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Intrinsic brackets and the $L_\infty$-deformation theory of bialgebras

Martin Markl
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Abstract

We show that there exists a Lie a bracket on the cohomology of any type of (bi)algebras over an operad or a PROP, induced by a strongly homotopy Lie structure on the defining cochain complex, such that the associated "quantum" master equation captures deformations. This in particular implies the existence of a Lie bracket on the Gerstenhaber-Schack cohomology of a bialgebra that extends the classical intrinsic bracket on the Hochschild cohomology, giving an affirmative answer to an old question about the existence of such a bracket.

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Martin Markl. 2004-11-20. Intrinsic brackets and the $L_\infty$-deformation theory of bialgebras. https://arxiv.org/abs/math/0411456

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