Directly Finite Modules of 𝔰𝔩2𝔡

Abstract

In this article, we define and construct directly finite modules of 𝔰𝔩2𝔡, an (ℕ, max)-graded infinite dimensional analogue of 𝔰𝔩2 that arises naturally in deep matrix Lie algebras and their equivalent formulations (𝒞*-algebras, Leavitt algebras). Constructing 𝔰𝔩2𝔡 as the direct limit of , we use direct limits of modules to define directly finite modules, and give several results refining the direct limit construction. We determine necessary and sufficient conditions for when a -module is cyclic. This is then used to determine all directly finite and cyclic 𝔰𝔩2𝔡-modules. Lastly, we give explicit formulas for irreducible highest weight modules of 𝔰𝔩2𝔡.