The cyclic Deligne conjecture for spaces, chain complexes and Hopf algebras

The cyclic Deligne conjecture for spaces, chain complexes

and Hopf algebras

Clemens Berger

University of Nice 28 April, 2009

Luminy CIRM

1joint work with Michael Batanin (Sydney)

Hochschild cochains C^•(A;A)

Gerstenhaber structure C^•(A;A^∗)

Batalin-Vilkovisky structure The dg- Deligne conjecture Multiplicative operads

The coloured operad for multiplicative operads Condensation of coloured operads

The cobar complex of a bialgebra The topological Deligne conjecture Braid and ribbon-braid groups

Coxeter geometry of permutation groups The categorical Deligne conjecture The Drinfeld double of a Hopf algebra

Hochschild cochains

Definition

For a (unital associative)K-algebraA andA-bimoduleM, the Hochschild cochain complexof Awith coefficients in M is given by

Cⁿ(A;M) =HomK(A^⊗n,M), n≥0, where forf ∈Cⁿ(A;M),

(∂_if)(a₁, . . . ,a_n+1) =







a1f(a2, . . . ,an) i = 0;

f(a₁, . . . ,a_ia_i+1, . . . ,a_n) i = 1, . . . ,n;

f(a₁, . . . ,a_n)a_n+1 i =n+ 1.

(s_if)(a1, . . . ,an−1) =f(a1, . . . ,a_i,1_A,a_i+1, . . . ,an−1).

The Hochschild cohomologyHH^•(A;M) is the cohomology of the cochain complex of the cosimplicialK-module C^•(A;M).

Cup and brace operations onC^•(A;A)

There is acup product

− ∪ −:C^m(A;A)⊗_KCⁿ(A;A)→C^m+n(A;A) (f ∪g)(a₁, . . . ,a_m+n) =f(a₁, . . . ,a_m)g(a_m+1, . . . ,a_m+n) and abrace operation

−{−}:C^m(A;A)⊗_KCⁿ(A;A)→C^m+n−1(A;A) wheref{g}(a₁, . . . ,am+n−1) is defined by

X

1≤i≤m

(−1)^{(i−1)(n−1)}f(a₁, . . . ,ai−1,g(a_i, . . . ,ai+n−1),a_i+n, . . . ,am+n−1).

The bracket{f,g}=f{g} −(−1)^{(|f|−1)(|g|−1)}g{f} induces a Lie bracket of degree−1 on HH^•(A;A).

Gerstenhaber structure

Definition

AGerstenhaber K -algebra(H,∪,{−,−}) is a graded-commutative K-algebra with Lie bracket of degree−1 such that

{f,g ∪h}={f,g} ∪h+ (−1)^|f|(|g|−1)g∪ {f,h}.

Proposition (Gerstenhaber ’63)

For any algebraA, the Hochschild cohomologyHH^•(A;A) is a Gerstenhaber algebra.

Theorem (F. Cohen ’72)

For any fieldK, the homologyH•(D₂;K) of the little disks operad is the operad for GerstenhaberK-algebras.

Corollary

For any based space (X,∗), the homologyH•(Ω²X;K) is a GerstenhaberK-algebra.

Connes’ coboundary onC^•(A;A^∗)

ForA^∗ =Hom_K(A,K), the adjunction

HomK(A^⊗n,A^∗)∼=HomK(A^⊗n+1,K)

induces a cyclic operatorτ_n onCⁿ(A;A^∗) of order n+ 1. These cyclic operators are compatible with the simplicial operators:

τn+1∂_i =∂i−1τn i >0, τn−1s_i =s_i−1τn i >0.

It results a covariant functor on Connes’cyclic category

∆C →Mod_K : [n]7→Cⁿ(A;A^∗).

In particular,C^•(A;A^∗) is a mixed complex

C⁰(A;A^∗)C¹(A,A^∗)C²(A;A^∗)· · · andHH^•(A;A^∗) has a differential ∆ of degree −1:

∆ⁿ:HHⁿ(A,A^∗)→HHⁿ⁻¹(A;A^∗).

Batalin-Vilkovisky structure

Definition

ABatalin-Vilkovisky algebrais a Gerstenhaber algebra (H,∪,{−,−}) with a differential ∆ of degree −1 such that

(−1)^|f|{f,g}= ∆(f ∪g)−(∆f ∪g)−(−1)^|f|(f ∪∆g).

Asymmetric K -algebra Ais aK-algebra equipped with an isomorphism ofA-bimodules A∼=A^∗, i.e. a symmetric exact pairing<−,−>:A⊗_KA→K such that <ab,c>=<a,bc>.

Proposition (Menichi ’04)

For any symmetric algebraA, the Hochschild cohomology HH^•(A,A) is a Batalin-Vilkovisky algebra.

Theorem (Getzler ’94)

For any fieldK, the homology H•(fD2,K) of the framed little disks operad is the operad for Batalin-VilkoviskyK-algebras.

The dg- Deligne conjecture

Theorem (MS ’02, KS ’02, Vo ’02, Ta ’04, BF ’04) The Hochschild cochain complex of an algebraA admits a

C•(D₂)-action inducing the Gerstenhaber structure onHH^•(A;A).

