 +

(12)

u

)= 1 +

(12)

u+

(13)

u+ v+

(23)

v+

(132)

u(u+ v)+

(123)

uv+

(132)

v(u+ v)+

(13)

uv(u+ v)

−(

1 +(12)

u+

(13)

u+ v+

(23)

v+

(123)

u(u+ v)+

(132)

uv+

(123)

v(u+ v)+

(13)

uv(u+ v)

)=

1

uv((132) + (123))− 1

uv((123) + (132)) = 0

Finally, as before, let T 13(u) and T 23(u) respectively denoteN∑

i,j=1

eij ⊗ 1⊗ tij(u) andN∑

i,j=1

1⊗

eij ⊗ tij(u), and R12(u) denote 1 + 1u

N∑i,j=1

eij ⊗ eji ⊗ 1, where 1 is the unit of Y (glN ) (and 1 is

short for 1⊗ 1⊗ 1).These are all elements of gl⊗2

N ⊗ Y (glN )[[u−1]].

Proposition 3.1.5. The definition relations of the algebra Y (glN ) are equivalent to the followingsingle relation in gl⊗2

N ⊗ Y (glN )[[u−1]]

R12(u− v)T 13(u)T 23(v) = T 23(v)T 13(u)R12(u− v). (3.11)

Proof. We have

σ12T 13(u)T 23(v) =(∑

eij ⊗ eji ⊗ 1)(∑

ekl ⊗ 1⊗ tkl(u))

(1⊗ epq ⊗ tpq(v))

=(∑

eil ⊗ eji ⊗ tjl(u))

(1⊗ epq ⊗ tpq(v))

=∑i,j,l,q

eil ⊗ ejq ⊗ tjl(u)tip(v) =∑i,j,k,l

eij ⊗ ekl ⊗ tkj(u)til(v)

Whereas

31

T 23(v)T 13(u)σ12 =∑i,j,k,l

eij ⊗ ekl ⊗ tkj(v)til(u)

Hence

σ12T 13(u)T 23(v)− T 23(v)T 13(u)σ12 =∑i,j,k,l

eij ⊗ ekl ⊗ (tkj(u)til(v)− tkj(v)til(u))

We recognize the left-hand side of the relation (3.7) (up to a sign). Next we have

(u− v)((T 13(u)T 23(v)− T 23(v)T 13(u)

)=

(u−v)[ (∑

eij ⊗ 1⊗ tij(u))(∑

1⊗ ekl ⊗ tkl(v))−(∑

1⊗ ekl ⊗ tkl(v))(∑

eij ⊗ 1⊗ tij(u)) ]

= (u− v)∑i,j,k,l

eij ⊗ ekl (tij(u)tkl(v)− tkl(v)tij(u))

If we combine everything we have, we get

(u− v)(R12(u− v)T 13(u)T 23(v)− T 23(v)T 13(u)R12(u− v)

)=∑

i,j,k,l

eij ⊗ ekl ⊗ ((u− v)[tij(u), tkl(v)] + tkj(u)til(v)− tkj(v)til(u))

It is clear that (3.11) is equivalent to (3.7), which is equivalent to (3.4), the defining relationsof the Yangian.

Remark 3.1.6. The relation (3.11) is usually called the RTT relation. We can note that it resemblesthe relation of the R-matrix (3.10).

3.2 Yangian in complex rank

3.2.1 DefinitionWe are going to define the Yangian of g, which we defined in Section 2.3, and is the equivalent ofglN in Deligne’s category Rep(Glt). We are going to follow the definition of Y (g) given in Section7 of [8], which is an interpolation of the RTT presentation, in complex rank.

Let us recall that g is the object g = V ⊗ V ∗ = • ⊗ ◦ = •◦, of the category Rep(Glt).First, let us define the tensor algebra A := T(

⊕i≥0(V ⊗ V ∗)(i)), which is not technically an

object of the category Rep(Glt), but a Ind-object. Then let Ti be the coevaluation map

Ti = 1 −→ (V ∗ ⊗ V )⊗ (V ⊗ V ∗)(i) ∈ Hom(1, (V ∗ ⊗ V )⊗A).

Then let T (u) := 1 + T0u−1 + T1u

−2 + · · · be the generating function of Ti. Let σ be thepermutation σ : V ⊗ V → V ⊗ V and let R(u) := 1 + σ

u , just as before (but in the general case wedo not have basis to give another definition of σ).

Let us consider the series

Q(u, v) := (u− v)(R12(u− v)T 13(u)T 23(v)− T 23(v)T 13(u)R12(u− v))

=∑

Qi,juivj

32

The coefficients Qi,j are elements of Hom(1, (V ∗ ⊗ V ) ⊗ (V ∗ ⊗ V ) ⊗ A), but they can be seen asmorphisms

Qij = (V ⊗ V ∗)⊗ (V ⊗ V ∗) −→ A

Let J be the ideal of A generated by the images of all the Qij .

Definition 3.2.1. The algebra Y (g) := A/J is called the Yangian of g.

Remark 3.2.2. • We can see that it is the same definition as in the classical case, but withoutusing basis vectors, which we do not have in complex rank.

• Y (g) is generated by the images of the coefficients Ti of T (u), seen as morphisms V ⊗V ∗ −→A. Thus from now on, for all the morphisms of A, we are going to study only their actionson T (u).

3.2.2 Hopf algebra structureAs promised before, the Yangian is supposed to be an Hopf algebra. Y (g) as we defined has indeedan Hopf algebra structure.

Proposition 3.2.3. The Yangian Y (g) is an Hopf algebra with

Coproduct ∆ : T (u) 7→ T 12(u)T 13(u),

Antipode S : T (u) 7→ T−1(u),

Counit ε : T (u) 7→ 1.

Remark 3.2.4. If we write the coproduct using the definition in the classical case we get

∆ : t(r)ij 7→

∑k

r∑s=0

t(s)ik ⊗ t

(r−s)kj (3.12)

Indeed, T 12(u)T 13(u) =(∑

eij ⊗ t(p)ij u−p ⊗ 1

)(∑ekl ⊗ 1⊗ t(q)kl u−q

)=∑eil ⊗ t(p)ij ⊗ t

(q)jl u

−q−p =∑i,l

eil ⊗∑k

(+∞∑r=0

(r∑s=0

t(s)ik ⊗ t

(r−s)kl

)u−r

),

which is exactly the result we wanted.The counit is defined by ε : t

(r)ij 7→ 0 for r ≥ 1.

Proof. First of all, we have to verify that (Y (g),∆, ε) is a bialgebra.For the associativity of the coproduct, it is enough to see that

((∆⊗ id) ◦∆)T (u) = ((id⊗∆) ◦∆)T (u) = T 12(u)T 13(u)T 14(u)

where the definition of T 14(u) is the obvious one.Furthermore ((ε ⊗ id) ◦∆)T (u) = (ε ⊗ id)T 12(u)T 13(u) = T (u), and the same goes for id⊗ε.

Hence the counit axiom is satisfied, and Y (g) is a bialgebra.

33

Next, in order for Y (g) to be an Hopf algebra, we also have to check that the following diagramcommutes (where m is the product and η : C→ Y (g) is the unit 1 7→ 1)

Y (g)⊗ Y (g)S⊗id

// Y (g)⊗ Y (g)

m

&&

Y (g)

∆

99

ε //

∆

&&

Cη

// Y (g)

Y (g)⊗ Y (g)id⊗S

// Y (g)⊗ Y (g)

m

99

We have T (u)∆7−→ T 12(u)T 13(u)

S⊗id7−−−→ (T 12)−1(u)T 13(u)m7−→ 1 = (η ◦ ε)T (u), and the same goes

for the lower part of the diagram. Thus (Y (g),∆, S, ε) is indeed an Hopf algebra.

34

Chapter 4

Representations of the Yangian Y (glN ),where N is an integer

In this chapter we consider the Yangian of glN , Y (glN ), where N is an integer, with classicaldefinitions of glN and Rep(GlN ). We are going to study the representations L of Y (glN ). If L andM are representations of Y (glN ), then their tensor product space L⊗M can be equipped with thestructure of a Y (glN )-module also, using the coproduct ∆, as Y (glN ) is an Hopf algebra.

In the theory of irreducible representations of Y (glN ), as in the case of representations ofreductive Lie algebras, there is a classification of finite-dimensional representations, using highestweights. See [13] or [2] for further reference.

