When Ext is a Batalin–Vilkovisky algebra

Abstract

We show under what conditions the complex computing general Ext-groups carries the structure of a cyclic operad such that Ext becomes a Batalin-Vilkovisky algebra. This is achieved by transferring cyclic cohomology theories for the dual of a (left) Hopf algebroid to the complex in question, which asks for the notion of contramodules introduced along with comodules by Eilenberg-Moore half a century ago. Another crucial ingredient is an explicit formula for the inverse of the Hopf-Galois map on the dual, by which we illustrate recent categorical results and answer a long-standing open question. As an application, we prove that the Hochschild cohomology of an associative algebra A is Batalin-Vilkovisky if A itself is a contramodule over its enveloping algebra A b A op . This is, for example, the case for symmetric algebras and Frobenius algebras with semisimple Nakayama automorphism. We also recover the construction for Hopf algebras.

Higher Brackets on Cyclic and Negative Cyclic (Co)Homology

“…In the case of Frobenius or (twisted) Calabi-Yau algebras (cf. Example 6.17), one can even define a homotopy calculus structure of Hochschild cochains over themselves: the main difficulty here consists in finding a cyclic operator on C ‚ pA, Aq which does not always exist but does so if A is what is called an anti Yetter-Drinfel'd contramodule over A e " A b A op , see [Ko2] for a treatment. The corresponding calculus structure can then be obtained by the specific form of the cyclic operator (see op.…”

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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].

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A Categorical Approach to Cyclic Cohomology of Quasi-Hopf Algebras and Hopf Algebroids

“…Again, roughly, the generalization from Hopf algebras consists of replacing the ground field k, over which everything is tensored, by a noncommutative ring R. Thus the objects underlying representations of a Hopf algebroid consist not of vector spaces over k, but of bimodules over R. It seems that Hopf algebroids provide an example of a construct that is the most noncommutative in noncommutative geometry. We point out that a definition of cyclic cohomology with an analogue of stable anti-Yetter-Drinfeld contramodule coefficients has been given by Kowalzig in [15]. Our goal here is to explain the formulas that appear there as an inevitable consequence of the general machinery.…”

“…As was observed in [2] or [15], we can equip a right contramodule M with a structure of a left R l -module via:…”

“…Let us state the definition of an anti-Yetter-Drinfeld contramodule over a Hopf algebroid. Notice that the definition is slightly different from the one used in [15].…”