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The Frobenius properad is Koszul
Stockholm University, Faculty of Science, Department of Mathematics.
ETH, Zurich.
ETH, Zurich.
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We show Koszulness of the prop governing involutive Lie bialgebras and also of the props governing non-unital and unital-counital Frobenius algebras, solving a long-standing problem. This gives us  minimal models for  their deformation complexes, and for deformation complexes of their algebras which are discussed in detail.

  Using an operad of graph complexes we prove, with the help of an earlier result of one of the authors, that there is a highly non-trivial action  of the  Grothendieck-Teichm\"uller group $GRT_1$ on (completed versions of) the minimal models of the properads governing Lie bialgebras and involutive Lie bialgebras by automorphisms.  As a corollary one obtains a large class of universal deformations of any (involutive) Lie bialgebra and any Frobenius algebra, parameterized by elements of the Grothendieck-Teichm\"uller Lie algebra.

We also prove that, for any given homotopy involutive Lie bialgebra structure in a vector space, there is an associated homotopy Batalin-Vilkovisky algebra structure on the associated Chevalley-Eilenberg complex.

Keyword [en]
Operads, Grothendieck-Teichmueller group, Lie bialgebras
National Category
Algebra and Logic
Research subject
URN: urn:nbn:se:su:diva-108183OAI: diva2:755228
Available from: 2014-10-14 Created: 2014-10-14 Last updated: 2014-10-20Bibliographically approved

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