Koszul categories of $sl(\infty)$-, $o(\infty)$-, $sp(\infty)$-modules

Various categories of integrable modules over the classical infinite- dimensional Lie algebras $sl(\infty), o(\infty), sp(\infty)$ have been studied in recent years. In this talk we construct a category based on the mixed tensor algebra (the tensor algebra of the natural and conatural modules). The indecomposable injectives in the category turn out to be simply indecomposable direct summands in the tensor algebra. In addition, we show that this category is antiequivalent to the category of locally unitary finite-dimensional modules over a Koszul algebra. This yields a remarkable equivalence of the corresponding categories for $o(\infty)$ and $sp(\infty)$. The talk is based on joint work with Elizabeth Dan-Cohan and Vera Serganova.