Representation categories of Mackey Lie algebras as universal monoidal categories

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Let K be an algebraically closed field of characteristic 0. We study a monoidal category Tα which is universal among all symmetric K-linear monoidal categories generated by two objects A and B such that A has a, possibly transfinite, filtration. We construct Tα as a category of representations of the Lie algebra glᵐ(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 injective objects of Tα and prove that the category Tα is Koszul. We pay special attention to the case where the filtration on A is finite. In this case α=ₜ for t≥ ₀.