The representation theory of Brauer categories II: Curried algebra

Abstract

A representation of $\mathfrak{gl}(V)=V\otimes V^{\ast }$ is a linear map $\mu :\mathfrak{gl}(V)\otimes M\to M$ satisfying a certain identity. By currying, giving a linear map $\mu$ is equivalent to giving a linear map $a:V\otimes M\to V\otimes M$, and one can translate the condition for $\mu$ to be a representation into a condition on $a$. This alternate formulation does not use the dual of $V$ and makes sense for any object $V$ in a tensor category $\mathcal{C}$. We call such objects representations of the curried general linear algebra on $V$. The currying process can be carried out for many algebras built out of a vector space and its dual, and we examine several cases in detail. We show that many well-known combinatorial categories are equivalent to the curried forms of familiar Lie algebras in the tensor category of linear species; for example, the titular Brauer category is the curried form of the symplectic Lie algebra. This perspective puts these categories in a new light, has some technical applications, and suggests new directions to explore.