The Frobenius properad is Koszul

Abstract

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-Teichmüller group $GR{T}_1$ 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-Teichmüller 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.

References in corpus (6)

• Modular operads and Batalin-Vilkovisky geometry
• Homological algebra related to surfaces with boundary
• Algebra+Homotopy=Operad
• A Hopf algebra quantizing a necklace Lie algebra canonically associated to a quiver
• The role of string topology in symplectic field theory
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Cited by in corpus (7)

• Toric varieties of Loday's associahedra and noncommutative cohomological field theories
• On quantizable odd Lie bialgebras
• Calculus of multilinear differential operators, operator $L_{\infty }$-algebras and ${IBL}_{\infty }$-algebras
• Mirror symmetry for the Tate curve via tropical and log corals
• Batalin-Vilkovisky structures on moduli spaces of flat connections
• The MV formalism for ${\mathrm{IBL}}_{\infty }$- and ${\mathrm{BV}}_{\infty }$-algebras
• Cumulants, Koszul brackets and homological perturbation theory for commutative $B{V}_{\infty }$ and $IB{L}_{\infty }$ algebras