4.1 Highest weight representations

Definition 4.1.1. A representation L of Y (glN ) is called a highest weight representation if thereexist a cyclic vector ζ in L (such that Y (glN ).ζ = L) which satisfies

tij(u).ζ = 0 for 1 ≤ j < i ≤ Ntii(u).ζ = λi(u)ζ for 1 ≤ i ≤ N

where λi(u) is a formal series

λi(u) = 1 + λ(1)i u−1 + λ

(2)i u−2 + · · · ∈ 1 + u−1C[[u−1]].

The vector ζ is called the highest weight vector of L and the tuple λ(u) := (λ1(u), λ2(u), . . . , λN (u))is called the highest weight of L.

This definition is inspired from the classical representation theory of semi-simple Lie algebras,and as in that case, we have can define the Verma module.

Definition 4.1.2. Let λ(u) := (λ1(u), λ2(u), . . . , λN (u)) be a tuple of formal series λi(u) ∈ 1 +u−1C[[u−1]]. The Verma module M(λ(u)) is the quotient Y (gln)/J , where J is the left ideal ofY (glN ) generated by all the coefficients of the series tij(u) for 1 ≤ j < i ≤ N and tii(u) − λi(u)for 1 ≤ i ≤ N .

Lemma 4.1.3. Let 1λ(u) be the image of 1 ∈ Y (glN ) in the quotient. Given any order on the setof generators t(r)ij , with 1 ≤ i < j ≤ N and r ≥ 1, the elements

t(r1)i1j1· · · t(rm)

imjm1λ(u), m ≥ 0,

with ordered products of the generators, form a basis of M(λ(u)).

Proof. This follows from the Poincaré-Birkhoff-Witt theorem in the Y (glN ) case : given an arbi-trary linear order on the set of generators t(r)ij , any element of the algebra Y (glN ) can be uniquelywritten as a linear combination of ordered monomials in these generators. See [13] for details.

35

As with a semi-simple Lie algebra,M(λ(u)) possesses a unique maximal proper submodule, thesum of all proper submodules. Hence we can define the irreducible highest weight representations.

Definition 4.1.4. The irreducible highest weight representation L(λ(u)) of Y (gln) is defined asthe quotient of M(λ(u)) by its unique maximal proper submodule.

Proposition 4.1.5. Every finite-dimensional irreducible representation L of the Yangian Y (gln)is isomorphic to a unique irreducible highest weight representation L(λ(u)).

Proof. First of all, we can see that with the embedding

Eij 7−→ t(1)ji , (4.1)

every representation of Y (glN ) is also a glN -module.Remark 4.1.6. This is indeed an algebra morphism, as [Eij , Ekl] = δjkEil − δilEkj and by (3.5)[t

(1)ji , t

(1)lk ] = δjkEli − δilEjk.

Let us recall some notations regarding representations of glN . Let h be the diagonal Cartansubalgebra of glN , and h∗ its dual space. Let ε1, . . . , εN be the basis vectors of h∗ dual to the basisvectors of h, E11, . . . , ENN , respectively. Every weight (in the classical sense) µ of L can be writtenµ = µ1ε1 + · · ·+µN εN . If α and β are two weights of L, we say that α precedes β, and write α < β,if β−α is nonzero, and can be written as a Z+-linear combination of the simple roots εi− εj , withi < j.

Let L be a finite-dimensional irreducible representation of Y (glN ). Let us introduce the follow-ing subspace of L,

L0 := { ζ ∈ L | tij(u).ζ = 0, 1 ≤ j < i ≤ N }.

The first step of the proof of Proposition 4.1.5 is to show that L0 6= {0}.As we said before, L is also a representation of glN , let us consider the set of weights of L. It

is a finite set as L is finite-dimensional, and has a maximal weight µ with respects to the orderingintroduced above. The corresponding weight vector ζ belongs to L0. Indeed, for 1 ≤ j < i ≤ Nand k = 1, . . . , N

Ekktij(u)ζ = [Ekk, tij(u)]ζ + tij(u) (Ekkζ)

By identifying the elements Eij of glN with their images via the embedding (4.1) we have

[Ekk, tij(u)] = [t(1)kk , tij(u)]

= δkjtik(u)− δiktkj(u) by taking the coefficient of v0 in (3.7)

Thus Ekktij(u)ζ = (δkjtik(u)− δiktkj(u))ζ + µ(Ekk)tij(u)ζ = (µ+ εj − εi)(Ekk)tij(u)ζHence, if tij(u)ζ 6= 0 then it is a non zero vector of L of weight µ+ εj − εi, which is > µ with

the ordering defined above (if j<i). This is a contradiction with the maximality of µ. Thus ζ ∈ L0.

Next step is to show that L0 is invariant under the action of all elements tkk(u), and conse-quently under the action of all elements t(r)kk . Let ζ be an element of L0.

[tij(u), tkk(v)]ζ =1

u− v(tkj(v)tik(u)− tkj(u)tik(v))ζ = 0 if i < k

= −[tkk(v), tij(u)]ζ =1

u− v(tik(u)tkj(v)− tik(v)tkj(u))ζ = 0 if k < j

Thus if i < j, tij(u)tkk(v)ζ = [tij(u), tkk(v)]ζ = 0, and tkk(v)ζ ∈ L0.

36

Furthermore, (3.7) implies that[tii(u), tii(v)] = 0

And(u− v)[tii(u), tjj(v)] = tji(v)tij(u)− tji(u)tij(v)

Hence, as operators on L0, the elements tkk(u), for k = 1, . . . , N are pairwise commutative, andso are the elements trkk, for k = 1, . . . , N and r ≥ 1.

Let ζ be a simultaneous eigenvector of L0 for these operators. ζ generates L because L issupposed to be irreducible. So ζ is a highest weight vector with respects to definition 4.1.1, witha certain weight λ(u).

By Lemma 4.1.3, the vector space L is spanned by the elements

t(r1)i1j1· · · t(rm)

imjmζ, m ≥ 0, ik < jk

with ordered products of the generators. Based on what we did earlier we know that t(r)ij ζ is ofweight µ+ εj − εi, which is < µ with respects to the partial order we defined above, if i < j. Thisimplies that the weight space Lµ is one-dimensional and spanned by the vector ζ. Moreover, if νis a weight of L and ν 6= µ then ν < µ. Hence the highest vector ζ of L is determined uniquely, upto a constant factor.

Hence we have shown that L is a highest weight representation. In order to prove that it isisomorphic to a unique L(λ(u)), one must use the same arguments as for a semisimple Lie algebra.As it is not the center of the discussion, we will not give details about that.

We now have that every finite-dimensional irreducible representation of Y (glN ) is isomorphicto a unique L(λ(u)). Thus to obtain a classification of irreducible representations of Y (glN ), weonly have to know which L(λ(u)) is finite-dimensional. We will begin by studying the case ofrepresentations of Y (gl2), and in the following section we will give the criteria for L(λ(u)) to befinite-dimensional.

4.2 Representations of Y (gl2)

4.2.1 A word on representations of gl2Let α, β be complex numbers and let L(α, β) denote the irreducible highest weight representationof the Lie algebra gl2 of weight (α, β). Let ζ denote the highest vector of L(α, β), then we have

E11ζ = aζ, E22ζ = βζ, E21ζ = 0

It is well known that if α−β ∈ Z+, then L(α, β) is finite-dimensional and that the vectors (E21)rζ,for r = 0, 1, . . . , αβ form a basis of L(α, β). If α− β /∈ Z+, then the vectors (E21)rζ, where r runsover all nonnegative integers, are linearly independent and form a basis of L(α, β), which is notfinite-dimensional.

It can be equipped with a Y (gl2)-module structure via some evaluation morphisms, for anz ∈ C, evz : Y (gl2)→ U(gl2) is defined (on the formal series (3.6))by

tij(u) 7→ δij +Ejiu− z

. (4.2)

As stated, evz is actually a morphism from Y (gl2)[[u1]] to U(gl2)[[u−1]], but by taking each com-ponent of the formal series in u−1, we get a morphism form Y (gl2) to U(gl2). Indeed,

evz(tij(u)) = δij +Ejiu− z

= δij +Ejiu

1

1− zu

= δij + Eji

+∞∑k=0

zku−k−1 (4.3)

37

Hence the formula (4.2) is just a simpler way to write the following morphism

evz : Y (gl2) → U(gl2),

t(r)ij 7→ Ejiz

r−1 if r ≥ 1,

t(0)ij = δij 7→ δij .

