Abstract
Let H be an algebraic group scheme over a field k acting on a commutative k-algebra A which is a unique factorisation domain. We show that, under certain mild assumptions, the monoid of nonzero H-stable principal ideals in A is free commutative. From this we deduce, in certain special cases, results about the monoid of nonzero semi-invariants and the algebra of invariants. We use an infinitesimal method which allows us to work over an arbitrary base field.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Benson, D.J.: Polynomial invariants of finite groups. In: London Mathematical Society Lecture Note Series, vol. 190. Cambridge University Press, Cambridge (1993)
Borel, A.: Linear Algebraic Groups, 2nd edn. Graduate Texts in Mathematics, vol. 126. Springer, New York (1991)
Bourbaki N.: Algèbre Commutative, Chap. 5–7. Hermann, Paris (1975)
Demazure M., Gabriel P.: Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs. Masson and Cie, Éditeur, Paris. North-Holland Publishing Co., Amsterdam (1970)
Donkin S.: Infinitesimal invariants of algebraic groups. J. Lond. Math. Soc. (2) 45(3), 481–490 (1992)
Grosshans, F.D.: Algebraic homogeneous spaces and invariant theory. In: Lecture Notes in Mathematics, vol. 1673. Springer, Berlin (1997)
Jantzen J.C.: Representations of Algebraic Groups, 2nd ed. American Mathematical Society, Providence (2003)
Jantzen, J.C.: Representations of Lie algebras in prime characteristic, notes by Iain Gordon, pp. 185–235 in: A. Broer (ed.) Representation theories and algebraic geometry, Proceedings Montreal 1977 (NATO ASI Series C, vol. 514), Dordrecht etc (Kluwer) (1998)
Kac V., Weisfeiler B.: Coadjoint action of a semi-simple algebraic group and the center of the envelo** algebra in characteristic p. Indag. Math. 38(2), 136–151 (1976)
Popov V.L.: Contractions of actions of reductive algebraic groups. Math. USSR-Sb. 58(2), 311–335 (1987)
Premet A.A., Tange R.H.: Zassenhaus varieties of general linear Lie algebras. J. Algebra 294(1), 177–195 (2005)
Tange R.H.: Infinitesimal invariants in a function algebra. Can. J. Math. 61(4), 950–960 (2009)
Tange R.H.: The Zassenhaus variety of a reductive Lie algebra in positive characteristic. Adv. Math. 224, 340–354 (2010)