Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-24T07:41:09.974Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

REFERENCES

1.Anick, D. J., On the homology of associative algebras, Trans. Am. Math. Soc. 296 (2) (1986), 641–659.Google Scholar
2.Auslander, M. and Reiten, I., Representation theory of artin algebras IV, invariants given by almost split sequences, Commun. Algebra 5 (5) (1977) 443–518.Google Scholar
3.Coutinho, S. C., A primer of algebraic D-modules, London Mathematical Society Students Texts, vol. 33 (Cambridge University Press, Cambridge, UK, 1995)Google Scholar
4.Dade, E. C., Group-graded rings and modules, Math. Z. 174 (3) (1980), 241–262.Google Scholar
5.Erdmann, K. and Wildon, M. J., Introduction to Lie algebras, Springer Undergraduate Mathematics Series (Springer, New York, NY, 2006).Google Scholar
6.Green, E. L. and Martínez Villa, R., Koszul and Yoneda algebras, in Representation theory of algebras, CMS Conference Proceedings, Cocoyoc, Mexico, 1994, vol. 18 (American Mathematical Society, Providence, RI, 1996), 247–297.Google Scholar
7.Green, E. L., and Martínez-Villa, R., Koszul and Yoneda algebras. II. Algebras and modules, II, CMS Conference Proceedings, Geiranger, Norway, 1996, vol. 24 (American Mathematical Society, Providence, RI, 1998), 227–244.Google Scholar
8.Le Bruyn, L. and Smith, S. P., Homogenised sl(2). Proc. Amer. Math. Soc. 118 (3) (1993).Google Scholar
9.Martinez-Villa, R. and Mondragon, J., On the homogenised Weyl algebra (preprint, 31 Oct. 2012), arXiv:1210.8207v1[math.RA].Google Scholar
10.Martinez-Villa, R. and Mondragon, J., Skew group algebras, invariants and Weyl algebras (preprint, 5 Nov. 2012), arXiv:1211.0981v1 [math.RA].Google Scholar
11.Martínez-Villa, R., and Zacharia, D., Approximations with modules having linear resolutions, J. Algebra 266 (2) (2003), 671–697.Google Scholar
12.Mazorchuk, V., Lectures on sℓ2(ℂ)-modules (Imperial College Press, London, 2010).Google Scholar
13.Miličić, D., Lectures on algebraic theory of D-modules (University of Utah, Salt Lake City, UT, 1986).Google Scholar
14.Popescu, N., Abelian categories with applications to rings and modules, LMS Monographs, vol. 3 (Academic Press, Waltham, MA, 1973).Google Scholar
15.Ringel, C. M., Cones, in Representation theory of algebras, CMS Conference Proceedings, Cocoyoc, Mexico, 1994, vol. 18 (American Mathematical Society Providence, RI, 1996), 583–586.Google Scholar
16.Smith, P. S., Some finite dimensional algebras related to elliptic curves, in Rep. theory of algebras and related topics, CMS Conference Proceedings, vol. 19 (American Mathematical Society, Providence, RI, 1996), 315–348.Google Scholar
17.Yamagata, K., Frobenius algebras, in Handbook of algebra, vol. 1 (North-Holland, Amsterdam, Netherlands, 1996), 841–887.Google Scholar