Batalin-Vilkovisky structures on Ext and Tor

Kowalzig, N. and Krähmer, U. (2014) Batalin-Vilkovisky structures on Ext and Tor. Journal für die Reine und Angewandte Mathematik (Crelles Journal), 2014(697), pp. 159-219. (doi: 10.1515/crelle-2012-0086)

Full text not currently available from Enlighten.


This article studies the algebraic structure of homology theories defined by a left Hopf algebroid U over a possibly noncommutative base algebra A, such as for example Hochschild, Lie algebroid (in particular Lie algebra and Poisson), or group and etale groupoid (co)homology. Explicit formulae for the canonical Gerstenhaber algebra structure on Ext_U(A,A) are given. The main technical result constructs a Lie derivative satisfying a generalised Cartan-Rinehart homotopy formula whose essence is that Tor^U(M,A) becomes for suitable right U-modules M a Batalin-Vilkovisky module over Ext_U(A,A), or in the words of Nest, Tamarkin, Tsygan and others, that Ext_U(A,A) and Tor^U(M,A) form a differential calculus. As an illustration, we show how the well-known operators from differential geometry in the classical Cartan homotopy formula can be obtained. Another application consists in generalising Ginzburg's result that the cohomology ring of a Calabi-Yau algebra is a Batalin-Vilkovisky algebra to twisted Calabi-Yau algebras.

Item Type:Articles
Glasgow Author(s) Enlighten ID:Kowalzig, Dr Niels and Kraehmer, Dr Ulrich
Authors: Kowalzig, N., and Krähmer, U.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Journal für die Reine und Angewandte Mathematik (Crelles Journal)
Journal Abbr.:J. Reine Angew. Math.
Publisher:Walter de Gruyter
ISSN (Online):1435-5345
Published Online:12 December 2012
Related URLs:

University Staff: Request a correction | Enlighten Editors: Update this record