(4.4)

Lemma 4.2.1. The formula (4.2) defines an algebra morphism from Y (gl2) to U(gl2).

Proof. We have to check that, for any z ∈ C, the definition of evz is consistent with the definingrelations of the Yangian Y (gl2). We are going to show that the definition (4.2) is consistent withthe relation (3.7) of the Yangian.

evz([tij(u), tkl(v)]) = evz(

1

u− v(tkj(v)til(u)− tkj(u)til(v)

)=

1

u− v

((δkj +

Ejkv − z

)(δil +

Eliu− z

)−(δkj +

Ejku− z

)(δil +

Eliv − z

))=

1

u− v

(δkjEli

(1

u− z− 1

v − z

)+ δilEjk

(1

v − z− 1

u− z

))=

1

u− v(δilEjk − δkjEli)

u− v(u− z)(v − z)

= (δilEjk − δkjEli)1

(u− z)(v − z)

[evz(tij(u)), evz(tkl(v))] =

[δij +

Ejiu− z

, δkl +Elkv − z

]= [Eji, Elk]

1

(u− z)(v − z)

Definition 4.2.2. Let V be a representation of U(gl2). The representation of Y (gl2) obtained bytaking the composition of the evaluation morphism evz and of the action of U(gl2) on V is called theevaluation representation of V and is denoted ev∗z(V ). It is also called a pull-back representation.

Remark 4.2.3. This definition can be generalized. The formula (4.2) also defines an algebra mor-phism from Y (glN ) to U(glN ), for N ≥ 3. Hence every U(glN )-module can be endowed with thestructure of a Y (glN )-module. The evaluation representations are denoted the same.

Let us note that via the evaluation morphism, L(α, β) is an irreducible highest weight repre-sentation, with highest vector ζ and highest weight (αi(u), βi(u)), where

α(u) = 1 +α

u−z

β(u) = 1 +β

u− z.

Hence the notation L(α, β) or L(λ) where λ = (α, β) is consistent, as a highest weight representa-tion of both gl2 and Y (gl2).

4.2.2 Prerequisites for the classification theoremIn the rest of the section, we are going to consider an irreducible highest weight representationL(λ(u)) of Y (gl2), with weight (λ1(u), λ2(u)).

Proposition 4.2.4. If L(λ(u)) is finite-dimensional, then there is a formal series of the form

f(u) = 1 + f1u−1 + f2u

−2 + · · · , fk ∈ C

such that f(u)λ1(u) and f(u)λ2(u) are polynomials in u−1.

38

Proof. The mapping T (u) 7→ λ2(u)−1T (u) is an automorphism of Y (gl2) (the image satisfies theRTT relation). By twisting the action of Y (gl2) on L(λ(u)) by this automorphism, we get airreducible highest weight representation of weight (ν(u), 1), with ν(u) = λ1(u)/λ2(u), hence iso-morphic to L(ν(u), 1). Thus, we can consider that the highest weight of L(λ(u)) as the formλ(u) = (ν(u), 1), without loss of generality.

Let ζ denote the highest weight vector of L(ν(u), 1). Since this representation is finite-dimensional, the vectors t(r)12 ζ, r ≥ 1 are linearly dependent. Hence there is a integer m and

c1, . . . , cm ∈ C, such that cm 6= 0 andm∑r=1

crt(r)12 ζ = 0. According to Lemma 4.1.3, the vector

ξ =

m∑r=1

crt(r)12 1λ(u)

is non-zero, and its image in the quotient L(ν(u), 1) is zero, hence ξ is a non-zero vector of K, theunique maximal proper submodule of M(ν(u), 1).

Using (3.5), in M(ν(u), 1), we have

t(k)21 t

(r)12 1λ(u) =

min{k,r}∑a=1

(t(k+r−a)11 t

(a−1)22 − t(a−1)

11 t(k+r−a)22

)1λ(u) = ν(k+r−1)

1λ(u)

So t(k)21 ξ =

m∑r=1

crt(k)21 t

(r)12 1λ(u) =

m∑r=1

crν(k+r−1)

1λ(u) ∈ C1λ(u).

Hence, t(k)21 ξ = 0 for all k ≥ 1, otherwise 1λ(u) would be in K, and we have the relations, for

all k ≥ 1m∑r=1

crν(k+r−1) = 0

They imply

ν(u)

(m∑r=1

crur−1

)=

+∞∑k=0

crν(k)ur−k−1 =

m−1∑s=0

usm∑r=1

crν(r−s−1)

︸ ︷︷ ︸=0 if s≤−1 or r−s−1<0

=

m−1∑s=0

usm∑

r=s+1

crν(r−s−1)

So ν(u)(c1 + c2u+ · · ·+ cmum) = b1 + b2u+ · · ·+ bmu

m−1, with bm = cm. Thus, taking

f(u) := c−1m

m∑r=1

cru−m−r

we get that f(u)ν(u) and f(u)1 are monic polynomials in u−1.

This proposition implies that by taking the composition of the action of Y (gl2) on L(λ(u)) withan appropriate automorphism, we get another highest weight representation of Y (gl2) for whichboth components of the highest weight are polynomials in u−1. From now on, we are going toassume that this is the case.

We write the decompositions

λ1(u) = (1 + α1u−1) . . . (1 + αmu

−1)

λ2(u) = (1 + β1u−1) . . . (1 + βmu

−1)

where {αi}1≤i≤m, {βi}1≤i≤m are complex numbers.

39

Lemma 4.2.5. Suppose that for every 1 ≤ i ≤ m−1 the following condition holds : if the multiset{αp − αq | i ≤ p, q ≤ m} contains non-negative integers, then αi − βi is minimal amongst them.Then the representation L(λ1(u), λ2(u)) of Y (gl2) is isomorphic to the tensor product module

L(α1, β1)⊗ L(α2, β2)⊗ . . .⊗ L(αm, βm).

We admit this result. The complete proof can be found in [13]. What is interesting aboutthis lemma is that the condition is not restrictive. Indeed, if some of the differences αp − βq arenonnegative integers, choose a minimal amongst them and re-enumerate it α1 − β1, then proceedby induction, considering all differences αp − βq.

The following Lemma is also going to be useful in the proof of the classification theorem.

Lemma 4.2.6. The tensor product module L = L(α1, β1) ⊗ L(α2, β2) ⊗ · · · ⊗ L(αp, βp) has asubmodule which is a highest weight module of weight (α(u), β(u)), with

α(u) = (1 + α1u−1)(1 + α2u

−1) · · · (1 + αpu−1),

β(u) = (1 + β1u−1)(1 + β2u

−1) · · · (1 + βpu−1).

Proof. For 1 ≤ i ≤ p, let ζi be the highest weight vector of L(αi, βi). We form the elementζ = ζ1 ⊗ ζ2 · · · ⊗ ζp ∈ L. Let L′ be the Y (gl2)-submodule of L generated by ζ.

L′ = Y (gl2).ζ (4.5)

Then, by the definition of the coproduct ∆ in Y (gl2), we have for i, j ∈ {1, 2},

tij(u).ζ =∑

(i1,...,ip)∈{1,2}pt1i1(u).ζ1 ⊗ ti1i2(u).ζ2 ⊗ · · · ⊗ tipj(u).ζp. (4.6)

Hence t12(u).ζ = 0, because we can not have 2 ≤ i1 ≤ i2 ≤ · · · ≤ ip ≤ 1. And

t11(u).ζ = t11(u).ζ1 ⊗ t11(u).ζ2 ⊗ · · · ⊗ t11(u).ζp =

p∏i=1

(1 + αiu−1)

t22(u).ζ = t22(u).ζ1 ⊗ t22(u).ζ2 ⊗ · · · ⊗ t22(u).ζp =

p∏i=1

(1 + βiu−1)

Thus, L′ is indeed a submodule of L which is a highest weight module of weight (α(u), β(u)).

4.2.3 The classification theoremTheorem 4.2.7. The irreducible highest weight representation L(λ1(u), λ2(u)) of Y (gl2) is finite-dimensional if and only if there exists a monic polynomial P (u) in u, such that

λ1(u)

λ2(u)=P (u+ 1)

P (u)(4.7)

And in this case, P is unique.

