Representation categories of mackey lie algebras as universal monoidal categories

Let K be an algebraically closed field of characteristic 0. We study a monoidal category T _α 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 ⊗ B → 1. WeconstructT _α as a category of representations of the Lie algebra gl ^M (V _* ,V) consisting of endomorphisms of a fixed diagonalizable pairing V _* ⊗V → K of vector spaces V _* and V of dimension α. Here α is an arbitrary cardinal number. We describe explicitly the simple and the injec-tive objects of T _α and prove that the category T _α is Koszul. We pay special attention to the case where the filtration on A is finite. In this case α = N _t for t ∈ Z≥0.