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Infinitesimal Invariants in a Function Algebra

Published online by Cambridge University Press:  20 November 2018

Rudolf Tange
Rudolf Tange*
Affiliation:
Department of Mathematics, University of York, York, YO10 5DD, UK, e-mail: [email protected]
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Abstract

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Let $G$ be a reductive connected linear algebraic group over an algebraically closed field of positive characteristic and let $\mathfrak{g}$ be its Lie algebra. First we extend a well-known result about the Picard group of a semi-simple group to reductive groups. Then we prove that if the derived group is simply connected and $\mathfrak{g}$ satisfies a mild condition, the algebra $K{{[G]}^{\mathfrak{g}}}$ of regular functions on $G$ that are invariant under the action of $\mathfrak{g}$ derived from the conjugation action is a unique factorisation domain.

Keywords

20G1513F15
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 61 , Issue 4 , 01 August 2009 , pp. 950 - 960
Copyright
Copyright © Canadian Mathematical Society 2012

References

[1] Borel, A., Linear Algebraic Groups. Second edition. Graduate Texts in Mathematics 126, Springer-Verlag, New York, 1991.Google Scholar
[2] Bourbaki, N., Groupes et Algèbres de Lie. Chaps. 4–6, Hermann, Paris, 1968.Google Scholar
[3] Brown, K. A. and Gordon, I., The ramification of centres: Lie algebras in positive characteristic and quantised enveloping algebras. Math. Z. 238(2001), no. 4, 733–779.Google Scholar
[4] Séminaire C., Chevalley 1956–1958, Classification des groupes de Lie algébriques. Vol. 2, Exposé 15.Google Scholar
[5] Donkin, S., Infinitesimal invariants of algebraic groups. J. London Math. Soc. 45(1992), no. 3, 481–490.Google Scholar
[6] Fossum, R. and Iversen, B., On Picard groups of algebraic fibre spaces. J. Pure Appl. Algebra 3(1973), 269–280.Google Scholar
[7] Friedlander, E. and Parshall, B. J., Rational actions associated to the adjoint representation. Ann. Sci. É cole Norm. Sup. 20(1987), no. 2, 215–226.Google Scholar
[8] Iversen, B., The geometry of algebraic groups. Advances in Math. 20(1976), no. 1, 57–85.Google Scholar
[9] Jantzen, J. C., Representations of Algebraic Groups. Pure and Applied Mathematics 131. Academic Press, Boston, MA, 1987.Google Scholar
[10] Marlin, R., Anneaux de Grothendieck des variétés de drapeaux. Bull. Soc. Math. France 104(1976), no. 4, 337–348.Google Scholar
[11] Popov, V. L., Picard groups of homogeneous spaces of linear algebraic groups and one-dimensional homogeneous vector fiberings. Izv. Akad. Nauk SSSR Ser. Mat. 38(1974), 294–322 (Russian); Math. USSR. Izv. 8(1974), no. 2, 301–327 (English translation).Google Scholar
[12] Premet, A. A. and Tange, R. H., Zassenhaus varieties of general linear Lie algebras. J. Algebra 294(2005), no. 1, 177–195.Google Scholar
[13] Steinberg, R., Automorphisms of classical Lie algebras. Pacific J. Math. 11(1961), 1119–1129.Google Scholar
[14] Steinberg, R., Lectures on Chevalley Groups. Notes prepared by John Faulkner and Robert Wilson. Yale University, New Haven, CN, 1968.Google Scholar
[15] Tange, R. H., The centre of quantum sln at a root of unity. J. Algebra 301(2006), no. 1, 425–445.Google Scholar
[16] Tange, R. H., The symplectic ideal and a double centraliser theorem. J. Lond. Math. Soc. 77(2008), no. 3, 687–699.Google Scholar