Proof. Suppose that L(λ1(u), λ2(u)) is finite-dimensional. By proposition 4.2.4, there is a formalseries f(u) such that

f(u)λ1(u) = 1 + a1u−1 + · · ·+ amu

−m = (1 + α1u−1) . . . (1 + αmu

−1)

f(u)λ2(u) = 1 + b1u−1 + · · ·+ bmu

−m = (1 + β1u−1) . . . (1 + βmu

−1)

40

for some m ≥ 0, where {αi}1≤i≤m, {βi}1≤i≤m are complex numbers. As we saw before, we canassume that the condition of lemma 4.2.5 is satisfied, by re-enumerating these parameters if nec-essary.

Then L(λ1(u), λ2(u)) is isomorphic to L(α1, β1)⊗ L(α2, β2)⊗ . . .⊗ L(αm, βm). As it is finite-dimensional, necessarily αi− βi are nonnegative integers for all 1 ≤ i ≤ m (see introduction of thesection).

Then the polynomial

P (u) :=

m∏i=1

(u+ βi)(u+ βi + 1) . . . (u+ αi − 1)

is well defined and satisfies (4.7). Indeed,

P (u+ 1)

P (u)=

m∏i=1

(u+ βi + 1)(u+ βi + 2) . . . (u+ αi)

(u+ βi)(u+ βi + 1) . . . (u+ αi − 1)=

m∏i=1

(u+ αi)

m∏i=1

(u+ βi)=f(u)λ1(u)

f(u)λ2(u)=λ1(u)

λ2(u)

For the reverse implication, suppose that the condition (4.7) holds for a polynomial P (u) =(u+ γ1) . . . (u+ γp). Set

µ1(u) = (1 + (γ1 + 1)u−1) . . . (1 + (γp + 1)u−1), µ2(u) = (1 + γ1u−1) . . . (1 + γpu

−1)

and consider the tensor product module

L := L(γ1 + 1, γ1)⊗ L(γ2 + 1, γ2)⊗ . . .⊗ L(γp + 1, γp).

L is obviously finite-dimensional. From Lemma 4.2.6, we know that under these conditions, L hasa submodule which is a highest weight module of weight (µ1(u), µ2(u)). This submodule is thenfinite-dimensional and so is its irreducible quotient L(µ1(u), µ2(u)). Since

λ1(u)

λ2(u)=P (u+ 1)

P (u)=µ1(u)

µ2(u)

Consider the automorphism of Y (gl2), T (u) 7→ f(u)T (u), where f(u) = λ1/µ1 = λ2/µ2. Itscomposition with the representation L(µ1(u), µ2(u)) is isomorphic to L(λ1(u), λ2(u)), which isthen finite-dimensional.

The only thing left to prove is that the polynomial P is unique. Take another monic polynomialQ which satisfies (4.7). Then we have

P (u+ 1)

P (u)=Q(u+ 1)

Q(u)

which means that the fraction P (u)/Q(u) is periodic in u, which is only possible if it is a constant,hence if P = Q.

Definition 4.2.8. The polynomial P is called the Drinfeld Polynomial of the finite-dimensionalrepresentation L(λ1(u), λ2(u)).

The relation (4.7) is the key of Theorem 4.2.7, hence we introduce a simpler notation. Fromnow on, we will write

λ1(u) −→ λ2(u) (4.8)

if there exists a monic polynomial P (u) such that (4.7) is satisfied.

41

4.3 Irreducible representations of Y (glN)

4.3.1 A criteria for finite-dimensional representationsWe now have the tools we need to prove the classification theorem for irreducible representationsof Y (glN ). Let us recall that every irreducible representation of Y (glN ) is isomorphic to a highestweight representation L(λ(u)), where λ(u) is a N -tuple of formal series λ(u) = (λ1(u), . . . , λN (u)),where each of the formal series λi(u) is of the form

λi(u) = 1 + λ(1)i u−1 + λ

(2)i u−2 + · · · ∈ C[[u−1]]

Theorem 4.3.1. The irreducible highest weight representation L(λ(u)) of the Yangian Y (glN ) isfinite-dimensional if and only if the following relations are satisfied

λ1(u) −→ λ2(u) −→ · · · −→ λN (u). (4.9)

Proof. First, suppose that dimL(λ(u)) < +∞.For 0 ≤ k ≤ N − 2, let us consider the morphism

Y (gl2) −→ Y (glN )

tij(u) 7−→ ti+k,j+k(u), for i, j = 1, 2

Y (gl2) acts on L(λ(u)) via this morphism and the action of Y (glN ). Let ζ be the highest weightvector of L(λ(u)) (as a Y (glN )-module). The Y (gl2)-module Y (gl2)ζ is a highest weight moduleof weight (λk+1(u), λk+2(u)). It is finite-dimensional, as a submodule of L(λ(u)). Hence, itsirreducible quotient is also finite-dimensional, and by Theorem 4.2.7, we have λk+1(u) −→ λk+2(u).This being true for all k ∈ {0, . . . , N − 2}, we have proven the first implication.

Conversely, suppose that we have λ1(u) −→ λ2(u) −→ · · · −→ λN (u). Then, for 1 ≤ i ≤ N −1,there exists a monic polynomial Pi, such that

λi(u)

λi+1(u)=Pi(u+ 1)

Pi(u)(4.10)

We introduce some notations. Let ki be the degree of the polynomial Pi. We write

Pi(u) = (u+ γ(1)i )(u+ γ

(2)i ) . . . (u+ γ

(ki)i ), γ

(l)i ∈ C. (4.11)

For 1 ≤ i ≤ N , let

µi(u) = u−kP1(u) · · ·Pi−1(u)Pi(u+ 1) · · ·PN−1(u+ 1) (4.12)

= (1 + µ(1)i u−1)(1 + µ

(2)i u−1) · · · (1 + µ

(k)i u−1) (4.13)

where k = k1 + · · ·+ kN−1. It is defined in order to have

µi(u)

µi+1(u)=Pi(u+ 1)

Pi(u)(4.14)

Consider L(µ(u)), the highest weight representation of Y (glN ) of weight µ(u) = (µ1(u), . . . , µN (u)).We have

µi(u) = u−k(u+ γ(1)1 ) . . . (u+ γ

(k1)1 )(u+ γ

(1)2 ) . . . (u+ γ

(k2)2 ) . . . (u+ γ

(ki−1)i−1 )(u+ 1 + γ

(1)i ) . . .

(u+ 1 + γ(ki)i ) . . . (u+ 1 + γ

(kN )N )

= (1 + γ(1)1 u−1) . . . (1 + γ

(ki−1)i−1 u−1)(1 + u−1 + γ

(1)i u−1) . . . (1 + u−1 + γ

(kN )N u−1)

42

Hence for 1 ≤ j ≤ N − 1 and 1 ≤ l ≤ kj , we have

µ(k1+···+kj−1+l)i =

{γ

(l)j if j < i

γ(l)j + 1 if j ≥ i

Let us consider µ(r) = (µ(r)1 , . . . , µ

(r)N ) =

γ(l)j + 1, . . . , γ

(l)j + 1︸ ︷︷ ︸

j times

, γ(l)j , . . . , γ

(l)j︸ ︷︷ ︸

r−j times

, for r = k1 + · · · +

kj−1 + l (l < kj). The glN -module L(µ(r)) is finite-dimensional, as µ(r)i − µ

(r)i+1 ∈ Z+, for 1 ≤ i ≤

N − 1. HenceL(µ(1))⊗ L(µ(2))⊗ · · · ⊗ L(µ(k))

is also finite-dimensional. It has the structure of a Y (glN )-module, via the evaluation morphismev0 (4.2) for example, as in 4.2.1. As in Lemma 4.2.6, the tensor product of the highest weightvectors of L(µ(r)) generates a highest weight representation of Y (glN ) with highest weight µ(u)(the proof of Lemma 4.2.6 extends clearly to the case of Y (glN )-modules). This representation isfinite-dimensional, and so is its irreducible highest weight representation, which is isomorphic toL(µ(u)). Thus we conclude that L(µ(u)) is finite-dimensional.

