Representation categories of Mackey Lie algebras as universal monoidal categories

摘要

Let K be an algebraically closed field of characteristic 0. We study a monoidal category T-alpha which is universal among all symmetric K-linear monoidal categories generated by two objects A and B such that A is equipped with a possibly transfinite filtration together with a pairing A circle times B - 1. We construct T-alpha as a category of representations of the Lie algebra gl(M) (V-*, V) consisting of endomorphisms of a fixed diagonalizable pairing V-*circle times V - K of vector spaces V-* and V of dimension alpha. Here alpha is an arbitrary cardinal number. We describe explicitly the simple and the injective objects of T-alpha and prove that the category T-alpha is Koszul. We pay special attention to the case where the filtration on A is finite. In this case alpha = aleph(t) for t is an element of Z(= 0).