GoBack - Temple Universitylorenz/talks/PrimeIdeals.pdf · enveloppante, Bull. Soc. Math. France 109...

GoBack

Prime ideals and group actions“Hopf Algebras and Related Topics” – USC 02/15/2009

Martin LorenzTemple University, Philadelphia

Thank you

Prime ideals and group actions USC 02/15/2009

Downloaded By: [Temple University] At: 15:03 28 January 2009

Thank you

Overview

• Background: enveloping algebras and quantized coordinatealgebras

Overview

• Tool: actions of algebraic groups, a brief reminder

Overview

• Tool: the Amitsur-Martindale ring of quotients

Overview

• Rational and primitive ideals

Overview

• Rational and primitive ideals

• Stratification of the prime spectrum

References

• “Group actions and rational ideals”,Algebra and Number Theory 2 (2008), 467-499

• “Algebraic group actions on noncommutative spectra”,posted at http://arXiv.org/abs/0809.5205

Both articles & the pdf file of this talk available on my web page:

http://math.temple.edu/˜lorenz/

I will work / base field k = k

Background

Enveloping algebras

Goal : For R = U(g), the enveloping algebra of a finite-dim’l Liealgebra g, describe

Prim R = {primitive ideals of R}

kernels of irreducible (generally infinite-dimensional)

representations R → Endk(V )

Enveloping algebras

Jacques Dixmier (* 1924)

in Reims, Dec. 2008

• former secretary of Bourbaki

• Ph.D. advisor of A. Connes, M. Duflo, . . .

• author of several influential monographs:

Les algebres d’operateurs dans l’espace hilbertien:algebres de von Neumann, Gauthier-Villars, 1957

Les C∗-algebres et leurs representations,Gauthier-Villars, 1969

Algebres enveloppantes, Gauthier-Villars, 1974

Dixmier’s Problem 11

from Algebres enveloppantes, 1974 :

• Problem 11 for k solvable, char k = 0Dixmier , Sur les ideaux generiques dans les algebres enveloppantes,Bull. Sci. Math. (2) 96 (1972), 17–26. existence: (a)

Borho, Gabriel, Rentschler , Primideale in Einhullenden auflosbarer Lie-Algebren,Springer Lect. Notes in Math. 357 (1973). uniqueness: (b)

• for noetherian or Goldie rings R / char k = 0:Mœglin & Rentschler

Orbites d’un groupe algebrique dans l’espace des ideaux rationnels d’une algebreenveloppante, Bull. Soc. Math. France 109 (1981), 403–426.

Sur la classification des ideaux primitifs des algebres enveloppantes, Bull. Soc. Math.France 112 (1984), 3–40.

Sous-corps commutatifs ad-stables des anneaux de fractions des quotients desalgebres enveloppantes; espaces homogenes et induction de Mackey, J. Funct. Anal.69 (1986), 307–396.

Ideaux G-rationnels, rang de Goldie, preprint, 1986.

• Problem 11 for k solvable, char k = 0Dixmier , Sur les ideaux generiques dans les algebres enveloppantes,Bull. Sci. Math. (2) 96 (1972), 17–26. existence: (a)

Borho, Gabriel, Rentschler , Primideale in Einhullenden auflosbarer Lie-Algebren,Springer Lect. Notes in Math. 357 (1973). uniqueness: (b)

• for noetherian or Goldie rings R / char k = 0:Mœglin & Rentschler

Orbites d’un groupe algebrique dans l’espace des ideaux rationnels d’une algebreenveloppante, Bull. Soc. Math. France 109 (1981), 403–426.

Sur la classification des ideaux primitifs des algebres enveloppantes, Bull. Soc. Math.France 112 (1984), 3–40.

Sous-corps commutatifs ad-stables des anneaux de fractions des quotients desalgebres enveloppantes; espaces homogenes et induction de Mackey, J. Funct. Anal.69 (1986), 307–396.

Ideaux G-rationnels, rang de Goldie, preprint, 1986.

• under weaker Goldie hypotheses / char k arbitrary:N. Vonessen

Actions of algebraic groups on the spectrum of rational ideals,J. Algebra 182 (1996), 383–400.

Actions of algebraic groups on the spectrum of rational ideals. II,J. Algebra 208 (1998), 216–261.

