2016/10/04 by Kowalzig, Niels · 1 citation
#Algebraic Topology (math.AT) #FOS: Mathematics #K-Theory and Homology (math.KT) #Quantum Algebra (math.QA) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1610.01229
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 ⊗ Aop. This is, for example, the case for symmetric algebras and Frobenius algebras with semisimple Nakayama automorphism. We also recover the construction for Hopf algebras.