Higher Brackets on Cyclic and Negative Cyclic (Co)Homology

Abstract

An analogous statement for (left) Hopf algebroids pU, Aq, that is, when Ext U pA, Aq happens to be a Batalin-Vilkoviskiȋ algebra appears to be more involved [Ko2]; this includes the case for Hochschild cohomology as pA e , Aq can be seen as a Hopf algebroid (but not as a Hopf algebra). It is known that the Gerstenhaber structure on Hochschild cohomology does not always extend (or rather restrict) to a Batalin-Vilkoviskiȋ algebra structure but only in certain cases as, for example, symmetric algebras, certain Frobenius algebras, as well as (twisted) Calabi-Yau algebras [Tr,LaZhZi,Gi,KoKr2]; a sufficiency criterion when this is the case can be found in [Ko2].On the other hand, if U is a braided (in the sense of quasi-triangular) bialgebra over K, then the Gerstenhaber bracket on Ext ‚ U pK, Kq turns out to be zero [Tai, Rem. 5.4]. At least for a cocommutative Hopf algebra, this can be directly seen from Eq. (1.1) together with the fact that in this case one obtains B " 0, as mentioned in [Me2, p. 321] again. In general, the possible cohomological vanishing of the Gerstenhaber bracket can be considered a consequence from the fact that the binary operation up to homotopy on the cochain complex inducing it, is homotopic to zero. This homotopy induces in turn a degree´2 Lie bracket on cohomology, which together with the cup product makes it an e 3 -algebra. The existence of an e 3 -algebra structure on the cohomology of a braided Hopf algebra (or rather bialgebra) has been conjectured in [Me2, Conj. 25], inspired by a conjecture attributed to Kontsevich in [Sh2] that this is the case for the Gerstenhaber-Schack cohomology of a Hopf algebra. This in turn amounts to an Ext-group in the category of tetramodules (or Hopf bimodules) [Tai, Cor. 3.9], or, in the finite dimensional case, equivalently to an Ext-group over the Drinfel'd double as a braided bialgebra. In [Sh2, Sh1], Shoikhet developed an approach based on n-fold monoidal abelian categories that implies that the Gerstenhaber-Schack cohomology groups indeed do carry the structure of an e 3 -algebra. More recently, Ginot and Yalin [GiYa] showed how to obtain this from the E 3 -algebra structure on the higher Hochschild complex of a E 2 -algebras given by the solution to the higher Deligne conjecture. This indeed implies the existence of an E 3 -structure on the deformation complex of a dg bialgebra, but an explicit expression for the degree´2 Lie bracket remains elusive.There is, however, quite an explicit degree´2 Lie bracket on the negative cyclic cohomology HC ‚ pAq of, for example, a symmetric algebra A, obtained by Menichi [Me1] generalising the construction of Chas-Sullivan's string topology bracket [ChSu].Since negative cyclic cohomology is related to Hochschild cohomology by a long exact sequence (that is, a version of the so-called SBI sequence), it is therefore tempting to presume that a possible braiding implies suitable vanishings such that the Chas-Sullivan-Menichi bracket can be transferred to the Hochschild cohomology, inducing the Lie bracket of the ...

Gravity algebra structure on the negative cyclic homology of Calabi–Yau algebras

“…Moreover, we think Theorem 1.1 and its corollaries have their own virtue, and it is better to write a separate paper. Finally, we mention that when we were finishing the current paper, we learned of the preprint of D. Fiorenza and N. Kowalzig [12], where the Lie algebra structure on the negative cyclic homology are also studied. Nevertheless, our primary goals and main interests are quite different.…”

Homotopy Cartan calculus and inner deformations of $A_\infty$-algebras

“…(As above, all permutations preserve the cyclic order of a's.) In a slightly different notation an explicit expression for T (A, B) was given in [13] and [24,Ex. 3.13]; in the former paper it is denoted by ρ.…”

“…Then, the following relations hold: The reader can verify these identities either directly, using the definitions of Sec. 4, or consult the papers [13], [24]. Comparing Rels.…”

A noncommutative calculus on the cyclic dual of Ext

“…For use in the main text, we will give some explicit formulae, see [Ko1] and also [FiKo,§6]. For ϕ P Oppq, ψ P Opqq, and x P Mpnq ι ϕ x " pµ ˝2 ϕq ‚ 0 x, L ϕ x " n´p`1 ř i"1 p´1q pp´1qpi´1q ϕ ‚ i x `p ř i"1 p´1q npi´1q`p´1 ϕ ‚ 0 t i´1 pxq, S ϕ x " n´p`1 ř j"1 n´p`1 ř i"j p´1q npj´1q`pp´1qpi´1q e ‚ 0 `ϕ ‚ i t j´1 pxq ˘, T ϕ,ψ pxq " p´1 ř j"1 p´1 ř i"j p´1q npj´1q`pq´1qpi´jq`p pϕ ˝p´i`j ψq ‚ 0 t j´1 pxq.…”