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HIGHER DEFORMATIONS OF LIE ALGEBRA REPRESENTATIONS II

Part of: Lie algebras and Lie superalgebras Linear algebraic groups and related topics

Published online by Cambridge University Press:  02 June 2020

MATTHEW WESTAWAY Open the ORCID record for MATTHEW WESTAWAY [Opens in a new window]
MATTHEW WESTAWAY*
Affiliation:
School of Mathematics, University of Birmingham, Birmingham, B15 2TT, UK email [email protected]
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Abstract

Steinberg’s tensor product theorem shows that for semisimple algebraic groups, the study of irreducible representations of higher Frobenius kernels reduces to the study of irreducible representations of the first Frobenius kernel. In the preceding paper in this series, deforming the distribution algebra of a higher Frobenius kernel yielded a family of deformations called higher reduced enveloping algebras. In this paper, we prove that the Steinberg decomposition can be similarly deformed, allowing us to reduce representation theoretic questions about these algebras to questions about reduced enveloping algebras. We use this to derive structural results about modules over these algebras. Separately, we also show that many of the results in the preceding paper hold without an assumption of reductivity.

MSC classification

Primary: 17B10: Representations, algebraic theory (weights)
Secondary: 17B35: Universal enveloping (super)algebras 17B50: Modular Lie (super)algebras 20G05: Representation theory
Type
Article
Information
Nagoya Mathematical Journal , Volume 244 , December 2021 , pp. 232 - 255
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Copyright
© 2020 Foundation Nagoya Mathematical Journal

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References

Brown, K. and Goodearl, K., Homological aspects of Noetherian PI Hopf algebras or irreducible modules and maximum dimension , J. Algebra 198 (1997), 240–265.CrossRefGoogle Scholar
Brown, K. and Goodearl, K., Lectures on Algebraic Quantum Groups, Advanced Courses in Mathematics, CRM Barcelona, Birkäuser, Basel, 2002.CrossRefGoogle Scholar
Brown, K. and Gordon, I., The ramification of centres: Lie algebras in positive characteristic and quantised enveloping algebras , Math. Z. 238 (2001), 733–779.CrossRefGoogle Scholar
Chamberlain, S., Integral bases for the universal enveloping algebras of map algebras, Ph.D. thesis, University of California Riverside, 2011.Google Scholar
Friedlander, E. and Parshall, B., Modular representation theory of Lie algebras , Amer. J. Math. 110 (1988), 1055–1093.CrossRefGoogle Scholar
Jantzen, J., Representations of Algebraic Groups, Pure and Applied Mathematics 131 , Academic Press, Boston, MA, 1987.Google Scholar
Jantzen, J., “ Representations of Lie algebras in prime characteristic ”, in Proc. NATO ASI Representation Theory and Algebraic Geometry, Montreal (1997), (ed. Broer, A.) Kluwer, Dordrecht, 1998, 185–235.Google Scholar
Jantzen, J., “ Representations of Lie algebras in positive characteristic ”, in Representation Theory of Algebraic Groups and Quantum Groups, Adv. Stud. Pure Math. 40 , Math. Soc. Japan, Tokyo, 2004, 175–218.CrossRefGoogle Scholar
McConnell, J. and Robson, J., Noncommutative Noetherian Rings, Revised edition, Graduate Studies in Mathematics 30 , Amer. Math. Soc., Providence, RI, 2001.CrossRefGoogle Scholar
Montgomery, S., Hopf Algebras and their Actions on Rings, CBMS Regional Conference Series in Mathematics 82 , Amer. Math. Soc., Providence, RI, 1993.CrossRefGoogle Scholar
Premet, A., Irreducible representations of Lie algebras of reductive groups and the Kac–Weisfeiler conjecture , Invent. Math. 121 (1995), 79–117.CrossRefGoogle Scholar
Rowen, L., Polynomial Identities in Ring Theory, Pure and Applied Mathematics 84 , Academic Press, Harcourt Brace Jovanovich, New York, London, 1980.Google Scholar
Rowen, L., Ring Theory, Student edition, Academic Press, Boston, MA, 1991.Google Scholar
Schneider, H., Representation theory of Hopf Galois extensions , Israel J. Math. 72 (1990), 196–231.CrossRefGoogle Scholar
Steinberg, R., Representations of algebraic groups , Nagoya Math. J. 22 (1963), 33–56.CrossRefGoogle Scholar
Sweedler, M., Hopf algebras with one grouplike element , Trans. Amer. Math. Soc. 127 (1967), 515–526.Google Scholar
Westaway, M., Higher deformations of Lie algebra representations I, preprint, arXiv:1807.00660, to appear.Google Scholar
Witherspoon, S., Clifford correspondence for finite-dimensional Hopf algebras , J. Algebra 218 (1999), 608–620.CrossRefGoogle Scholar