THE HOMOGENISED ENVELOPING ALGEBRA OF THE LIE ALGEBRA sl(2,C)

MARTINEZ-VILLA, ROBERTO

Glasgow mathematical journal, Tome 56 (2014) no. 3, pp. 551-568

Résumé

In this paper, we study the homogenised 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 self-injective of radical five zeros, and the structure of B! is given. We describe next the category of homogenised 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 Martínez-Villa and Zacharia (Approximations with modules having linear resolutions, J. Algebra266(2) (2003), 671–697)] that all graded stable components of a self-injective Koszul algebra are of type ZA∞. Here, we characterise the graded B!-modules corresponding to the Koszul duality to homogenised Verma modules, and prove that these 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.

Détail

DOI : 10.1017/S0017089514000032

Mots-clés : Primary 16S30, 17B35, Secondary 17B10