Quantum groups

Goal : For R = Oq(kn),Oq(Mn),Oq(G) . . . a quantized coordinatering, describe

Spec R = {prime ideals of R} ⊇ Prim R

Quantum groups

Goal : For R = Oq(kn),Oq(Mn),Oq(G) . . . a quantized coordinatering, describe

Spec R = {prime ideals of R} ⊇ Prim R

Typically, some algebraic torus T acts rationally by k-algebraautomorphisms on R; so have

Spec R −→ SpecT R = {T -stable primes of R}

P 7→ P : T =⋂

Quantum groups

T -stratification of Spec R( Goodearl & Letzter, . . . ; see the monograph by Brown & Goodearl )

Spec R =⊔

I∈SpecT R

SpecI R

{P ∈ Spec R | P : T = I}

Quantum groups

T -stratification of Spec R( Goodearl & Letzter, . . . ; see the monograph by Brown & Goodearl )

Spec R =⊔

I∈SpecT R

SpecI R

Tool: Algebraic Groups

Definition

There an anti-equivalence of categories

{affine algebraicgroups /k }

comm. affine reducedHopf k-algebras

G = Homk-alg(k[G], k) ↔ k[G]

Definition

There an anti-equivalence of categories

{affine algebraicgroups /k }

comm. affine reducedHopf k-algebras

G = Homk-alg(k[G], k) ↔ k[G]

Equivalently, affine algebraic groups are precisely the closedsubgroups of GLn for some n:

GLn, SLn, Tn, Un, Dn, On, . . .all finite groups

Rational representations

For any affine algebraic k-group G,

G-modules ≡ k[G]-comodules

Comodule structure map ∆M : M → M ⊗ k[G]

m 7→∑

m0 ⊗m1

Rational representations

For any affine algebraic k-group G,

G-modules ≡ k[G]-comodules

Comodule structure map ∆M : M → M ⊗ k[G]

m 7→∑

m0 ⊗m1

We obtain a linear representation ρM : G→ GL(M) by

g.m =∑

m0〈g,m1〉

Representations arising in the way are called rational.

Notation

For the remainder of this talk,

R denotes an associative k-algebra (with 1)

G is an affine algebraic k-group acting rationally on R;so R is a k[G]-comodule algebra.

Equivalently, we have a rational representation

ρ = ρR : G→ Autk-alg(R) ⊆ GL(R)

Tool: The Amitsur-Martindale ringof quotients

The definition

In brief,

Qr(R) = lim−→I∈E

Hom(IR, RR)

where E = E (R) is the filter of all I E R such that l. annR I = 0.

The definition

In brief,

Hom(IR, RR)

Explicitly, elements of Qr(R) are equivalence classes of rightR-module maps

f : IR → RR (I ∈ E ) ,

the map f being equivalent to f ′ : I ′

R → RR (I ′ ∈ E ) if f = f ′ onsome J ⊆ I ∩ I ′, J ∈ E .

The definition

In brief,

Hom(IR, RR)

Addition and multiplication of Qr(R) come from addition andcomposition of R-module maps.

The definition

In brief,

Hom(IR, RR)

Sending r ∈ R to the equivalence class of λr : R→ R, x 7→ rx,yields an embedding of R as a subring of Qr(R).

Extended centroid and central closure

Defs & Facts : • The extended centroid of R is defined by

C(R) = Z Qr(R)

Here Z denotes the center. If R is prime thenC(R) is a k-field.

Extended centroid and central closure

Defs & Facts : • The extended centroid of R is defined by

C(R) = Z Qr(R)

Here Z denotes the center. If R is prime thenC(R) is a k-field.

• Put R = RC(R) ⊆ Qr(R). The algebra R issaid to be centrally closed if R = R. If R issemiprime then R is centrally closed.

Original references

for prime rings R:

W. S. Martindale, III , Prime rings satisfying a generalizedpolynomial identity, J. Algebra 12 (1969), 576–584.

for general R:

S. A. Amitsur , On rings of quotients, Symposia Math., Vol. VIII,Academic Press, London, 1972, pp. 149–164.

Examples

• If R is simple , or a finite product of simple rings, then

Qr(R) = R

Examples

• If R is simple , or a finite product of simple rings, then

Qr(R) = R

• For R semiprime right Goldie ,

Qr(R) = {q ∈ Qcl(R) | qI ⊆ R for some I ⊳ R with annR I = 0}

classical quotient ring of R

In particular,

C(R) = ZQcl(R)

Examples

• R = U(g)/I a semiprime image of the enveloping algebra of afinite-dimensional Lie algebra g. Then

Qr(R) = { ad g-finite elements of Qcl(R) }

Rational Ideals

Definition

Want : an intrinsic characterization of “primitivity”, ideally

in detail . . .

