Higher Structures on Modules over Operads · Copenhagen, 12-10-2015 Niels Kowalzig (U Roma La...

Higher Structures on Modules over Operads

Niels Kowalzig

U Roma La Sapienza

Copenhagen, 12-10-2015

Niels Kowalzig (U Roma La Sapienza) Higher Structures on Modules over Operads Copenhagen, 12-10-2015 1 / 44

Motivation: Hochschild cohomology and homology

Let A be an associative algebra over a commutative ring k , setAe := A⊗k A

op, and let M be an Ae-module.

Define the Hochschild chain and cochain complexes (Hochschild,1945) with coefficients in M as

C •(A,M) = Homk(A⊗k•,M), C•(A,M) := M ⊗k A⊗k•,

with respective differentials β and b that are defined using themultiplication in A and the Ae-module structure on M , giving riseto Hochschild cohomology H•(A,M) and homology H•(A,M).H•(A,M) is an interesting object (depending on your interests)since it classifies infinitesimal deformations of the multiplicativestructure of A.


Higher structures on Hochschild (co)homology

Gerstenhaber (1963) discovered that H•(A,M) carries more structure:

that of a graded commutative multiplication `;that of a graded Lie bracket [., .] of degree −1;such that [., .] is a graded derivation of `.

For this to hold, M needs to satisfy certain conditions.Rinehart (1963), Daletski-Gel’fand-Tsygan (1989), Nest-Tsygan(1998), and partially others (Connes, Getzler, Goodwillie) noticed (forM := A) that there is even more structure to be unveiled on the pair(H•(A,A),H•(A,A)):

a graded H•(A,A)-module structure ι on H•(A,A);a graded H•(A,A)-Lie algebra module structure L on H•(A,A);such that with the cyclic differential B one has the homotopy formula

L = [B, ι].

Some of the above structure can be already seen on the (co)chain level

(C •(A,A),C•(A,A)), others need a “homotopic correction”.Niels Kowalzig (U Roma La Sapienza) Higher Structures on Modules over Operads Copenhagen, 12-10-2015 3 / 44

Digression: cyclic objects

We briefly recall the concept of cyclic objects and cyclic homology:

Let C be a category and Λ Connes’ cyclic category (a combinationof the simplicial category and the cyclic groups). A para-cyclicobject in M is a functor X : Λop → C, that is, a simplicial object(X•, d•, s•) together with morphisms tn : Xn → Xn subject to

di tn =

tn−1di−1 if 1 ≤ i ≤ ndn if i = 0,

si tn =

tn+1si−1 if 1 ≤ i ≤ nt2n+1sn if i = 0.

A para-cyclic object is called cyclic if

tn+1n = id.

Define Hochschild operator, norm operator, extra degeneracy:

b :=∑n

j=0(−1)jdj , N :=∑n

j=0(−1)nt jn+1, s−1 := tn+1sn,,

and define Connes’ cyclic operator or Connes’ coboundary:B := (1− (−1)ntn+1)s−1N .


Digression: mixed complexes and cyclic homology

These operators fulfill B2 = 0,Bb + bB = 0, and b2 = 0, hence eachcyclic object gives rise to a mixedcomplex: define BC•,•X byBCp,q(X ) := Xq−p for 0 ≤ p ≤ q andzero otherwise:

The homology of the zeroth column isthe Hochschild homology

HH•(X ) := H•(C•X , b),

whereas cyclic homology is defined tobe the homology of the total complex:

HC•(X ) := H•(TotBC•,•X , b + B).

X2 X1 X0

X1 X0

X0.

?

b

?

b

?

b

?

b

?

b

B B

?

b

B


Gerstenhaber algebras, more precisely

Definition

Let k be a commutative ring. A Gerstenhaber algebra over k is agraded commutative k-algebra (V ,`)

V =⊕p∈N

V p, α ` β = (−1)pqβ ` α ∈ V p+q, α ∈ V p, β ∈ V q

with a graded Lie bracket ·, · : V p+1 ⊗k Vq+1 → V p+q+1 on the

desuspension

V [1] :=⊕p∈Z

V p+1

for which all operators γ, · satisfy the graded Leibniz rule

γ, α ` β = γ, α ` β + (−1)pqα ` γ, β, γ ∈ V p+1, α ∈ V q.


