Reference : The Frobenius operad is Koszul
Scientific journals : Article
Physical, chemical, mathematical & earth Sciences : Mathematics
The Frobenius operad is Koszul
Camos, Ricardo [University of Zurich > Institute of Mathematics]
Merkulov, Sergei mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
Willwacher, Thomas [University of Zurich > Institute of Mathematics]
Duke Mathematical Journal
Duke University Press
Yes (verified by ORBilu)
[en] operads, properads ; homological algebra, graph complexes ; Grothendieck-Teichmueller group
[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 [W3], that there is a highly non-trivial action of the Grothendieck-Teichm¨uller group GRT 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-Teichmueller 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.

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