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

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

2004-11-20

arxiv.org/abs/math/0411456v7

Mathematics

Algebraic Topology

The final version. To appear in Journal of Homotopy and Related Structures

Scientific paper

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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