Gerstenhaber modules

DefinitionA Gerstenhaber module over V is a graded (V ,`)-module (Ω,a),

Ω =⊕n∈Z

Ωn, ιαx := α a x ∈ Ωn−p, α ∈ V p, x ∈ Ωn

with a representation of the graded Lie algebra (V [1], ·, ·)

L : V p+1 ⊗k Ωn → Ωn−p, α⊗k x 7→ Lα(x)

which satisfies for α ∈ V p+1, β ∈ V q, x ∈ Ω the mixed Leibniz rule

β a Lα(x) = β, α a x + (−1)pqLα(β a x)


Batalin-Vilkovisky modules = differential calculi

DefinitionA Gerstenhaber module is Batalin-Vilkovisky if there is a k-lineardifferential

B : Ωn → Ωn+1, BB = 0

such that Lα is for α ∈ V p given by the homotopy formula

Lα(x) = B(α a x)− (−1)pα a B(x).

A pair (V ,Ω) of a Gerstenhaber algebra and of a Batalin-Vilkoviskymodule over it is also called a (noncommutative) differentialcalculus.


The very classical example

X a compact smooth manifold, k := C. Then

V :=∧

A DerC(A), A := C∞(X ,C)

is a Gerstenhaber algebra w.r.t. the Schouten-Nijenhuis bracketwhich for vector fields is just the commutator.The differential forms Ω(X ) on X are a Gerstenhaber module overV . The module structure is given by the contraction ι of adifferential form by a multivector field and the action L is the Liederivative of a differential form along a multivector field.The Cartan homotopy or magic formula, that is,

L = [ι, d]

says that this Gerstenhaber module is BV with respect to theexterior derivative B := d.


The now classical example

Let A be an associative algebra over a field k (or at least let A beprojective over a commutative ring k). Then its Hochschildcohomology

V := H•(A,A) = Ext•Ae(A,A), Ae := A⊗k Aop

is canonically a Gerstenhaber algebra (Gerstenhaber’s originalexample 1963).The Hochschild homology

Ω := H•(A,A) = TorAe

• (A,A)

is canonically a BV module over V (Rinehart 1963, Connes,Getzler, Goodwillie, and ultimately Nest-Tsygan in the 80s and90s).


The (sort-of) universal example

Theorem (K.-Krahmer 2012)If U is a left Hopf algebroid which is right projective over its base algebraA and M is a stable anti Yetter-Drinfel’d (SaYD) module over U, then(

Ext•U(A,A),TorU• (M,A))

carries a canonical structure of a noncommutative differential calculus.

This contains the examples above, but also applies to groups, groupoids,Lie algebras, Lie algebroids, Poisson algebras, etc. An extension is

Theorem (K., 2013)

If (U,A) and M is as above, and N a braided commutativeYetter-Drinfel’d algebra such that M ⊗A N is SaYD, then(

Ext•U(A,N),TorU• (M,N))

carries a canonical structure of a noncommutative differential calculus.


QuestionHow to encode this algebraic structure as compact as possible,possibly even without the bialgebroid framework?

For example, one already knows from Gerstenhaber-Schack,McClure-Smith (and probably others) that if O is an operad withmultiplication, then it gives rise to a cosimplicial structure andthe respective cohomology H•(O) carries the structure of aGerstenhaber algebra.If one furthermore (in some sense) dualises the notion of a cyclicoperad (in a sense to be specified below), one obtains:

Theorem (K., 2013)

Let O be an operad (in k-Mod) with multiplication and M a cyclicunital (comp) O-module. Then M carries the structure of a cyclick-module and the pair

(H•(O),H•(M)

)defines a noncommutative

differential calculus.


Operads: the fundamental example

Consider the space O := Homk(A⊗k•,A) of Hochschild cochains withvalues in A.

For φ ∈ Op and ψ ∈ Oq, define the operation

φiψ(a1, . . . , ap+q−1) := φ(a1, . . . , ai−1, ψ(ai , . . . , ai+q−1), ai+q, . . . , ap+q−1),

that is, insertion of ψ into φ at the ith slot.This apparently works replacing A by any vector space V , but in theHochschild case there is a distinguished element µ ∈ O2 given by

µ(a, b) := ab, a, b ∈ A,

the multiplication in A, which will serve in a moment.The operation is not (strictly speaking) associative, but sort of in amore general sense:


Associativity behaviour

φ

χψ

=

φ

χψ

Figure: Parallel composition axiom


φ

ψ

χ

=

φ

ψ

χ

Figure: Sequential composition axiom


Operads: formal definition

Definition (“Partial” definition)

A (non-Σ, unital) operad (in k-Mod) is a sequence Onn≥0 ofk-modules with an identity element 1 ∈ O1 and k-bilinear operationsi : Op ⊗Oq → Op+q−1 such that for ϕ ∈ Op, ψ ∈ Oq, andχ ∈ Or : ϕ i ψ = 0 if p < i or p = 0,

(ϕ i ψ) j χ =

(ϕ j χ) i+r−1 ψ if j < i ,

ϕ i (ψ j−i+1 χ) if i ≤ j < q + i ,

(ϕ j−q+1 χ) i ψ if j ≥ q + i ,

ϕ i 1 = 1 i ϕ = ϕ for i ≤ p

The operad O is called operad with multiplication if there exists amultiplication µ ∈ O2 and a unit e ∈ O0 such that

µ 1 µ = µ 2 µ, and µ 1 e = µ 2 e = 1.


Gerstenh. algebras from operads w. multiplication

Theorem (Gerstenhaber-Schack/McClure-Smith)Each operad with multiplication defines a cosimplicial k-module. Itscohomology forms a Gerstenhaber algebra.

The explicit structure maps read as follows: for ϕ ∈ Op, ψ ∈ Oq, set

ϕψ :=∑p

i=1(−1)|q||i |ϕ i ψ ∈ Op+q−1, |n| := n − 1,

and define their Gerstenhaber bracket by

ϕ,ψ := ϕψ − (−1)|p||q|ψϕ.The graded commutative product is given by the cup product

ϕ ` ψ := (µ 2 ϕ) 1 ψ ∈ Op+q.

Finally, the coboundary of the cosimplicial k-module results as

δϕ = µ, ϕ,

and then the triple (O, δ,`) forms a DG algebra.Niels Kowalzig (U Roma La Sapienza) Higher Structures on Modules over Operads Copenhagen, 12-10-2015 17 / 44

A remark for algebraic topologists

As is well-known, the cup product is only commutative up tohomotopy. More precisely,

(−1)p−1ϕδψ − (−1)p−1δ(ϕψ) + δϕψ = ϕ ` ψ − (−1)pqψ ` ϕ.

Thinking of `=:`0 and =:`1, this is the Steenrod relation indegree 1.


(Comp) modules over operads: the fundamental

example

For an associative k-algebra A, define

M := C•(A) := A⊗k•+1 and O := C •(A) := Homk(A⊗k•,A).

There is an obvious collection of maps •i : O ⊗kM→M given by

ϕ •i (a0, . . . , an) :=(a0, . . . , ai−1, ϕ(ai , . . . , ai+p−1), ai+p, . . . , an

),

for i = 1, . . . , n − p + 1 and n ∈ N , which satisfies similar“associativity” relations as those for operads:


(Comp) modules over operads: formal definition

DefinitionA comp or opposite module M over an operad O (called O-module forshort) is a sequence Mnn≥0 ∈ k-Mod together with k-linear operations

•i : Op ⊗kMn →Mn−p+1, for i = 1, . . . , n − p + 1, 0 ≤ p ≤ n,

and zero if p > n, and satisfying for ϕ ∈ Op, ψ ∈ Oq, and x ∈Mn

ϕ •i(ψ •j x

)=

ψ •j

(ϕ •i+q−1 x

)if j < i ,

(ϕ j−i+1 ψ) •i x if j − p < i ≤ j ,

ψ •j−p+1

(ϕ •i x

)if 1 ≤ i ≤ j − p,

where p > 0, q ≥ 0, n ≥ 0. An O-module is called unital if

1 •i x = x , for i = 1, . . . , n, for all x ∈Mn,


Comp modules depicted

Keeping the fundamental example in mind, a comp module could bedepicted as:

0

x

1 · · · i · · ·i + p − 1 · · · n

ϕ

Figure: Comp modules


Cyclic comp modules: the fundamental example

For

M := C•(A) and O := C •(A)

as above, apparently even the map

ϕ •0 (a0, . . . , an) :=(ϕ(a0, . . . , ap−1), ap, . . . , an

)makes sense. With the usual cyclic operator

t(a0, . . . , an) := (an, a0, . . . , an−1)

one observes the formula

t(ϕ •i (a0, . . . , an)