Definition

“coeur”

“Herz”

“heart”

“core”

Definition : • Recall that C(R/P ) is a k-field for any P ∈ Spec R.We call P rational if C(R/P ) = k.

Definition

“coeur”

“Herz”

“heart”

“core”

Definition : • Recall that C(R/P ) is a k-field for any P ∈ Spec R.We call P rational if C(R/P ) = k.

• Put Rat R = {P ∈ Spec R | P is rational }; so

Rat R ⊆ Spec R

Connection with irreducible representations

Given an irreducible representation f : R→ Endk(V ), let P = Ker fbe the corresponding primitive ideal of R.

• There always is an embedding of k-fields

C(R/P ) → Z (EndR(V ))

• There always is an embedding of k-fields

C(R/P ) → Z (EndR(V ))

• Typically , EndR(V ) = k (“weak Nullstellensatz”); in this case

PrimR ⊆ RatR

Examples

The weak Nullstellensatz holds for

• R any affine k-algebra, k uncountable Amitsur

• R an affine PI-algebra Kaplansky

• R = U(g) “Quillen’s Lemma”

• R = kΓ with Γ polycyclic-by-finite Hall, L.

• many quantum groups: Oq(kn), Oq(Mn(k)), Oq(G), . . .

Examples

The weak Nullstellensatz holds for

• R any affine k-algebra, k uncountable Amitsur

• R an affine PI-algebra Kaplansky

• R = U(g) “Quillen’s Lemma”

• R = kΓ with Γ polycyclic-by-finite Hall, L.

• many quantum groups: Oq(kn), Oq(Mn(k)), Oq(G), . . .

In fact, in all these examples except the first, it has been shownthat, under mild restrictions on k or q,

PrimR = RatR

Group action: G-primes

The G-action on R yields actions on { ideals of R }, Spec R, RatR,. . . . As usual, G\? denotes the orbit sets in question.

Definition : A proper G-stable ideal I ⊳ R is called G-prime ifA,B E

G-stabR , AB ⊆ I ⇒ A ⊆ I or B ⊆ I. Put

G-Spec R = {G-prime ideals of R}

Group action: G-primes

Prop n The assignment γ : P 7→ P :G =⋂

g.P yields

surjections

Spec R

can.��

γ // // G-Spec R

G\ Spec R

77 77ooooooooooo

Group action: G-rational ideals

Definition : Let I ∈ G-Spec R. The group G acts on C(R/I)and the invariants C(R/I)G are a k-field. We call IG-rational if C(R/I)G = k. Put

G-RatR = {G-rational ideals of R}

Group action: G-rational ideals

Definition : Let I ∈ G-Spec R. The group G acts on C(R/I)and the invariants C(R/I)G are a k-field. We call IG-rational if C(R/I)G = k. Put

G-RatR = {G-rational ideals of R}

The following result solves Dixmier’s Problem # 11 (a),(b) forarbitrary algebras.

Theorem 1 G\RatRbij.−→ G-RatR

∈ ∈

G.P 7→⋂

Noncommutative spectra

Spec R

{{{{xxxxxxxxxxxxxxxxxxxxx

γ : P 7→P :G=⋂

g∈G g.P

"" ""FFFFFFFFFFFFFFFFFFFFF

xxxxxxxxxx

||||xxxxxxxxxx

EEEEEEEE

"" ""EEEE

G\ Spec R // // G-Spec R

G\RatR∼=

//� ?

G-RatR� ?

Spec R

γ : P 7→P :G=⋂

g∈G g.P

xxxxxxxxxx

||||xxxxxxxxxx

EEEEEEEE

"" ""EEEE

G\RatR∼=

//� ?

G-RatR� ?

Spec R carries the Jacobson-Zariski topology: closed subsets arethose of the form V(I) = {P ∈ Spec R | P ⊇ I} where I ⊆ R.

Spec R

γ : P 7→P :G=⋂

g∈G g.P

xxxxxxxxxx

||||xxxxxxxxxx

EEEEEEEE

"" ""EEEE

G\RatR∼=

//� ?

G-RatR� ?