Theorem (KS ’06, TZ ’06, Ka ’07, BB ’09)

The Hochschild cochain complex of a symmetric algebraA admits aC•(fD2)-action inducing theBV-structure onHH^•(A;A).

Proposition (Gerstenhaber-Voronov ’95, Menichi ’04) The Hochschild cochain complex ofAis isomorphic to the deformation complexof the endomorphism operadEndA of A.

IfAis symmetric, thenEnd_A is multiplicative cyclic.

Proof.

Cⁿ(A;A) =Hom(A^⊗n,A) =EndA(n)3µn. Forf ∈EndA(n),

∂0f =µ2◦₁f,∂nf =µ2◦₀f,∂_if =f ◦_iµ2 if 0<i <n.

IfAis symmetric thenEnd_A is cyclic and τ_n(µ_n) =µ_n.

Multiplicative operads

Definition

A multiplicative (cyclic) operadis a non-symmetric (cyclic) operad Oequipped with a map of (cyclic) operads Ass → O.

A muliplicative (cyclic) operadO has an underlying cosimplicial (cocyclic) objectO^•. In a closed monoidal category E equipped withδ : ∆→ E, thedeformation complex of O isHom_∆(δ^•,O^•).

Example

ForE =Ch(Z) andδ_Z: ∆→Ch(Z) : [n]7→N∗(∆[n];Z) we get C^•(A;A) =Hom_∆(δ^•_Z,End^•_A).

Theorem (Kaufmann ’07, BB ’09)

For any multiplicative chain operadO, the deformation complex of Oadmits a C•(D2)-action. If O is multiplicative cyclic, this action extends to aC•(fD₂)-action.

The coloured operad for multiplicative operads

LetL₂(n1, . . . ,n_k;n) be the set of iso-classes of planar rooted trees withn leaves and a bipartite vertex-set such that:

1. one part of the vertex-set is in bijection with{1, . . . ,k};

2. the vertex with label i has arity ni;

3. each edge has at least one labelled extremity;

4. unlabelled vertices have arity 6= 1.

LetC_[n]=Z/(n+ 1)Zand put

L^cyc₂ (n₁, . . . ,n_k;n) =L₂(n₁, . . . ,n_k;n)×C_[n1]× · · · ×C_[nk]. L₂ and L^cyc₂ are N-coloured operads for an evident substitution of trees into trees; inL^cyc₂ , the cyclic permutations distinguish for each labelled vertex one of its incident edges, the neutral element stands for the edge closest to the root of the tree.

Lemma

L₂-algebras are multiplicative operads; L^cyc₂ -algebras are

multiplicative cyclic operads. The category of unary operations of L₂ (resp. L^cyc₂ ) is ∆ (resp. ∆C).

Condensation of coloured operads

Unary operations of a coloured operad act covariantly on inputs and contravariantly on the output; therefore:

L₂(−,· · · ,−;−) : ∆^op× · · · ×∆^op×∆→Sets.

Givenδ_Z: ∆→Ch(Z) we canrealizemultisimplicially, and totalize the resulting cosimplicial chain complex. This yields

ξ(L₂, δ_Z)(k) :=Hom_∆(δ^•,|L₂(

k

z }| {

−,· · · ,−;•)|_δ⊗k Z

), k ≥0.

Proposition (Day-Street ’03, McClure-Smith ’04, BB ’09) ξ(L₂, δ_Z) (resp. ξ(L^cyc₂ , δ^cyc

Z )) is a chain operad acting on the deformation complex of any multiplicative (cyclic) operad.

Theorem (BB ’09)

As chain operads we haveξ(L₂, δ_Z)∼C•(D2) and ξ(L^cyc₂ , δ^cyc

Z )∼C•(fD₂).

The cobar complex of a bialgebra

Theorem (cf. Gerstenhaber-Schack ’92, Menichi ’04) The cobar complex ΩAof a bialgebra (resp. involutive Hopf algebra)Ahas an action by ξ(L₂, δ_Z) (resp. ξ(L^cyc₂ , δ^cyc_Z )). Its homologyH•(ΩA;Z) is a Gerstenhaber (resp. BV-) algebra.

Proof.

The bialgebraA is a comonoid in the monoidal category of A-modules. Therefore: (ΩA)n=A^⊗n∼=HomA(A,A^⊗n).This Z-linear operad is multiplicative via the diagonal of A. IfA has an involutive antipode then the operad is multiplicative cyclic.

Remark

(a) ΩC•(ΩX;Z)∼C•(Ω²X;Z) (Adams). Theξ(L₂, δ_Z)-action on ΩC•(ΩX;Z) corresponds to the C•(D₂)-action onC•(Ω²X;Z).

(b) IfA is involutive, theξ(L^cyc₂ , δ^cyc

Z )-action induces a cocyclic structure on ΩAyieldingHC^•(A) of Connes-Moscovici ’99.

