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KRULL DIMENSION OF AFFINOID ENVELOPING ALGEBRAS OF SEMISIMPLE LIE ALGEBRAS

Published online by Cambridge University Press:  01 October 2013

KONSTANTIN ARDAKOV  and
IAN GROJNOWSKI
KONSTANTIN ARDAKOV
Affiliation:
School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom
IAN GROJNOWSKI
Affiliation:
Department of Pure Mathematics and Mathematical Statistics (DPMMS), University of Cambridge, Cambridge CB3 0WB, United Kingdom
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Abstract

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Using Beilinson–Bernstein localisation, we give another proof of Levasseur's theorem on the Krull dimension of the enveloping algebra of a complex semisimple Lie algebra. The proof also extends to the case of affinoid enveloping algebras.

Keywords

14G2216S3032C38
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 55 , Issue A: New developments in Noncommutative Algebra and its applications. A conference in honour of Kenny Brown's and Toby Stafford's 60th birthdays. , October 2013 , pp. 7 - 26
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013 

References

REFERENCES

1.Ajitabh, K., Smith, S. P. and Zhang, J. J., Auslander–Gorenstein rings, Comm. Algebra 26 (7) (1998), 2159–2180.CrossRefGoogle Scholar
2.Ardakov, K. and Wadsley, S. J., On irreducible representations of compact p-adic analytic groups, preprint, Ann. Math. 178 (2) (2013), 453–557.CrossRefGoogle Scholar
3.Ardakov, K., Krull dimension of Iwasawa algebras, J. Algebra 280 (1) (2004), 190–206.Google Scholar
4.Beilinson, A. and Bernstein, J., Localisation de $\mathfrak{g}$-modules, C. R. Acad. Sci. Paris. Sér.I Math. 292 (1) (1991), 15–18.Google Scholar
5.Beilinson, A. and Bernstein, J., A proof of Jantzen conjectures. In Gel'fand Seminar, I. M., Advances in Soviet Mathematics, vol. 16 (American Mathematical Society, Providence, RI, 1993), 1–50.Google Scholar
6.Bernstein, J. and Lunts, V., A simple proof of Kostant's theorem that $U(\mathfrak{g})$ is free over its center, Amer. J. Math. 118 (5) (1996), 979–987.Google Scholar
7.Berthelot, P., $\mathscr{D}$-modules arithmétiques I. opèrateurs diff'erentiels de niveau fini, Ann. Sci. École Norm. Sup. (4) 29 (2) (1996), 185–272.Google Scholar
8.Bezrukanvikov, R., Mirkovic, I. and Rumynin, D., Localization of modules for a semisimple Lie algebra in prime characteristic, Ann. Math. 167 (3) (2008), 945–991.Google Scholar
9.Bezrukavnikov, R., Braverman, A. and Positselskii, L., Gluing of abelian categories and differential operators on the basic affine space, J. Inst. Math. Jussieu 1 (4) (2002), 543–557.CrossRefGoogle Scholar
10.Demazure, M., Invariants symétriques entiers des groupes de Weyl et torsion, Invent. Math. 21 (1973), 287–301.Google Scholar
11.Dixmier, J., Enveloping algebras, Graduate Studies in Mathematics, vol. 11 (American Mathematical Society, Providence, RI, English edition, 1996).Google Scholar
12.Grothendieck, A., Éléments de géométrie algébrique. I. Le langage des schémas, Inst. Hautes Études Sci. Publ. Math. 4 (1960), 228.Google Scholar
13.Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, vol. 52 (Springer-Verlag, New York, 1997).Google Scholar
14.Levasseur, T., Sur la dimension de Krull de l'algèbre enveloppante d'une algèbre de Lie semi-simple. In Paul Dubreil and Marie-Paule Malliavin Algebra Seminar, 34th Year (Paris, 1981), Lecture Notes in Mathematics, vol. 924 (Springer, Berlin, 1982), 173–183.Google Scholar
15.Levasseur, T., Krull dimension of the enveloping algebra of a semisimple Lie algebra, Proc. Amer. Math. Soc. 130 (12) (2002), 3519–3523 (electronic).Google Scholar
16.Levasseur, T. and Stafford, J. T.. Differential operators and cohomology groups on the basic affine space, Studies in Lie theory, Progress in Mathematics, vol. 243 (Birkhäuser/Boston, MA, 2006), 377–403.Google Scholar
17.Li, H. and Van Oystaeyen, F., Zariskian filtrations (Kluwer Academic Publishers, Dordrecht, the Netherlands, 1996).Google Scholar
18.McConnell, J. C. and Robson, J. C., Noncommutative Noetherian rings, Graduate Studies in Mathematics, vol. 30 (American Mathematical Society, Providence, RI, revised edition, 2001). With the cooperation of L. W. Small.Google Scholar
19.Schneider, P. and Teitelbaum, J., Algebras of p-adic distributions and admissible representations, Invent. Math. 153 (1) (2003), 145–196.Google Scholar
20.Smith, S. P., Krull dimension of the enveloping algebra of sl(2, C), J. Algebra 71 (1) (1981), 189–194.Google Scholar
21.Smith, S. P., Krull dimension of factor rings of the enveloping algebra of a semisimple Lie algebra, Math. Proc. Cambridge Philos. Soc. 93 (3) (1983), 459–466.CrossRefGoogle Scholar
22.Weibel, C. A., An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38 (Cambridge University Press, Cambridge, 1994).Google Scholar