։ is a surjection whose target has the final topology,

→ is an inclusion whose source has the induced topology, and∼= is a homeomorphism, from Thm 1

Spec R

γ : P 7→P :G=⋂

g∈G g.P

xxxxxxxxxx

||||xxxxxxxxxx

EEEEEEEE

"" ""EEEE

G\RatR∼=

//� ?

G-RatR� ?

Next, we turn to Spec R and the map γ . . .

Stratification of the primespectrum

Goal #1

Recall : γ : Spec R։ G-Spec R, P 7→ P : G =⋂

g∈G g.P , yields theG-stratification of Spec R

Spec R =⊔

I∈G-Spec R

SpecI R

Goal #1 : describe the G-strata

SpecI R = γ−1(I) = {P ∈ Spec R | P :G = I}

Goal #1

For simplicity, I assume G to be connected ; so k[G] is a domain.In particular,

G-Spec R = SpecG R = {G-stable primes of R}

The rings TI

For a given I ∈ G-Spec R, put Fract k[G]

TI = C(R/I)⊗ k(G)

This is a commutative domain, a tensor product of two fields.

The rings TI

TI = C(R/I)⊗ k(G)

G-actions: • on C(R/I) via the given action ρ : G→ Autk-alg(R)

• on k(G) by the right and left regular actionsρr : (x.f)(y) = f(yx) and ρℓ : (x.f)(y) = f(x−1y)

• on TI by ρ⊗ ρr and Id⊗ρℓ ←− commute

The rings TI

TI = C(R/I)⊗ k(G)

PutSpecG TI = {(ρ⊗ ρr)(G)-stable primes of TI}

Stratification Theorem

Theorem 2 Given I ∈ G-Spec R, there is a bijection

c : SpecI R −→ SpecG TI

having the following properties:

(a) G-equivariance: c(g.P ) = (Id⊗ρℓ)(g)(c(P ));

(b) inclusions: P ⊆ P ′ ⇐⇒ c(P ) ⊆ c(P ′);

(c) hearts: C (TI/c(P )) ∼= C (R/P ⊗ k(G)) as k(G)-fields;

(d) rationality: P is rational ⇐⇒ TI/c(P ) = k(G).

Stratification Theorem

Theorem 2 There is a bijection

c : SpecI R −→ SpecG TI

having the following properties:

(a) G-equivariance: c(g.P ) = (Id⊗ρℓ)(g)(c(P ));

(b) inclusions: P ⊆ P ′ ⇐⇒ c(P ) ⊆ c(P ′);

(c) hearts: C (TI/c(P )) ∼= C (R/P ⊗ k(G)) as k(G)-fields;

(d) rationality: P is rational ⇐⇒ TI/c(P ) = k(G).

Cor : Rational ideals are maximal in their strata

Goal #2: Finiteness of G-Spec R

Assume that R sats the Nullstellensatz : weak Nullstellensatz &Jacobson property

e.g., R affine noetherian / uncountable k, affine PI, . . .semiprime =

⋂primitives

Assume that R sats the Nullstellensatz : weak Nullstellensatz &Jacobson property

Prop n The following are equivalent:

(a) G-Spec R is finite;

(b) G has finitely many orbits in RatR;

(c) R sats (1) ACC for G-stable semiprime ideals,(2) the Dixmier-Mœglin equivalence, and(3) G-Rat R = G-Spec R.

locally closed = primitive = rational

Example : If G is an algebraic torus then a sufficient condition forthe equality G-Spec R = G-Rat R is

dimk Rλ ≤ 1 for all λ ∈ X(G)

For a commutative domain R, this is also necessary.

Cor (classical) Let R be an affine commutative domain / k

and let G be an algebraic k-torus acting rationally on R. Then:

G-Spec R is finite ⇐⇒ dimk Rλ ≤ 1 for all λ ∈ X(G).

Local closedness

Recall: locally closed = open ∩ closed

Theorem 3 If P ∈ RatR then:

{P} loc. cl. in Spec R ⇐⇒ {P :G} loc. cl. in G-Spec R

Local closedness

Recall: locally closed = open ∩ closed

Theorem 3 If P ∈ RatR then:

{P} loc. cl. in Spec R ⇐⇒ {P :G} loc. cl. in G-Spec R

Cor If P ∈ Rat R is loc. closed in Spec R then the orbit G.Pis open in its closure in RatR.

Rudolf Rentschler (PhD 1967 Munich)

GoBack - Temple Universitylorenz/talks/PrimeIdeals.pdf · enveloppante, Bull. Soc. Math. France 109...