But we have, by definition, (4.14). Hence, for 1 ≤ i ≤ N − 1

µi(u)

µi+1(u)=

λi(u)

λi+1(u). (4.15)

If we let f(u) be the formal series

f(u) =λ1(u)

µ1(u)= · · · = λN (u)

µN (u)(4.16)

then the composition of the representation of Y (glN ) on L(µ(u)) and the automorphism of Y (glN )defined by T (u) 7→ f(u)T (u) is isomorphic to L(λ(u)). Thus L(λ(u)) is also finite-dimensional.

4.3.2 Structure of finite-dimensional irreducible representationsThe following result states that with only the evaluation representations, we can get all the irre-ducible representations of Y (g), and so all the representations of Y (g).

Let V denote the gln-module CN . If ΛiV is the ith exterior product of V , and z ∈ C, letΛiV (z) denote the pull-back ev∗z(ΛiV ), via the evaluation morphism evz defined in 4.2.1.

Proposition 4.3.2. Every finite-dimensional irreducible representation of Y (glN ) is isomorphicto a subquotient of a tensor product of evaluation representations, of the form

m⊗i=1

ΛiV (z(1)i )⊗ · · · ⊗ ΛiV (z

(ri)i )⊗

n⊗j=1

ΛjV ∗(w(1)j )⊗ · · · ⊗ ΛjV ∗(w

(sj)j ) (4.17)

where m,n ≥ 0 and r1, . . . , rm, s1, . . . , sn ≥ 0 are integers, and z(k)i , w

(l)j ∈ C.

If ~r = (r1, . . . , rm) and ~s = (s1 . . . , sn), z = {z(k)i }1≤i≤m,1≤k≤ri and w = {w(l)

j }1≤j≤n,1≤l≤sj ,then let us introduce the notation

~E(~r,~s, z, w) :=

m⊗i=1

ΛiV (z(1)i )⊗ · · · ⊗ ΛiV (z

(ri)i )⊗

n⊗j=1

ΛjV ∗(w(1)j )⊗ · · · ⊗ ΛjV ∗(w

(sj)j ). (4.18)

43

Proof. Via the evaluation morphism (4.2), ΛiV (z) is an irreducible representation of Y (glN ) of

weight λ(u) = (1 +1

u− z, . . . , 1 +

1

u− z︸ ︷︷ ︸i

, 1 . . . , 1), and ΛjV ∗(w) is an irreducible representation of

Y (glN ) of weight µ(u) = (1 . . . , 1, 1− 1

u− w, . . . , 1− 1

u− w︸ ︷︷ ︸i

).

Using the coproduct ∆ of the Hopf algebra Y (glN ), given by (3.12), we can compute the actionof Y (glN ) on ζk ⊗ ζl, seen as an element of ΛkV (z1)⊗ ΛlV (z2). Suppose k ≤ l.

tij(u)ζk ⊗ ζl =

N∑l=1

til(u)ζk ⊗ tlj(u)ζl = 0 if i < j

=

(1 +

1

u− z1

)(1 +

1

u− z2

)if i = j ≤ k ≤ l

=

(1 +

1

u− z2

)if k < i = j ≤ l

= 1 if i = j > l

Hence, Y (gln)ζk ⊗ ζl is a highest weight representation of Y (gln) of weight((1 +

1

u− z1

)(1 +

1

u− z2

), . . . ,

(1 +

1

u− z1

)(1 +

1

u− z2

),(

1 +1

u− z2

), . . . ,

(1 +

1

u− z2

), 1, . . . , 1

).

Following the same logic, the action of Y (gln) onm⊗i=1

ζ⊗rii ⊗n⊗j=1

ξ⊗sjj , seen as an element of

~E(~r,~s, z, w) spans a highest weight representation of weight (if N is a larger integer)(m∏i=1

ri∏k=1

(1 +

1

u− z(k)i

),

m∏i=2

ri∏k=1

(1 +

1

u− z(k)i

), . . . ,

rm∏k=1

(1 +

1

u− z(k)m

), 1, . . . , 1,

sn∏l=1

(1− 1

u− w(l)n

),

sn∏l=1

(1− 1

u− w(l)n

) sn−1∏l=1

(1− 1

u− w(l)n−1

), . . . ,

n∏j=1

sj∏l=1

(1− 1

u− w(l)j

) .

(4.19)

Hence, there is a quotient of Y (gln)m⊗i=1

ζ⊗rii ⊗n⊗j=1

ξ⊗sjj which is an irreducible highest weight

representation of weight (4.19). As Y (gln)m⊗i=1

ζ⊗rii ⊗n⊗j=1

ξ⊗sjj is a submodule of ~E(~r,~s, z, w), there

is a subquotient (quotient of a submodule) of ~E(~r,~s, z, w) which is an irreducible highest weightrepresentation of weight (4.19).

We have the criteria of Theorem 4.3.1 to see if is it a finite-dimensional representation. If λ(u)denotes the weight (4.19), then λk(u)/λk+1(u) is of the following form.

44

λk(u)

λk+1(u)=

ri∏k=1

(1 +

1

u− z(k)i

)=

ri∏k=1

(u+ 1− z(k)i )

ri∏k=1

(u− z(k)i )

, or (4.20)

=

ri∏k=1

(1 + 1

u−z(k)i

)sj∏l=1

(1− 1

u−w(l)j

) =

ri∏k=1

(u+ 1− z(k)i )

sj∏l=1

(u− w(l)j )

sj∏l=1

(u− 1− w(l)j )

ri∏k=1

(u− z(k)i )

, or (4.21)

=

sj∏l=1

(1− 1

u− w(l)j

)=

sj∏l=1

(u− w(l)j )

sj∏l=1

(u− 1− w(l)j )

. (4.22)

We have explicit monic polynomials Pk(u) such that

λk(u)

λk+1(u)=Pk(u+ 1)

Pk(u).

For (4.20), Pk(u) =ri∏k=1

(u− z(k)i ), for (4.21), Pk(u) =

sj∏l=1

(u−1−w(l)j )

ri∏k=1

(u− z(k)i ), and for (4.22),

Pk(u) =sj∏l=1

(u − 1 − w(l)j ). Hence ~E(~r,~s, z, w) has a subquotient which is a finite-dimensional

irreducible representation of Y (gln).Conversely, L(λ(u)) is the finite-dimensional irreducible representation of Y (gln) of weight λ(u).

Then by Theorem 4.3.1, there are monic polynomials Pk(u) such that λk(u)/λk+1(u) = Pk(u +1)/Pk(u). Which means that λk(u)/λk+1(u) is of the form (4.20) (the three results (4.20),(4.21)and (4.22) are of the same form). Thus λ(u) is of the form (4.19) and L(λ(u)) is a subquotient ofa certain ~E(~r,~s, z, w).

45

Chapter 5

Representations of Y (glt)

First of all, to be able to study them, we have to know what is a representation of Y (glt) (or Y (g)for short), when t is not necessarily an integer. It cannot be a vector space on which Y (g) actsbecause Y (g) is not an algebra in the usual sense. In the last chapter, we saw that there was anembedding Eij 7→ t

(1)ji thanks to which every Y (glN )-module was also a glN -module. It is then

logical to look for representations of Y (g) as objects of the category Rep(Glt). Hence we can givea definition similar to what exists for Rep(Glt), as seen in 2.3.3.

Definition 5.0.1. A representation of Y (g) is an object E of the category Rep(Glt), togetherwith a morphism of Hom(Y ⊗ E,E).

We cannot easily adapt the method we have seen in the last chapter to the general case of Y (g).The main reason is that we lack an equivalent for the notion of weight. Y (glt) has only one typeof generator, T (u), so we cannot imitate the glN case with (Ei, Fi, Hi)-type basis.

Nevertheless, there would be a way to introduce a highest weight theory on Rep(glt), via aninclusion of the type

St n (C∗)t∈Rep(St)

⊂ Glt.

And then we would use the functors

Rep(Y (glt)) −→ Rep(Glt) −→ Rep(St n (C∗)t)

The problem with this method is that the theory is not built yet on Rep(Glt), so it is not ready tobe used in a more complex case. Hence it is not an approach we chose to investigate any further.

We are now going to consider different approaches to the problem, from giving explicit examplesof representations of Y (g), to the beginning of a more detailed study.

5.1 Evaluation representations

We can easily obtain a family of representations of Y (g), but taking the pull-backs of the repre-sentations of g, hence elements of Rep(Glt), as in 4.2.1.

