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

Published online by Cambridge University Press:  22 August 2014

ROBERTO MARTINEZ-VILLA
ROBERTO MARTINEZ-VILLA*
Affiliation:
Centro de Ciencias Matematicas, UNAM Morelia, Michoacán, Mexico e-mails: mvilla@matmor.unam.mx
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Abstract

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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.

Keywords

Primary 16S3017B35Secondary 17B10
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 56 , Issue 3 , September 2014 , pp. 551 - 568
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

References

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