The Frobenius properad is Koszul

Abstract

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 [W3], that there is a highly non-trivial action of the Grothendieck-Teichmüller 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-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.

Moduli Spaces of (Bi)algebra Structures in Topology and Geometry

“…Two motivating examples are the realizations of Poincaré duality of oriented closed manifolds as homotopy Frobenius algebra structures at the cochain level, and realizations of the Lie bialgebra structure on string homology at the chain level. Let us note that an explicit realization has been recently obtained in [13] (with interesting relationships with symplectic field theory and Lagrangian Floer theory), using a notion of homotopy involutive Lie bialgebra which actually matches with the minimal model of the associated properad obtained in [9] (see [13,Remark 2.4]). However, classification and deformation theory of such structures, as well as the potential new invariants that could follow, are still to be explored.…”

Moduli stacks of algebraic structures and deformation theory

“…These operations induce the Poincaré duality at the cohomology level. The properads F rob and ucF rob are proved to be Koszul respectively by Corollary 2.10.1 and Theorem 2.10.2 in [9], hence there are explicit resolutions. We thus get a moduli stack of "cochain-level Poincaré duality" ucF rob n ∞ {C * M } on an n-dimensional compact oriented manifold M .…”

“…If the moduli space BiLie ⋄,2−n ∞ {C S 1 * (LM )} is not empty, then we get a moduli stack of chain-level string operations on M . Moreover, the properad of involutive Lie bialgebras is Koszul by Theorem 2.8 of [9], with the properad of Frobenius bialgebras as Koszul dual, hence a small explicit model for BiLie ⋄,2−n ∞ . 5.3.…”

On Quantizable Odd Lie Bialgebras

“…This additional constraint is satisfied in many interesting examples studied in homological algebra, string topology, symplectic field theory, Lagrangian Floer theory of higher genus, and the theory of cohomology groups H(M g,n ) of moduli spaces of algebraic curves with labelings of punctures skewsymmetrized [D,ES,C,CS,CFL,Sc,CMW,MW1]. In the odd case the most interesting for applications Lie bialgebras have c = 1, d = 0.…”

“…They have been introduced in [M1] and have seen applications in Poisson geometry, deformation quantization of Poisson structures [M2] and in the theory of cohomology groups H(M g,n ) of moduli spaces of algebraic curves with labelings of punctures symmetrized [MW1]. The homotopy and deformation theories of even/odd Lie bialgebras and also of involutive Lie bialgebras have been studied in [CMW,MW2]. A key tool in those studies is a minimal resolution of the properad governing the algebraic structure under consideration.…”