When Ext is a BatalinVilkovisky algebra
Abstract
We show under what conditions the complex computing general Extgroups carries the structure of a cyclic operad such that Ext becomes a BatalinVilkovisky 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 EilenbergMoore half a century ago. Another crucial ingredient is an explicit formula for the inverse of the HopfGalois map on the dual, by which we illustrate recent categorical results and answer a longstanding open question. As an application, we prove that the Hochschild cohomology of an associative algebra A is BatalinVilkovisky if A itself is a contramodule over its enveloping algebra A \otimes 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2016
 arXiv:
 arXiv:1610.01229
 Bibcode:
 2016arXiv161001229K
 Keywords:

 Mathematics  Rings and Algebras;
 Mathematics  Algebraic Topology;
 Mathematics  KTheory and Homology;
 Mathematics  Quantum Algebra
 EPrint:
 36 pages