)= ϕ •i+1 t(a0, . . . , an),


Cyclic O-modules

Crucial is then the following notion dual to that of a cyclic operad:

Definition (K., 2013)

A para-cyclic unital (comp) O-module is a unital (comp) O-module Mas before, equipped with two additional structures:

an extra (k-linear) comp module map

•0 : Op ⊗kMn →Mn−p+1, 0 ≤ p ≤ n + 1,

declared to be zero if p > n + 1, and such that the above associativityrelations extends to i = 0;a morphism t :Mn →Mn such that for ϕ ∈ Op and x ∈Mn

t(ϕ •i x) = ϕ •i+1 t(x), i = 0, . . . , n − p.

A para-cyclic O-module is called cyclic if tn+1 = id is true on Mn.


Cyclic comp modules depicted

0

x

· · · i · · ·i + p − 1· · ·n − 1

n

ϕ

0

x

· · · i + 1· · ·i + p · · ·n − 1

n

ϕ

=

Figure: The relation t(ϕ •i x) = ϕ •i+1 t(x) for cyclic comp modules


Homology of cyclic O-modules

Proposition (K., 2013)

Let (O, µ) be an operad with multiplication. A cyclic unital O-moduledefines a cyclic k-module (d•, s•, t•), where for x ∈Mn

di (x) = µ •i x , i = 0, . . . , n − 1,dn(x) = µ •0 t(x),sj(x) = e •j+1 x , j = 0, . . . , n.

As usual, one defines the Hochschild or simplicial boundary operatorb :=

∑ni=0(−1)idi , explicitly here

b :=∑n−1

i=0 (−1)iµ •i x + (−1)nµ •0 t(x),

and the norm operator, the extra degeneracy, and the cyclic differential

N :=∑n

i=0(−1)inti , s−1 := t sn, B = (id− t) s−1 N.


Homology of cyclic O-modules

DefinitionFor any cyclic unital O-module M over an operad (O, µ) withmultiplication, we call the homology of (M•, b), denoted by H•(M), its(Hochschild) homology, and the homology of the mixed complex(M•, b,B), denoted HC•(M), the cyclic homology of M.

We did not call in a purely random manner the action •0 an “extra”comp module map: one computes

e •0 x = s−1(x)

for x ∈Mn, hence the extra degeneracy. With this, one obtains forx ∈ Mn

B(x) =∑n

i=0(−1)ine •0 ti (x).


Cyclic O-modules as Gerstenhaber modules

Definition (K., 2013)

Let (O, µ) be an operad with multiplication and M a cyclic unitalO-module. Define

the cap product of ϕ ∈ Op with x ∈Mn by

ιϕ := ϕ a · :Mn →Mn−p, ϕ a x := (µ 2 ϕ) •0 x .

the Lie derivative Lϕ :Mn →Mn−p+1 of x ∈Mn along ϕ ∈ Op

with p < n + 1 by

Lϕx :=

n−p+1∑i=1

(−1)ζpi ϕ •i x +

p∑i=1

(−1)ξnp,iϕ •0 ti−1(x),

where ζpi and ξnp,i are sign functions.

Compare the formal similarity of the cap with the cup product.Niels Kowalzig (U Roma La Sapienza) Higher Structures on Modules over Operads Copenhagen, 12-10-2015 27 / 44

DG module and DG Lie algebra module structures

Dual to the fact that (O, δ,`) is a DG algebra, one has:

Theorem (K., 2013)

The triple (M−•, b,a) defines a left DG module over (O, δ,`),

ιϕ ιψ = ιϕ`ψ, [b, ιϕ] = ιδϕ,

where ϕ ∈ Op, ψ ∈ O. The Lie derivative defines a DG Lie algebrarepresentation of (O[1], ., .) on M−•,

[Lϕ,Lψ] = Lϕ,ψ.

Moreover, the simplicial differential on C•(M) is given by the Liederivative along the multiplication:

b = −Lµ and [b,Lϕ] + Lδϕ = 0.

All proven by tedious computations.Niels Kowalzig (U Roma La Sapienza) Higher Structures on Modules over Operads Copenhagen, 12-10-2015 28 / 44

The Gerstenhaber module

By the preceding formulae, both ιϕ and Lϕ descend to well definedoperators on the simplicial homology H•(M) as soon as ϕ is a cocycle. Inthis case, together with the preceding theorem, the operations ι and Lturn H•(M) into a module over the Gerstenhaber algebra H•(O):

Theorem (K., 2013)For any two cocycles ϕ ∈ Op, ψ ∈ Oq, the induced maps

Lϕ : H•(M)→ H•−p+1(M), ιψ : H•(M)→ H•−q(M)

satisfy

[ιψ,Lϕ] = ιψ,ϕ.

