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Geometry and Physics: Volume IA Festschrift in honour of Nigel Hitchin$
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Andrew Dancer, Jørgen Ellegaard Andersen, and Oscar García-Prada

Print publication date: 2018

Print ISBN-13: 9780198802013

Published to Oxford Scholarship Online: December 2018

DOI: 10.1093/oso/9780198802013.001.0001

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Deformation Theory of Lie Bialgebra Properads

Deformation Theory of Lie Bialgebra Properads

(p.219) 10 Deformation Theory of Lie Bialgebra Properads
Geometry and Physics: Volume I

Sergei Merkulov

Thomas Willwacher

Oxford University Press

This chapter presents the homotopy derivations of the properads governing even and odd Lie bialgebras, as well as involutive Lie bialgebras. The answer may be expressed in terms of the Kontsevich graph complexes. In particular, this shows that the Grothendieck–Teichmuller group acts faithfully and essentially transitively on the completions of the properads governing even Lie bialgebras and involutive Lie bialgebras, up to homotopy. This shows also that, in contrast to the even case, the properad governing odd Lie bialgebras admits precisely one non-trivial automorphism—the standard rescaling automorphism, and that it has precisely one non-trivial deformation, which we describe explicitly.

Keywords:   properad, Lie bialgebra, graph complex, homotopy derivation, deformation

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