Reference : Deformation theory of Lie bialgebra properads
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Deformation theory of Lie bialgebra properads
Merkulov, Sergei mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
Willwacher, Thomas [Department of Mathematics > Unicersity of Zurich]
Geometry and Physics: A Festschrift in honour of Nigel Hitchin
Oxford University Press
volume 1
Geometry and Physics: A Festschrift in honour of Nigel Hitchin
September-October 2016
Andrew Dancer, Jørgen Ellegaard Andersen, Oscar García-Prada
Cambridge, Aarhus
[en] homological algebra ; deformation theory ; Lie bialgebras
[en] We compute 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-Teichm\"uller 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 by 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.
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