A bit trickier to prove involving a recursive argument plus still a bunch of

truly unpleasant computations.


The Batalin-Vilkovisky module structure

We now give a sort of homotopy formula relating the cyclicdifferential with the Lie derivative and the cap product on the chainlevel. Since here the full cyclic bicomplex is “seen”, ιϕ needs a“cyclic correction” Sϕ, and we call the sum Iϕ := ιϕ + Sϕ the cycliccap product. Define for 0 ≤ p ≤ n

Sϕ :Mn →Mn−p+2,

Sϕ :=

n−|p|∑j=1

n−|p|∑i=j

(−1)ϑn,pj,i e •0

(ϕ •i t j−1(x)

),

with a certain sign function ϑn,pj ,i . If p > n, put Sϕ := 0.


The Batalin-Vilkovisky module structure

Theorem (K., 2013)

For any cochain ϕ ∈ O• in the normalised cochain complex O•, theCartan-Rinehart homotopy formula

Lϕ = [B + b, ιϕ + Sϕ]− ιδϕ − Sδϕ

= [B + b, Iϕ]− Iδϕ

holds on the normalised chain complex M•. For cocycles, thissimplifies on homology to

Lϕ = [B, ιϕ].


Examples I: associative algebras

For an associative k-algebra A, define

M := C•(A) := A⊗k•+1 and O := C •(A) := Homk(A⊗k•,A).

(ϕ i ψ)(a1, . . . , ap+q−1) := ϕ(a1, . . . , ai−1, ψ(ai , . . . , ai+q−1), ai+q, . . . , ap+q−1

),

µ(a⊗k b) := ab, 1 := idA(·), e := 1A,

ϕ •i (a0, . . . , an) :=(a0, . . . , ai−1, ϕ(ai , . . . , ai+p−1), ai+p, . . . , an

),

for i = 1, . . . , n − p + 1. This gives a C •(A)-module structure on C•(A).Extending this formula in the obvious way to i = 0 and with the usualcyclic operator

t(a0, . . . , an) := (an, a0, . . . , an−1).

this gives a cyclic C •(A)-module. The operators `,a,B,Lϕ, Sϕ are those

found in the work of Daletski, Gel’fand, Nest, Tamarkin, Tsygan and

those on which the construction of Kontsevich-Soibelman is based.Niels Kowalzig (U Roma La Sapienza) Higher Structures on Modules over Operads Copenhagen, 12-10-2015 32 / 44

Examples I.b: twisted Calabi-Yau algebras

Extending the previous example, consider for a σ ∈ Aut(A) the algebra Aitself as an (A,A)-bimodule Aσ via a I x J b := axσ(b) for a, b, x ∈ A, andset

M := C•(A,Aσ) := Aσ ⊗k A⊗k• and

O := C •(A) := Homk(A⊗k•,A).

In case σ is semisimple (diagonalisable), that is, if there is a subsetΣ⊆ k \ 0 and a decomposition of k-vector spaces

A =⊕

λ∈Σ Aλ, Aλ = a ∈ A | σ(a) = λa,

then(H•(A),H•(A,Aσ)

)forms a nc calculus again.

As a side remark, Nest-Tysgan suggested to consider H•(A,Aσ) as a

quantum analogue to the Fukaya category, while H•(A,Aσ) was related by

Kustermans, Murphy and Tuset to Woronowicz’s concept of covariant

differential calculi over compact quantum groups.Niels Kowalzig (U Roma La Sapienza) Higher Structures on Modules over Operads Copenhagen, 12-10-2015 33 / 44

The following is a slight generalisation of Ginzburg’s notion of aCalabi-Yau algebra (where σ = id):

DefinitionAn algebra A is a twisted Calabi-Yau algebra with modularautomorphism σ ∈ Aut(A) if the Ae-module A has (as an Ae-module) afinitely generated projective resolution of finite length and there existsd ∈ N and isomorphisms of right Ae-modules

ExtiAe(A,Ae) '

0 i 6= d ,Aσ i = d .

