A Koszul category of representations of finitary Lie algebras

Abstract

Abstract. We find for each simple finitary Lie algebra g a category Tg of integrable modules in which the tensor product of copies of the natural and conatural modules are injective. The objects in Tg 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 g. The category Tg 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 T o(∞) and T sp(∞) corresponding respectively to the orthogonal and symplectic finitary Lie algebras o(∞), sp(∞).

Stability Patterns in Representation Theory

“…We now give an application of Theorem 3.3.1: the computation of the Ext groups between simple objects of Rep(GL). This recovers [DPS,Corollary 6.5].…”

“…Later, in [DPS,Corollary 6.11], the equivalence was established at the level of abelian categories (ignoring the tensor structure). Our result is the common generalization of the two.…”

“…This result is closely related to [DPS,Corollary 5.2]. Specifically, we can think of (dwb) as a locally finite quiver with relations, and the path algebra of this quiver is isomorphic to the algebra …”

“…We say that a representation of GL(∞) is algebraic if it appears as a subquotient of a finite direct sum of the T n,m . This definition is somewhat ad hoc; see [DPS,Sections 3,4] for a more natural characterization of these representations. We denote by Rep(GL) the category of algebraic representations of GL(∞).…”

“…Classification of injectives. As an application of Theorem 3.2.11, we use 2.1.5 to obtain a description of the injective objects of Rep(GL), which recovers [DPS,Corollary 4.6].…”