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Let .g; OEp/ be a finite-dimensional restricted Lie algebra, defined over an algebraically closed field k of characteristic p > 0. The scheme of tori of maximal dimension of g gives rise to a finite group S.g/ that coincides with the Weyl group of g in case g is a Lie algebra of classical type. In this paper, we compute the group S.g/ for Lie algebras of Cartan type and provide applications concerning weight space decompositions, the existence of generic tori and polynomial invariants.

Abstract.By using an approach to the theory of Frobenius extensions that emphasizes notions related to associative forms, we obtain results concerning the trace and corestriction mappings and transitivity. These are employed to show that the extension of enveloping algebras determined by a subalgebra of a Lie superalgebra is a Frobenius extension, and to study certain questions in representation theory. IntroductionThe theory of Frobenius extensions, initiated independently by Kasch [10] and Nakayama-Tsuzuku [13], has, aside from its intrinsic value, proven to be a useful tool in the study of groups and modular Lie algebras. The twofold purpose of this note is to extend the abstract theory as well as to point out another application concerning the theory of Lie superalgebras.In § 1 we reformulate some of the basic features of the theory of Frobenius extensions and generalize various results of [2,15], including results on transitivity of Frobenius extensions. Our approach, which emphasizes associative forms, is particularly useful for the study of the trace and corestriction mappings. The second section is devoted to the investigation of extensions defined by the universal enveloping algebras of a Lie superalgebra L and a subalgebra K containing the space Lo of even elements. Our main result, namely that their respective enveloping algebras define a Frobenius extension, strikingly parallels recent work on restricted Lie algebras (cf. [3]).The irreducible modules of the so-called basic classical Lie superalgebras are obtained, according to [8], by inducing from representations of certain subalgebras. By showing that some of the essential features of this theory can be derived from the general principles of the theory of Frobenius extensions, we provide a new conceptual approach to various well-known results.Frobenius extensions can in particular be used to investigate the cohomology theory of the defining rings. It is, for instance, possible to introduce a complete relative cohomology theory that generalizes well-known principles from the theory of groups (cf. [4,11]). We shall not dwell on these aspects here, but rather

Abstract.By using an approach to the theory of Frobenius extensions that emphasizes notions related to associative forms, we obtain results concerning the trace and corestriction mappings and transitivity. These are employed to show that the extension of enveloping algebras determined by a subalgebra of a Lie superalgebra is a Frobenius extension, and to study certain questions in representation theory. IntroductionThe theory of Frobenius extensions, initiated independently by Kasch [10] and Nakayama-Tsuzuku [13], has, aside from its intrinsic value, proven to be a useful tool in the study of groups and modular Lie algebras. The twofold purpose of this note is to extend the abstract theory as well as to point out another application concerning the theory of Lie superalgebras.In § 1 we reformulate some of the basic features of the theory of Frobenius extensions and generalize various results of [2,15], including results on transitivity of Frobenius extensions. Our approach, which emphasizes associative forms, is particularly useful for the study of the trace and corestriction mappings. The second section is devoted to the investigation of extensions defined by the universal enveloping algebras of a Lie superalgebra L and a subalgebra K containing the space Lo of even elements. Our main result, namely that their respective enveloping algebras define a Frobenius extension, strikingly parallels recent work on restricted Lie algebras (cf. [3]).The irreducible modules of the so-called basic classical Lie superalgebras are obtained, according to [8], by inducing from representations of certain subalgebras. By showing that some of the essential features of this theory can be derived from the general principles of the theory of Frobenius extensions, we provide a new conceptual approach to various well-known results.Frobenius extensions can in particular be used to investigate the cohomology theory of the defining rings. It is, for instance, possible to introduce a complete relative cohomology theory that generalizes well-known principles from the theory of groups (cf. [4,11]). We shall not dwell on these aspects here, but rather

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