The Ischebeck spectral sequence leads to a Poincare-type duality

H•(A,A) ' Hd−•(A,Aσ),

and by a result of Ginzburg/Lambre one shows that the BV modulestructure on the RHS yields a BV algebra structure (which we probablywon’t discuss) on the LHS. Examples come from quantum groups,homogeneous spaces, the Podles quantum sphere.


Example II: noncommutative Poisson structures

Define O,M, i , e, 1, •j , t as in the example before, but replace themultiplication µ by a more general π ∈ C 2(A) that fulfils

π 1 π = π 2 π.

In case π is a cocycle, such a structure is called a noncommutativePoisson structure by Xu/Ginzburg, and can be seen as a translation ofthe property [π, π]SN = 0 that defines a Poisson structure on a manifold.The resulting simplicial differential b = −Lπ is the algebraic Brylinskiboundary that defines Poisson homology. Its original definition as

bπ := [ιπ, ddeRham],

is nothing than the Cartan-Rinehart homotopy. The resulting cosimplicialdifferential β = π, · is the algebraic Koszul-Lichnerowicz coboundarydefining Poisson cohomology. Sϕ and Lϕ look as before, but for `,aone obtains the new operators of Poisson cup and Poisson cap product:

(ϕ ` ψ)(a1, . . . , ap+q) =π(ψ(a1, . . . , aq), ϕ(aq+1, . . . , ap+q)

),

ϕ a (a0, . . . , an) =(π(a0, ϕ(a1, . . . , ap)), ap+1, . . . , an

),

which one may call the Poisson cup product and Poisson cap product,

respectively.


Example III: calculi for left Hopf algebroids

The idea is to find a sort of bialgebra over a noncommutative basering A. In a standard way, define A-rings (or A-algebras) and A-corings(or A-coalgebras) as monoids resp. comonoids in the category of(A,A)-bimodules. In particular, an Ae-ring U can be described by twomaps s : A→ U and t : Aop → U, called source and target.

Definition (Takeuchi, Lu, Xu)

A (left) A-bialgebroid is a k-module U that is both an Ae-ring (U, s, t)and an A-coring (U,∆, ε) such that the “coring-arising” (A,A)-bimodulestructure coincides with that of the “ring-arising” left Ae-modulestructure, and such that

∆ is a unital k-algebra morphism taking values in U ×A U;ε(s(a)t(b)u) = aε(u)b and ε(uv) = ε(us(εv)) = ε(ut(εv)).

Here, the Sweedler-Takeuchi subspace

U ×A U := ∑

i ui ⊗A u′i ∈ U ⊗A U |∑

i ui t(a)⊗A u′i =∑

i ui ⊗A u′i s(a),carries an algebra structure, in contrast to U ⊗A U.


Left Hopf algebroids

For a left bialgebroid (U ,A), define the (Hopf-)Galois map

β : U ⊗Aop U → U ⊗A U , u ⊗Aop v 7→ u(1) ⊗A u(2)v .

For bialgebras over fields, β is bijective if and only if U is a Hopfalgebra, and β−1(u ⊗k v) := u(1) ⊗k S(u(2))v , where S is theantipode of U . This motivates:

Definition (Schauenburg)

A left A-bialgebroid U is called a left Hopf algebroid (or ×A-Hopfalgebra) if β is a bijection.

We adopt a Sweedler-type notation

u+ ⊗Aop u− := β−1(u ⊗A 1)

for the so-called translation map β−1(· ⊗A 1) : U → U ⊗Aop U .Niels Kowalzig (U Roma La Sapienza) Higher Structures on Modules over Operads Copenhagen, 12-10-2015 37 / 44

Examples of left Hopf algebroids

For A = k , left Hopf algebroids are simply Hopf algebras.The enveloping algebra Ae = A⊗k A

op of an arbitrary (unital)k-algebra A is a left bialgebroid over A by means of

s(a) := a ⊗k 1, t(b) := 1⊗k b,∆(a ⊗k b) := (a ⊗k 1)⊗A (1⊗k b), ε(a ⊗k b) := ab,

and a left Hopf algebroid by

(a ⊗k b)+ ⊗Aop (a ⊗k b)− := (a ⊗k 1)⊗Aop (b ⊗k 1)

Observe: there is no notion of antipode for left Hopf algebroids.


