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.

Key Words:

• Cyclic modules
• Depth algebra
• Deep matrix algebra
• Direct limit of modules
• Directly finite module
• Leavitt algebra

2000 Mathematics Subject Classification:

• 17B10
• 17B65

ACKNOWLEDGMENTS

The author would like to thank Gene Abrams and Darren Funk-Neubauer for pointing out the connection between deep matrices and Leavitt algebras. Additionally, the author would like to thank the referee for his very helpful suggestions.

Notes

Communicated by A. Elduque.