Abstract
Let k be a commutative ring, G′⊃G finite affine algebraic k-groups, and H′⊃H the dual Hopfalgebras of the affine algebras of G′ resp. G. The main results of this paper are: (I) If k is semilocal (e.g. k a field) there is an H′-linear, H∥H′-colinear, unitary, augmented isomorphism H→H∥H′◯ H′, where H∥H′ is the coalgebra belonging to G/G′. (II) If the k-submodule of the fixelements of (H∥H′)* is isomorphic to k (e.g. k principal or semilocal), then H′⊃H is a Frobeniusextension of the second kind.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literatur
BIALYNICKI-BIRULA, A.: On the equivalence of integral representations of groups. Proc. Am. M. S. 26, 371–377 (1970)
BOURBAKI, N.: Algèbre commutative, Chap. 2. Actualités Sci. Indust. no. 1290. Paris: Hermann 1961.
DE MEYER, F. u. INGRAHAM, E.: Separable algebras over commutative rings. Lecture Notes in Mathematics 181. Berlin-Heidelberg-New York: Springer 1970.
DEMAZURE, M. u. GABRIEL, P.: Groupes algébriques. Tome 1. Paris-Amsterdam: Masson 1970.
CHASE S. u. SWEEDLER, M.: Hopf algebras and Galois theory. Lecture Notes in Mathematics 97. Berlin-Heidelberg-New York: Springer 1969.
HATTORI, A.: Rank element of a projective module. Nagoya Math. J. 25, 113–120 (1965)
KASCH, F.: Projektive Frobenius-Erweiterungen. Sitzungsberichte der Heidelberger Akademie der Wissenschaften, 89–109 (1960/61).
LARSON, R.: Characters of Hopf algebras. Journal of Alg. 17, 352–368 (1971)
LARSON, R.: Orders in Hopf algebras. Erscheint demnächst.
LARSON, R. u. SWEEDLER, M.: An associative orthogonal bilinear form for Hopf algebras. Am. J. Math. 91, 75–94 (1969).
MÜLLER, B.: Quasi-Frobeniuserweiterungen. Math. Z. 85, 345–368 (1964).
NAKAYAMA, T. u. TSUZUKU: On Frobenius extensions I. Nagoya Math. J. 17, 89–110 (1960).
OBERST, U. u. SCHNEIDER, H.-J.: Untergruppen formeller Gruppen von endlichem Index. Erscheint demnächst.
PAREIGIS, B.: Einige Bemerkungen über Frobeniuserweiterungen. Math. Ann. 153., 1–13 (1964)
PAREIGIS, B.: When Hopf algebras are Frobenius algebras. Journal of Alg. 18, 588–596 (1971)
SWEEDLER, M.: Hopf algebras. New York: Benjamin 1969.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Oberst, U., Schneider, HJ. Über Untergruppen Endlicher Algebraischer Gruppen. Manuscripta Math 8, 217–241 (1973). https://doi.org/10.1007/BF01297688
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01297688