Operads with multiplication from bialgebroids

Set

C •(U ,A) := HomAop(U⊗Aop•,A),

along with the structure maps

i : C p(U ,A)⊗ C q(U ,A)→ C |p+q|(U ,A), i = 1, . . . , p,

where |p| := p − 1, given by

(ϕ i ψ)(u1, . . . , up+q−1) := ϕ(u1, . . . , ui−1,Dψ(ui , . . . , ui+q−1), ui+q, . . . , up+q−1),

where Dψ(u1, . . . , uq) := ψ(u1(1), . . . , u

q(1)) u1

(2) · · · uq(2), together with

µ := ε(mU(·, ·)), 1 := ε(·), and e := 1A,

where mU is the multiplication map of U .


Gerstenhaber algebras from bialgebroids

Theorem (K.-Krahmer, 2012)

Let (U ,A) be a left bialgebroid. Then C •(U ,A) with the structuremaps given above forms an operad with multiplication.

Hence, by the theorem of Gerstenhaber-Schack/McClure-Smithmentioned before, one obtains:

Corollary

The cohomology groups H•(U ,A) (or Ext•U(A,A) for a certainA-projectivity of U over A) form a Gerstenhaber algebra.

For introducing cyclic modules over operads arising from bialgebroids,we need more structure:


Coefficients in cyclic homology

Start with a right U-module M that is simultaneously a leftU-comodule; one needs to know what happens if action isfollowed by coaction: M is an anti-Yetter- Drinfel’d module if

(mu)(−1) ⊗A (mu)(0) = u−m(−1)u+(1) ⊗A m(0)u+(2)

for all u ∈ U ,m ∈ M and stable if first coacting and thenacting yields m(0)m(−1) = m,(along with the fact that the induced A-module structures ofaction and coaction need to coincide).The category UaYDU of aYD modules is not monoidal but amodule category over the category U

UYD of Yetter-Drinfel’dmodules (to be introduced below ), that is, tensoring them witha YD module gives an aYD again.


Operad modules from left Hopf algebroids

Moreover, to get a cyclic operad module, (U,A) needs to be not only abialgebroid but a left Hopf algebroid. For M ∈ UaYDU , we set

C•(U,M) := M ⊗Aop U⊗Aop•,

and for the comp module maps

ϕ •i (m, u1, . . . , uk) := (m, u1, . . . , ui−1,Dϕ(ui , . . . , ui+p−1), ui+p, . . . , uk),

for i = 1, . . . , k − p + 1, and any ϕ ∈ Cp(U,A), where p ≥ 0, and, asbefore,

Dφ(u1, . . . , up) := ψ(u1(1), . . . , u

p(1)) u1

(2) · · · up(2).


Cyclic operad modules from left Hopf algebroids

Theorem (K., 2013)

Let (U,A) be a left Hopf algebroid and M an anti Yetter-Drinfel’d module.Then C•(U,M) forms a para-cyclic module over the operad C •(U,A)with multiplication w.r.t. the extra comp module map

ϕ •0 (m, u1, . . . , uk)

:=(m(0), u

1+, . . . , ϕ(uk−p+2

+ , . . . , uk+, uk− · · · u1

−m(−1)) I uk−p+1+

),

where ϕ ∈ Cp(U,A), along with the cyclic operator

t(m, u1, . . . , uk) := (m(0)u1+, u

2+, . . . , u

k+, u

k− · · · u1

−m(−1)).

If the anti Yetter-Drinfel’d module M is stable, then C•(U,M) is a cyclicunital module over the operad C •(U,A).


Hence, as said, one obtains a noncommutative differentialcalculus on the pair (cohomology, homology) and under certainprojectivity assumptions on the pair (ExtU ,Tor

U), which wasproven before (K.-Krahmer, 2012).

QuestionWhy is this interesting, in particular when looking at all these trulyhorrible formulae?

The answer is that many classical homology theories can betreated in a unified manner here:

in fact, inserting for the pair (U ,A) the pair (Ae ,A) yields theHochschild example, but (kG , k), (U(g), k),(VΩ1(X ), C∞(X )), (VL,A) yield group, Lie algebra, Poissonhomology, and Lie algebroid homology; also groupoids canbe treated.

The classical calculus structure from differential geometrywith Cartan’s homotopy formula L = [d , ι] is also contained.


Higher Structures on Modules over Operads · Copenhagen, 12-10-2015 Niels Kowalzig (U Roma La...

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