DIRECTLY FINITE MODULES OF (sic)l2δ

In this article, we define and construct directly finite modules of sl(2) , an max graded infinite dimensional analogue of sl(2) 2 that arises naturally in deep matrix Lie algebras and their equivalent formulations ( *- algebras, Leavitt algebras). Constructing sl(2) as the direct limit ofsl(2)(k), 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 sl(2)(k) - module is cyclic. This is then used to determine all directly finite and cyclic sl(2) - modules. Lastly, we give explicit formulas for irreducible highest weight modules of sl(2) 2 .