A Koszul category of representations of finitary Lie algebras

We find for each simple finitary Lie algebra gg a category TgTg of integrable modules in which the tensor product of copies of the natural and conatural modules is injective. The objects in TgTg can be defined as the finite length absolute weight modules, where by absolute weight module we mean a module which is a weight module for every splitting Cartan subalgebra of gg. The category TgTg is Koszul in the sense that it is antiequivalent to the category of locally unitary finite-dimensional modules over a certain direct limit of finite-dimensional Koszul algebras. We describe these finite-dimensional algebras explicitly. We also prove an equivalence of the categories To(∞)To(∞) and Tsp(∞)Tsp(∞) corresponding respectively to the orthogonal and symplectic finitary Lie algebras o(∞)o(∞), sp(∞)sp(∞).