The homogenized enveloping Algebra of the Lie Algebra sl(2,C)

TL;DR

This paper investigates the structure and representation theory of the homogenized enveloping algebra of sl(2,C), establishing its Koszul and Artin-Schelter regularity, and exploring its modules and duality properties.

Contribution

It introduces the homogenized algebra B of U(sl(2,C)), proves its Koszul and Artin-Schelter regularity, and characterizes its modules and duality structures.

Findings

B is Koszul and Artin-Schelter regular of global dimension four.

Homogenized Verma modules are Koszul of projective dimension two.

Identifies the structure of the Yoneda algebra B! and its modules.

Abstract

In this paper we study the homogenized algebra $B$ of the enveloping algebra $U$ of the Lie algebra sl(2,C). We look first to connections between the category of graded left $B$- modules and the category of $U$-modules, then we prove $B$ is Koszul and Artin-Schelter regular of global dimension four, hence its Yoneda algebra $% B^{!}$ is selfinjective of radical five zero, the structure of $B^{!}$ is given. We describe next the category of homogenized Verma modules, which correspond to the lifting to $B$ of the usual Verma modules over $U$, and prove that such modules are Koszul of projective dimension two. It was proved in [MZ] that all graded stable components of a selfinjective Koszul algebra are of type $ZA_{\infty}$, we characterize here the graded $B^{!}$% -modules corresponding under Koszul duality to the homogenized Verma modules, and prove that they are located at the mouth of a regular component, in this way we obtain a family of components over a wild algebra indexed by C. The paper ends with the description of a family of weight modules over $B$ which corresponds to the weight modules over $U$ and with a description of the category of $B$ - modules corresponding to the Gelfand's category $% \mathcal{O}$ of $U$-modules .