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

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 .