The representation theory of Brauer categories II: Curried algebra

A representation of gl (V) =V V^ is a linear map gl (V) M M satisfying a certain identity. By currying, giving a linear map is equivalent to giving a linear map a V M V M, and one can translate the condition for 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 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.