If we translate (4.2) in the matrix form, with the notations of 3.1.3, we obtain

T (u) =

N∑i,j=1

eij ⊗ tij(u) 7→ 1 +σ

u− z= R(u− z) (5.1)

As all these objects exist for Y (glt), we can use the same definition, in a more general setting.

Definition 5.1.1. The evaluation morphism evz : Y (glt)→ U(glt) is defined by

T (u) 7→ R(u− z). (5.2)

46

As the Yang R-matrix R satisfies the Yang-Baxter equation (3.10), evz defines an algebramorphism.Remark 5.1.2. σ : V ⊗ V → V ⊗ V can also be seen as a element of HomRep(Glt)(1, V ⊗ V ∗ ⊗V ⊗ V ∗) = Hom(1, g⊗ g) hence, the images of the coefficients of R(u− z) are indeed elements ofU(glt).

Definition 5.1.3. Let X be an object of Rep(Glt), and z ∈ C. The evaluation representationX(z) is the pullback ev∗zX of X, via the evaluation morphism evz, to a representation of Y (g). Wehave morphisms

Y (g)⊗X evz−−→ U(glt)⊗X → X. (5.3)

If X1, . . . , Xk are objects of Rep(Glt), and z1, . . . , zk are complex numbers, we can constructthe tensor product X1(z1)⊗· · ·⊗Xk(zk), which is also a representation of Y (g),using the coproduct∆ associated to the Hopf algebra structure of Y (g).Remark 5.1.4. Let us recall here that the two meanings of the notation L(λ), first the indecompos-able object of Rep(Glt) corresponding to the bipartition λ by (2.14), if λ is a partition, and thenthe irreducible highest weight representation of glN of weight λ, if λ is a tuple λ = (λ1, . . . , λN ),actually refers to the same object, if t = N is an integer, thanks to 2.4.5.

Hence, if λ be a bipartition of arbitrary size, and L(λ) denotes the highest weight representationof glN of weight λ = (λ1, . . . , λN ) then, as in the particular case of gl2,the evaluation representationof Y (glN ) obtained via the composition of (4.2) and the the action of glN is an irreducible highestweight representation of Y (glN ) of weight λ(u) = (λ1(u), . . . , λN ), where, for i ∈ {1, . . . , N}

λi(u) = 1 +λi

u− z∈ C[[u−1]]. (5.4)

We will write L(λ)z for this evaluation representation, and also more generally if L(λ) is theindecomposable object of Rep(Glt) corresponding to the bipartition λ.

5.2 Two conjectures

We are now going to present two results which show that evaluation representations are the keyto understand the classification of irreducible representations of the Yangian Y (g).

Unfortunately, these results will not be proven in this paper, but they could be ideas to expandthe work that we started here.

5.2.1 First conjectureLet λ1, . . . , λk be bipartitions of arbitrary size, and L(λ1), . . . , L(λk) the corresponding indecom-posable objects of Rep(Glt). For z1, . . . , zk ∈ C, we are going to consider the tensor product ofevaluation representations of indecomposables L(λ1)z1 ⊗ · · · ⊗ L(λk)zk .

Conjecture 5.2.1. For λ1, . . . , λk bipartitions of arbitrary size, the representation of Y (g),

L(λ1)z1 ⊗ · · · ⊗ L(λk)zk

is irreducible for generic parameters values of z1, . . . , zk.

This result was stated in [8]. It should not be too difficult to prove. We would first see thatwith the action of the loop algebra, it is already a irreducible module.

The question of knowing for exactly which values of z1, . . . , zk the Y (g)-module L(λ1)z1 ⊗· · ·⊗L(λk)zk is irreducible is a much more difficult question.

This result is classic in the context of representations of the Yangian Y (glN ), where N is aninteger, see [13] for a detailed proof.

47

5.2.2 Second conjectureThe second conjecture is the fact that Proposition 4.3.2 stays true in the case of the Yangian Y (g).

Conjecture 5.2.2. Every finite-dimensional irreducible representation of Y (g) is isomorphic to asubquotient of a tensor product of evaluation representations, of the form

m⊗i=1

ΛiV (z(1)i )⊗ · · · ⊗ ΛiV (z

(ri)i )⊗

n⊗j=1

ΛjV ∗(w(1)j )⊗ · · · ⊗ ΛjV ∗(w

(sj)j ) (5.5)

where m,n ≥ 0 and r1, . . . , rm, s1, . . . , sn ≥ 0 are integers, and z(k)i , w

(l)j ∈ C.

5.3 The invariant algebra Y (g)g

Another idea to study Y (g) and its representations is to look at the invariant algebra Y (g)g.

Definition 5.3.1. Let Y (g)g, or simply Y inv, denote the invariant algebra HomRep(Glt)(1, Y (g)).

Y inv is a simpler object to study than Y (g) because it is an ordinary algebra, thanks to thedefinition of the hom-spaces of Rep(Glt).

5.3.1 Representations of the invariant algebra

The following Proposition explains how to link representations of Y and representations of Y inv.

Proposition 5.3.2. If Y acts on E then Y inv acts on Hom(X,E), for any another object X inRep(Glt).

Proof. Let E be a representation of Y , then there is a morphism Y ⊗ E → E. Rep(Glt) is asemisimple category, as seen in 2.5 (we consider the case where t /∈ Z here), so E, as an object ofRep(Glt), can be uniquely written as the sum E =

⊕λ bipartition

L(λ) ⊗ Eλ, where Eλ are vectors

spaces, all but finitely many of them are zero (actually, as in a more usual framework, Eλ =Hom(L(λ), E)). We have also Y =

⊕L(λ′)⊗ Yλ′ .

Then there is a morphism⊕L(λ)⊗ L(λ′)⊗ Yλ ⊗ Eλ′ →

⊕L(λ′′)⊗ Eλ′′ . (5.6)

If we restrict at the source to L(λ) = 1, then Yλ = Hom(1, Y ) = Y inv, and we get a morphism⊕L(λ)⊗ Y inv ⊗ Eλ →

⊕L(λ′)⊗ Eλ′ . (5.7)

L(λ) and L(λ′) being irreducible objects, according to Schur’s Lemma, there are morphisms L(λ)→L(λ′) if and only if λ = λ′, and if so the morhism is a homothety. Then for every bipartition λ,we have a morphism

Y inv ⊗ Eλ → Eλ (5.8)

Hence we have shown that Y inv acts on Hom(L(λ), E), for every bipartition λ. As every object Xcan be written X =

⊕L(λ)⊗Xλ, we can easily build an action

Y inv ⊗Hom(X,E)→ Hom(X,E). (5.9)

Remark 5.3.3. In particular, Y inv acts on Hom(1, E).

48

5.3.2 Generators of the invariant algebraLet us recall that in the Yangian Y (g), we have the generating function

T (u) = 1 + T0u−1 + T1u

−2 + · · · , (5.10)

with Ti ∈ Hom(1, (V ∗ ⊗ V ) ⊗ Y ) = Hom(1, g∗ ⊗ Y ) ∼= (g⊗ Y )inv. Indeed, g∗ ∼= g are clearly

isomorphic in the category.Let r ∈ N and i1, . . . , ir ≥ 0 be a r-tuple of integers. We will use the following notation.

Ci1···ir := TrV (Ti1 · · ·Tir ) (5.11)

TrV : g→ C is a morphism of the category Rep(Glt), represented by • ◦ . Here we only applyTrV to the first component, in order to get an element of Y inv.Remark 5.3.4. It is well known that in the case of regular vector spaces, using the definitiong = V ⊗ V ∗, the trace map can be simply written

Tr: V ⊗ V ∗ → Cv ⊗ φ 7→ φ(v).

(5.12)

Ci1···ir is called a cyclic word. It can also be written using the generating functions

Ci1···ir = TrV (coeffu−i1−11 ···u−ir−1

rT (u1) · · ·T (ur)). (5.13)

Theorem 5.3.5. The cyclic words Ci1···ir , for i1, . . . , ir ≥ 0 generate Y inv.

It works exactly the same way as in a more classical setting of polynomial invariants of matrices,more precisely

Lemma 5.3.6. (Procesi’s Theorem) The ring of invariants of k n × n matrices, A1, . . . , Ak isgenerated by tr(Ai1 · · ·Air ), Ai1 · · ·Air running over all possible (noncommutative) monomials.

Proof. See [14].

