It seems that the Gerstenhaber-Schack (bi)complex of an associative bialgebra carries a homotopy e_3-algebra structure and a degree 2 Lie algebra bracket, up to homotopy. Does anyone know about a reference where these structures have been explicitly shown?
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$\begingroup$ When you say 'it seems...' do you mean you've read it somewhere? $\endgroup$– Fernando MuroCommented Nov 27, 2013 at 17:33
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1$\begingroup$ It seems that there's a sketch of the structure in: Hopf algebras, tetramodules, and n-fold monoidal categories Boris Shoikhet arxiv.org/pdf/0907.3335v2.pdf $\endgroup$– James GriffinCommented Nov 27, 2013 at 17:46
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1$\begingroup$ This has been partially written up by the same author in: arxiv.org/abs/1202.1938 $\endgroup$– James GriffinCommented Nov 27, 2013 at 17:47
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2 Answers
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It's done in Ginot-Yalin Deformation theory of bialgebras, higher Hochschild cohomology and formality (arXiv:1606.01504
More precisely, my undertsanding is that what's true by abstract non-sense is that under some (co)nilpotency condition there's an equivalence between $E_2$-algebras and (dg-)bialgebras, and that the deformation complex of an $E_2$-algebra is $E_3$. What is done in that paper apart from carefully writing this up, is identifying this with the standard GS complex explaining and precisely in what way t controls deformations of bialgebra in this setting.
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I don't know about e3, but the Lie up to homotopy is in "Intrinsic brackets and the $L_{\infty}$-deformation theory of bialgebras" by Martin Markl, https://arxiv.org/abs/math/0411456.
