A Koszul category of representations of finitary Lie algebras

Abstract

We find for each simple finitary Lie algebra $\mathfrak{g}$ a category ${\mathbb{T}}_{\mathfrak{g}}$ of integrable modules in which the tensor product of copies of the natural and conatural modules are injective. The objects in ${\mathbb{T}}_{\mathfrak{g}}$ 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 $\mathfrak{g}$. The category ${\mathbb{T}}_{\mathfrak{g}}$ 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 ${\mathbb{T}}_{o(\infty )}$ and ${\mathbb{T}}_{sp(\infty )}$ corresponding respectively to the orthogonal and symplectic finitary Lie algebras $o(\infty )$, $sp(\infty )$.

22 pages

References in corpus (2)

• Cartan subalgebras of root-reductive Lie algebras
• Categories of integrable $sl(\infty )$-, $o(\infty )$-, $sp(\infty )$-modules

Cited by in corpus (7)

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• Notes on Restricted Inverse Limits of Categories
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• Annihilators of highest weight $\mathfrak{sl}(\infty )$-modules
• On ideals in U$(\mathfrak{sl}(\infty ))$, U$(\mathfrak{o}(\infty ))$, U$(\mathfrak{sp}(\infty ))$
• Lie algebras of linear systems and their automorphisms