With the same reasoning as before, the space of Gl(V )-invariant vectors of V ⊗r ⊗ V ∗⊗r isthe space of linear maps V ∗⊗r ⊗ V ⊗r → C invariant under Gl(V ), which identifies naturally tothe space of Gl(V )-invariants of End(V )⊗r. From Schur-Weyl duality, we know that this space isgenerated by elements indexed by permutations of Sr. The isomorphism can be easily written, asfollows.

Φ : Sk → {V ∗⊗k ⊗ V ⊗k → C}Gl(V ),

σ 7→(φ1 ⊗ . . .⊗ φk ⊗ v1 ⊗ . . .⊗ vk 7→

k∏i=1

φσ(i)(vi)

).

We are going to see how the images of this morphism can be obtain by products of traces, as in5.3.6. First of all, if we work in g∗ = V ∗ ⊗ V , the product is

(φ⊗ v) · (ψ ⊗ w) = φ⊗ ψ(v)w. (5.14)

Hence,

(φ1 ⊗ v1) · (φ2 ⊗ v2) · · · · · (φk ⊗ vk) = (φ1 ⊗ φ2(v1)v2) · (φ3 ⊗ v3) · · · · · (φk ⊗ vk)

= (φ1 ⊗ φ2(v1)φ3(v2)v3) · · · · · (φk ⊗ vk)

= φ1 ⊗ φ2(v1)φ3(v2) · · ·φk(vk−1)vk

49

As tr(φ⊗ v) = φ(v), tr((φi1 ⊗ vi1) · · · (φir ⊗ vir )) = φi1(vir )φi2(vi2) · · ·φir (vir−1).Thus, let σ be a permutation of Sk, if we decompose σ in disjoint cycles

σ = (i11i12 · · · i1r1)(i21i

22 · · · i2r2) · · · (im1 im2 · · · imrm),

Then,

Φ(σ)(φ1⊗ . . .⊗φk⊗v1⊗ . . . vk) = tr((φi11 ⊗vi11) · · · (φi1r1 ⊗vi1r1 )) tr((φi21 ⊗vi21) · · · (φi2r2 ⊗vi2r2 )) · · ·

tr((φim1 ⊗ vim1 ) · · · (φimrm ⊗ vimrm )) (5.15)

We have shown that with products of traces of products, we can obtain all the invariants. TheTheorem 5.3.5 follows from the same reasoning.

5.4 Action of the invariant algebra

What we want to do now is to compute values of the action of the generators Ci1···ir on certainvector spaces.

5.4.1 Action on an evaluation representationBy the evaluation representations, T (u) acts on V (z) as

T (u)|V (z) = R(u− z) = 1 +σ

u− z. (5.16)

If we look at the classical case, σ =∑eij ⊗ eji, as an element of U(glN ), acts as the identity

on V (z), the same goes in the complex case.If we generalize this result we get, for n ≥ 1,

Tr(T (u1)T (u2) · · ·T (un))|V (z) = tr

((1 +

σ

u1

)(1 +

σ

u2

)· · ·(

1 +σ

un

)), (5.17)

with σ2 = 1, tr(1) = t and tr(σ) = 1.

Remark 5.4.1. Recall that we are only taking the trace on the first component, hence if we computetr(σ) in the classical case, we get

tr(σ) = tr(∑

eij ⊗ eji)

=∑

eii = 1. (5.18)

We will admit that the result in the same in our situation (see below for details on how to computetraces of permutations).

Proposition 5.4.2. We have the following result, for V the tautological object of Rep(Glt) andz ∈ C.

Tr(T (u1)T (u2) · · ·T (un))|V (z) =1

2

((t+ 1)

n∏i=1

(1 +

1

ui

)+ (t− 1)

n∏i=1

(1− 1

ui

))(5.19)

50

Proof.

Tr(T (u1)T (u2) · · ·T (un))|V (z) = tr

((1 +

σ

u1

)(1 +

σ

u2

)· · ·(

1 +σ

un

))= tr

(1 + σ

(1

u1+ · · ·+ 1

un

)+

1

u1u2+

1

u1u3+ · · · 1

un−1un+

σ

(1

u1u2u3+ · · ·+ 1

un−2un−1un

)+ · · ·+ σn

(1

u1u2 · · ·un

))

= t

[n/2]∑k=0

∑1≤i1<···<i2k

1

u1 · · ·u2k

+

[n+1/2]∑l=1

∑1≤i1<···<i2l−1

1

u1 · · ·u2l−1

Let us give name to these sums

Se =

[n/2]∑k=0

∑1≤i1<···<i2k

1

u1 · · ·u2kand So =

[n+1/2]∑l=1

∑1≤i1<···<i2l−1

1

u1 · · ·u2l−1.

We haven∏i=1

(1 +

1

ui

)= Se + So and

n∏i=1

(1− 1

ui

)= Se − So

Hence

Tr(T (u1)T (u2) · · ·T (un))|V (z) = tSe + So = t

(1

2

(n∏i=1

(1 +

1

ui

)+

n∏i=1

(1− 1

ui

)))+

1

2

(n∏i=1

(1 +

1

ui

)−

n∏i=1

(1− 1

ui

))

=1

2

((t+ 1)

n∏i=1

(1 +

1

ui

)+ (t− 1)

n∏i=1

(1− 1

ui

))

Remark 5.4.3. The result is a scalar, but it was expected. As we saw in 5.3.2, invariant functionsare generated by pairings of the V and V ∗ (and so by permutations). But there is only one pairingof V ⊗ V ∗ (or equivalently, one permutation of the set of one element).

Also, V (z) being an irreducible representation of Y (g) (as V in an irreducible representationof Rep(Glt)), by Schur’s Lemma, it was expected that the action would be scalar.

The action on one V (z), being scalar, is not the most interesting case. The situation becomesmore interesting when we look at the action on a tensor product of evaluation representations,thanks to the coproduct ∆.

5.4.2 Action on a tensor product of evaluation representationsLet us see how the invariant algebra acts on a tensor product of evaluation representations, of theform V (z1)⊗ V (z2)⊗ · · · ⊗ V (zn).

T (u)|V (z1)⊗V (z2)⊗···⊗V (zn) = R12(u− z1)R13(u− z2) · · ·R1(n+1)(u− zn) (5.20)

51

Indeed, the coproduct on Y (g) is defined by ∆(T (u)) = T 12(u)T 13(u).Hence the action of the invariant algebra is of the form

tr (T (u1)T (u2) · · ·T (ur))|V (z1)⊗V (z2)⊗···⊗V (zn) =

tr

(n+1∏i=2

R1i(u1 − zi−1)

n+1∏i=2

R1i(u2 − zi−1) · · ·n+1∏i=2

R1i(ur − zi−1)

)= tr

r∏i=1

n∏j=1

(1 +

σ1(j+1)

ui − zj

)where σ1(j+1) denotes the permutation between the first and the (j + 1)th component.

Remark 5.4.4. The products written here are ordered. As we are computing products of permu-tations, the order matters.

5.4.3 Traces of permutationsAs we saw above, we needed to know how to compute the traces of different permutations, in orderto get a result for the action of Y inv on a tensor product of evaluation representations.

Example 5.4.5. First, let us look at a simple example. Let θ be the permutation ( 1 2 3 ),

θ(x⊗ y ⊗ z) = z ⊗ x⊗ y. (5.21)

If we look at the classical case, this permutation would be written θ =∑eij ⊗ ejk ⊗ eki.

Hence tr θ =∑eik ⊗ eki = σ.

Example 5.4.6. Now we are going to look at a more complex example, θ = ( 1 2 4 )( 3 5 ), whichsends x⊗ y ⊗ z ⊗ u⊗ v to u⊗ x⊗ v ⊗ y ⊗ z. It can be written

θ =∑

eij ⊗ ejm ⊗ ekn ⊗ emi ⊗ enk. (5.22)

Hence, tr θ =∑eim ⊗ ekn ⊗ emi ⊗ enk = ( 1 3 )( 2 4 ).

As we see that the first element plays a different part, we are going to denote it 0 instead of 1.Hence

tr( 0 1 2 ) = ( 1 2 ) and tr(( 0 1 3 )( 2 4 )) = ( 1 3 )( 2 4 ).

We begin to see the result appearing.

Proposition 5.4.7. Let θ be a permutation of {0, 1, . . . , n}. We write the decomposition of θ indisjoint cycles, beginning by the cycle containing 0.

