15 October 2016 The Frobenius properad is Koszul
Ricardo Campos, Sergei Merkulov, Thomas Willwacher
Duke Math. J. 165(15): 2921-2989 (15 October 2016). DOI: 10.1215/00127094-3645116


We show the Koszulness of the properad governing involutive Lie bialgebras and also of the properads governing nonunital 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 nontrivial action of the Grothendieck–Teichmüller group GRT1 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 (involutive) Lie bialgebras and Frobenius algebras, parameterized by elements of the Grothendieck–Teichmüller Lie algebra. We also prove that for any given homotopy involutive Lie bialgebra structure on a vector space, there is an associated homotopy Batalin–Vilkovisky algebra structure on the associated Chevalley–Eilenberg complex.


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Ricardo Campos. Sergei Merkulov. Thomas Willwacher. "The Frobenius properad is Koszul." Duke Math. J. 165 (15) 2921 - 2989, 15 October 2016. https://doi.org/10.1215/00127094-3645116


Received: 20 October 2014; Revised: 29 October 2015; Published: 15 October 2016
First available in Project Euclid: 19 September 2016

zbMATH: 1360.18014
MathSciNet: MR3557276
Digital Object Identifier: 10.1215/00127094-3645116

Primary: 18D50
Secondary: 17B62 , 55P50

Keywords: Frobenius algebras , Grothendieck–Teichmüller group , involutive Lie bialgebras , operads , string topology

Rights: Copyright © 2016 Duke University Press


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Vol.165 • No. 15 • 15 October 2016
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