Operads, modules over walled Brauer categories, and Koszul complexes

Abstract

We investigate certain complexes that are associated to an operad $\mathscr{O}$ in $k$-vector spaces, where $k$ is a field of characteristic $0$. This exploits the study of modules over the $k$-linearization of the upward walled Brauer category, $k\mathsf{uwb}$ (respectively of the downward walled Brauer category, $k\mathsf{dwb}$). These are Koszul over $k(\mathbf{FB \times FB})$, where $\mathbf{FB}$ is the category of finite sets and bijections. We show that the Chevalley-Eilenberg complex for the Lie algebra of derivations $\mathrm{Der} (\mathscr{O} (V))$ of the free $\mathscr{O}$-algebra on a finite-dimensional vector space $V$ has a precursor given by the Koszul complex on an explicit module over $(k\mathsf{dwb})_-$ (a twisted $k$-linearization of $\mathsf{dwb}$); this module is constructed naturally from the operad $\mathscr{O}$. Following Dotsenko, we also consider the more general case where a {\em wheeled} term is included. This identification exploits functoriality with respect to the category of finite-dimensional $k$-vector spaces with morphisms taken to be split monomorphisms, together with the relationship with functors on the upward (and downward) walled Brauer category. We also exploit methods developed by Sam and Snowden for investigating the stabilization of the families of representations of the general linear groups associated to a functor on the category of split monomorphisms between $k$-vector spaces. We give a new perspective on the results of Dotsenko, who investigated the stable homology of Lie algebras of derivations and established a link with the wheeled bar construction for wheeled operads. In particular, we explain why one of the Koszul complexes that we consider should be considered as the appropriate form of hairy graph complex for operads, by analogy with the case of cyclic operads.