θ = ( 0 i1 . . . ip )( j1 . . . jq ) · · · ( k1 . . . kr ).

Then, if pgeq1 the trace of θ is

tr θ = ( i1 . . . ip )( j1 . . . jq ) · · · ( k1 . . . kr ). (5.23)

And if p = 0,tr θ = t( j1 . . . jq ) · · · ( k1 . . . kr ). (5.24)

The best way to understand this result is to look at some figures. Let us represent the permu-tation ( 0 1 3 )( 2 4 ) with the same types of schemes as in 2.1.1

52

( 0 1 3 )( 2 4 ) =

• • • • •

• • • • •

This figure can be translated as a (1, • ◦ • ◦ • ◦ • ◦ •◦)-diagram, hence as an invariant on(V ⊗ V ∗)(0) ⊗ (V ⊗ V ∗)(1) ⊗ (V ⊗ V ∗)(2) ⊗ (V ⊗ V ∗)(3) ⊗ (V ⊗ V ∗)(4).

( 0 1 3 )( 2 4 ) =• ◦ • ◦ • ◦ • ◦ • ◦(0) (1) (2) (3) (4)

Then, as we said in 5.3.2, the trace on V ⊗ V ∗ is the morphism • ◦ . Hence, by taking thetrace on the first component, we are going to get a (1, • ◦ • ◦ • ◦ •◦)-diagram, or an invariant on(V ⊗ V ∗)(1) ⊗ (V ⊗ V ∗)(2) ⊗ (V ⊗ V ∗)(3) ⊗ (V ⊗ V ∗)(4). We obtain

tr( 0 1 3 )( 2 4 ) =• ◦ • ◦ • ◦ • ◦ • ◦(0)

(1) (2) (3) (4)

=• ◦ • ◦ • ◦ • ◦(1) (2) (3) (4)

We have the same result as in the classical case, that tr( 0 1 3 )( 2 4 ) = ( 1 3 )( 2 4 ).

Proof. From these figures, it is clear that the cycles where 0 does not appear do not change underthe trace, so we can suppose that θ is a (p+ 1)-cycle, θ = ( 0 i1 . . . ip ). First, let us study the casewhen p ≥ 1. We follow the same reasoning as before

• ◦ ··· • ◦ ··· • ◦ ···(0)

(i1) (ip) 7→ ··· • ◦ ··· • ◦ ···(i1) (ip)

Hence, ip is send to i1 and the rest is unchanged. Thus tr( 0 i1 . . . ip ) = ( i1 . . . ip ).Let us now study the case when p = 0, i.e. when 0 is unchanged by the permutation. As we

saw above, the other cycles are not disrupted by taking the trace on the first component, thus wecan suppose that θ is the identity.

θ =• ◦ • ◦ ··· • ◦(0) (1) (n)

tr θ = • ◦ • ◦ ··· • ◦(0) (1) (n)

= t • ◦ ··· • ◦(1) (n)

Thus tr(id{0,1,...,n}) = t id{1,...,n}.

5.4.4 Action on the evaluation of an exterior product representationTo study a simple case, we are going to compute the action of specific cyclic words Ci1i2 =trV (Ti1Ti2) on an evaluation representation ΛnV (z).

Moreover, according to the Second Conjecture 5.2.2, these are the types of representationswhich generates the finite-dimensional irreducible representations, together with ΛnV ∗(z), henceit is important to know how the invariant algebra Y inv acts on them.

53

Theorem 5.4.8.

tr(T (u)T (v))|ΛnV (z) = t+ 2−(

1− 1

u− z

)n−(

1− 1

v − z

)n+

n∑k,l=1

(−1)k+l

ukvl

((t+ 1)

(n

k + l − 1

)−(n

k

)(n

l

))Proof. We are going to use the same notations as in Section 5.4.3, by denoting by 0 the firstelements of the permuted set for example. From Section 5.4.2, we know that

tr (T (u)T (v))|V (z)⊗n = tr

n∏i=1

(1 +

( 0 i )

u− z

) n∏j=1

(1 +

( 0 j )

v − z

) . (5.25)

As we are looking at an action on an exterior product, only the signatures of the resulting permu-tations will matter. And, as we have product of transpositions, we only need to keep track of theparity of the number of transpositions we multiply.

As we saw in Section 5.4.3, if θ is permutation leaving 0 unchanged, the resulting permutationwill have the same signature, and a factor t will appear. If 0 is permuted though, the signaturewill change. Hence, the question is to know which permutation will stabilize 0, and which will not,and count them.

First of all,

tr (T (u)T (v))|V (z)⊗n =

tr

n∑k=0

1

(u− z)k∑

1≤i1<···<ik≤n

( 0 ik · · · i1 )

n∑l=0

1

(v − z)l∑

1≤j1<···<jl≤n

( 0 jl · · · j1 )

As, ( 0 i1 )( 0 i2 ) · · · ( 0 ik ) = ( 0 ik · · · i1 ), for 1 ≤ i1 < · · · < ik ≤ n, and the same goes for the(jm)1≤m≤l. Hence,

tr (T (u)T (v)) =

n∑k,l=0

1

(u− z)k(v − z)ltr

∑1≤i1<···<ik≤n

( 0 ik · · · i1 )∑

1≤j1<···<jl≤n

( 0 jl · · · j1 )

(5.26)

A product of two permutations of the form ( 0 ik · · · i1 )( 0 jl · · · j1 ), with 1 ≤ i1 < · · · < ik ≤ nand 1 ≤ j1 < · · · < jl ≤ n leaves 0 unchanged if and only if

• either k = l = 0, and the result is the identity,

• or k and l are positive, and jl = i1.

Hence, if k, l ≥ 1, the products of permutations ( 0 ik · · · i1 )( 0 jl · · · j1 ) leaving 0 unchanged arein bijection with the sequences 1 ≤ j1 < · · · < jl = i1 < · · · ik ≤ n, of length k+ l− 1 in {1, . . . , n}.Thus their number is

(n

k+l−1

).

The total number of products of permutations ( 0 ik · · · i1 )( 0 jl · · · j1 ) is(nk

)(nl

). Recall that

if ( 0 ik · · · i1 )( 0 jl · · · j1 ) leaves 0 unchanged, then

tr(( 0 ik · · · i1 )( 0 jl · · · j1 )) = ε(( 0 ik · · · i1 )( 0 jl · · · j1 )) = t(−1)k+l. (5.27)

And if ( 0 ik · · · i1 )( 0 jl · · · j1 ) disrupt 0, then

tr(( 0 ik · · · i1 )( 0 jl · · · j1 )) = −ε(( 0 ik · · · i1 )( 0 jl · · · j1 )) = (−1)k+l−1. (5.28)

54

Thus,

tr (T (u)T (v)) =

n∑k,l=1

1

(u− z)k(v − z)l

(t(−1)k+l

(n

k + l − 1

)

+(−1)k+l−1

((n

k

)(n

l

)−(

n

k + l − 1

)))+

n∑k=1

1

(u− z)ktr

∑1≤i1<···<ik≤n

( 0 ik · · · i1 )

+

n∑l=1

1

(v − z)ltr

∑1≤j1<···<jl≤n

( 0 jl · · · j1 )

+ tr(id)

But

t(−1)k+l

(n

k + l − 1

)+ (−1)k+l−1

((n

k

)(n

l

)−(

n

k + l − 1

))=

(−1)k+l−1

((n

k + l − 1

)(t+ 1)−

(n

k

)(n

l

))And

n∑k=1

1

(u− z)ktr

∑1≤i1<···<ik≤n

( 0 ik · · · i1 )

=

n∑k=1

(−1)k−1

(u− z)k

(n

k

)= −

n∑k=1

(n

k

)(−1

u− z

)k= −

((1− 1

u− z

)n− 1

)= 1−

(1− 1

u− z

)n.

The same goes forn∑l=1

1(v−z)l tr

( ∑1≤j1<···<jl≤n

( 0 jl · · · j1 )

), and we have the result we predicted.

Remark 5.4.9. The exact same reasoning could be used to compute tr(T (u1)T (u2) · · ·T (ur))|ΛnV (z),but we would need the number of products of r permutations of the form ( 0 ik · · · i1 )( 0 jl · · · j1 ),with 1 ≤ i1 < · · · < ik ≤ n leaving 0 unchanged, the rest of the arguments are the same.

55

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