Documents

Transcript of GoBack - Temple Universitylorenz/talks/PrimeIdeals.pdf · enveloppante, Bull. Soc. Math. France 109...

La radiothérapie moderne des cancers des voies aéro ...(60 %), devant l’évolution métastatique (30 %) et la survenue de nouveaux cancers primitifs (10 %) [1, 2]. Afin d’améliorer

LYMPHOMES MALINS PRIMITIFS DU TUBE DIGESTIF · •Les lymphomes primitifs du tube digestif sont des lymphomes non-hodgkiniens (LNH) •Prolifération clonale de lymphocytes (phénotype

Fotokatalog Photographic Catalogue Catalogo fotograficofotokat/Fotokataloge/Bulloz_o_J_1.pdf · EN VENTE LES MUSBES DE AVIGNON (Primitifs fran~ais-XVIIIe siede). BESANC;ON (Dessins

perso.univ-rennes1.fr€¦ · Alg ebres de distributions et D-modules arithm etiques Christine Huyghe et Tobias Schmidt 9 octobre 2015 Abstract Let p be a prime number, V a complete

Scan - Temple Universitylorenz/papers/36.pdfpolycyclic-by-finite groups over right Noetherian rings S to determine the Goldie rank of S* F in certain cases. Introduction. In [6], S.

Un imitateur moderne des Primitifs flamands établi …Lázaro Galdiano (1862-1947).3 Catalogué par Emilio Camps Cazorla vers 1949, peu après la mort du collectionneur, dans un inventaire

Analyse morphologique, classification et protection des paysages : le cas de … · 2017. 2. 10. · primitifs. On les retrouve dans le cadastre, les défrichements agricoles, l'habitat,

GoBack - Temple Universitylorenz/talks/Raleigh.pdf · enveloppante, Bull. Soc. Math. France 109 (1981), 403–426. Sur la classiﬁcation des ideaux primitifs des alg´ ebres enveloppantes`,

Le Devin historien en Msopotamie Devin...les plus primitifs, se situaient sur une chaîne ininterrompue, qui traversait le cosmos de part en part, et à laquelle seule la science galiléenne

Domloup (35)€¦ · Web view2017/05/09 · Après s’être fait présenter les budgets primitifs de l’exercice 2016 et les décisions modificatives qui s’y rattachent, les

evolution - Project Overview | Project IRENE - Spotlight …digitalassets.lib.berkeley.edu/anthpubs/ucb/text/kas047... · · 2006-11-09indo-europ6enes et les Aryas primitifs, essai

Déficits immunitaires primitifs (DIP) : modèle pour l'étude de la susceptibilité génétique aux infections

tdr63b ||||| Analyse discriminante lin · A.B. Dufour, D. Chessel & J.R. Lobry 1 Les Iris de Fisher C’est un des jeux de donn ees les plus c el ebres de la statistique (?, ?).?

The 1902 exhibition, Les Primitifs flamands: scholarly ... · Andrée Hayum The 1902 exhibition, Les Primitifs flamands: scholarly fallout and art historical reflections 2 time, altogether,

GoBack - Temple Universitylorenz/talks/RingTheoreticMethods.pdf · Ring theoretic methods Ben-Gurion University 05/26/2010 For a ﬁxed nonsingular associative R-bilinear formβ:

arXiv:1008.1971v1 [math.RT] 11 Aug 2010 · 6.1. Invariances 24 6.2. Algèbres de groupes et graduations 26 6.3. Extensions triviales d’algèbres 28 6.4. Algèbres extérieures

De Baldus à Le Gray Primitifs de la photographie du XIXe siècle à … › wp-content › uploads › ... · 2020-04-08 · de Photographie en 1857, Baldus réalise en 1859 un nouvel

Algèbres de Lie en Physique D efinitions et propri et es Applications des alg ebres de Lie de dimension finie Applications des alg ebres de Lie de dimension infinie Alg ebres de Lie

L’Atelier du Regard - data.over-blog-kiwi.comdata.over-blog-kiwi.com/0/78/00/89/201312/ob_606e25_04-fiche... · L’Atelier du Regard : Giotto et les primitifs italiens Fresque

Université de Montréalbroera/Lie13a.pdf · LIE ALGEBRAS AND LIE GROUPS MAT6633 ABRAHAM BROER References [1] Bourbaki, Nicolas, El ements de math ematique. XXVI. Groupes et alg ebres