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Batalin–Vilkovisky structures on Ext and Tor

  • Niels Kowalzig EMAIL logo and Ulrich Krähmer
Published/Copyright: December 12, 2012
Journal für die reine und angewandte Mathematik (Crelles Journal)
From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2014 Issue 697

Abstract

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 étale groupoid (co)homology. Explicit formulae for the canonical Gerstenhaber algebra structure on ExtU(A,A) are given. The main technical result constructs a Lie derivative satisfying a generalised Cartan–Rinehart homotopy formula whose essence is that TorU(M,A) becomes for suitable right U-modules M a Batalin–Vilkovisky module over ExtU(A,A), or in the words of Nest, Tamarkin, Tsygan and others, that ExtU(A,A) and TorU(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.

Funding source: Excellence Network of the University of Granada (GENIL)

Funding source: Polish Government Grant

Award Identifier / Grant number: N201 1770 33

Funding source: Marie Curie PIRSES-GA-2008-230836 network

Funding source: Royal Society/RFBR joint project

Award Identifier / Grant number: JP101196/11-01-92612

It is our pleasure to thank Ryszard Nest, Boris Shoikhet, and Boris Tsygan for inspiring discussions and explaining to us some aspects of their work. Furthermore, we thank the referee for their careful reading and suggestions. N.K. acknowledges funding by the Excellence Network of the University of Granada (GENIL) and would like to thank the University of Glasgow for hospitality and support. U.K. furthermore acknowledges funding by the Polish Government Grant N201 1770 33, the Marie Curie PIRSES-GA-2008-230836 network and the Royal Society/RFBR joint project JP101196/11-01-92612, and thanks ITEP Moscow for hospitality.

Received: 2012-5-15
Revised: 2012-9-5
Published Online: 2012-12-12
Published in Print: 2014-12-1

© 2014 by De Gruyter

Articles in the same Issue

  1. Frontmatter
  2. Holomorphic one-forms, integral and rational points on complex hyperbolic surfaces
  3. Mordell–Weil groups of elliptic threefolds and the Alexander module of plane curves
  4. Reflexive differential forms on singular spaces. Geometry and cohomology
  5. Special cycles on unitary Shimura varieties II: Global theory
  6. Batalin–Vilkovisky structures on Ext and Tor
  7. Tangent cones and regularity of real hypersurfaces
https://doi.org/10.1515/crelle-2012-0086

Articles in the same Issue

  1. Frontmatter
  2. Holomorphic one-forms, integral and rational points on complex hyperbolic surfaces
  3. Mordell–Weil groups of elliptic threefolds and the Alexander module of plane curves
  4. Reflexive differential forms on singular spaces. Geometry and cohomology
  5. Special cycles on unitary Shimura varieties II: Global theory
  6. Batalin–Vilkovisky structures on Ext and Tor
  7. Tangent cones and regularity of real hypersurfaces
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