(c)ξ(L₂, δ_Z) contains the second filtration stage of the surjection operadof MS ’03, BF ’04 as a suboperad. Cyclic extension ?

The topological Deligne conjecture

There is a cosimplicial resp. cocyclic space

δ_top: ∆→Top: [n]7→∆ⁿ resp. δ^cyc_top : ∆C →Top: [n]7→∆ⁿ×S¹. Theorem (McClure-Smith ’04, Salvatore ’09, BB ’09)

The operadξ(L₂, δ_top) is weakly equivalent to D₂ and acts on the deformation complex of multiplicative operads in spaces.

The operadξ(L^cyc₂ , δ^cyc_top) is weakly equivalent tofD2 and acts on the deformation complex of multiplicative cyclic operads in spaces.

Remark (cf. Markl ’99, Salvatore-Wahl ’03, Salvatore ’09) fD₂(k)∼=D₂(k)×(S¹)^k, ξ(L^cyc₂ , δ^cyc_top)(k)∼=ξ(L₂, δ_top)(k)×(S¹)^k. Forn= 1:

fD(1)∼=D(1)oS¹,Hom_∆C(δ^cyc_top, δ^cyc_top)∼=Hom_∆(δtop, δtop)S¹. Proposition (Sinha ’06)

The simplicial 2-sphereS²= ∆[2]/∂∆[2] is anL₂-coalgebra in finite pointed sets. For a based space (X,∗), Ω²X is the deformation complex of the multiplicative operad (X,∗)^(S2,∗).

Braid and ribbon-braid groups

S_k denotes the permutation grouponk letters. S^±_k denotes the signed permutation grouponk letters.

S^±_k =S_k oS₂ =S_kn(S₂)^k acts onfD₂(k) =D₂(k)×(S¹)^k. Definition (Braid and ribbon-braid groups on k strands)

B_k = π₁(D₂(k)/S_k) RB_k = π₁(fD₂(k)/S^±_k) PBk = π1(D2(k)) PRBk = π1(fD2(k)) Proposition (Asphericity ofD2(k) and fD2(k))

D₂(k)/S_k = K(B_k,1) fD₂(k)/S^±_k = K(RB_k,1) D2(k) = K(PB_k,1) fD2(k) = K(PRB_k,1) Corollary

The coveringsD₂(k)→D₂(k)/S_k andfD₂(k)→fD₂(k)/S^±_k are classified by the short exact sequences 1→PBk →Bk →S_k →1 and 1→PRB_k →RB_k →S^±_k →1.

Problem

Describe the operad structure ofD₂ (resp. fD₂) in terms of the pure braid (resp. ribbon-braid) groups.

Coxeter geometry of permutation groups

B_k =<s1, . . . ,sk−1|(s_is_j)²= 1 if|i−j|>1 and (s_is_i+1)³ = 1>

Thepure Artin group PBk =Ker(Bk →S_k)∼=π1(C^k− A_Sk) whereA_Sk is the complexified braid arrangement.

TheSalvetti complex Sal_Sk is a partially ordered set of the same equivariant homotopy type asC^k− A_Sk.

Sal−: (Coxeter groups)→(posets)

is a functor commuting with finite products. Thus, (PB_k)k≥0 is a categorical operad. Similarly, (PRBk)k≥0 is a categorical operad.

Proposition

D₂ ∼K(PB,1) andfD₂ ∼K(PRB,1) as operads. Moreover, PB-algebras are braided strict monoidal categories; PRB-algebras are ribbon-braided (i.e. balanced) strict monoidal categories.

Corollary (B ’98, Salvatore-Wahl ’03)

The nerve of a braided (resp. ribbon-braided) strict monoidal category isE₂ (resp. framed E₂).

The categorical Deligne conjecture

Consider the cosimplicial category

δCat : ∆→Cat: [n]7→[n][n]⁻¹

Proposition

There are weak equivalences of categorical operads PB →^∼ ξ(L₂, δCat) and PRB →^∼ ξ(L^cyc₂ , δCat).

Definition

A central element of a monoidal categoryE is a pair (A,cA) where cA,−:A⊗ − ∼=− ⊗A andc_A,B⊗C = (1_B ⊗c_A,C)◦(c_A,B ⊗1_C).

The centerZE is the category of central elements.

Proposition

ForE =ModH,ZE 'ModDH whereDH is the Drinfeld double of the Hopf algebraH.

The Drinfeld double of a Hopf algebra

Proposition (Street ’04) ZE=Hom_∆(δ_Cat,EndE) Corollary

The center of a monoidal category is braided monoidal; in particular, the Drinfeld double of a Hopf algebra is “braided”.

Definition

An involutive category is a closed monoidal categoryE such that the duality functor (−)^∗ =Hom(−,I) is self-adjoint. A Hopf algebraH is called quasi-involutive ifMod_H is involutive.

Proposition

The categoryE_f of symmetric duality objects of an involutive categoryE has a multiplicative cyclic endomorphism-operadEndE_f. Corollary

The center ofE_f is ribbon-braided; in particular, the Drinfeld double of a quasi-involutive Hopf algebra is “ribbon-braided”.