Representations of restricted Lie algebras and families of associative ℒ-algebras
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Alexander Premet
Abstract
Let ℒ be an n-dimensional restricted Lie algebra over an algebraically closed field K of characteristic p > 0. Given a linear function ξ on ℒ and a scalar λ ∈ K, we introduce an associative algebra Uξ,λ (ℒ) of dimension pn over K. The algebra Uξ,1 (ℒ) is isomorphic to the reduced enveloping algebra Uξ (ℒ), while the algebra Uξ,0 (ℒ) is nothing but the reduced symmetric algebra Sξ (ℒ). Deformation arguments (applied to this family of algebras) enable us to derive a number of results on dimensions of simple ℒ-modules. In particular, we give a new proof of the Kac-Weisfeiler conjecture (see [41], [35]) which uses neither support varieties nor the classification of nilpotent orbits, and compute the maximal dimension of simple ℒ-modules for all ℒ having a toral stabiliser of a linear function.
© Walter de Gruyter
Articles in the same Issue
- Approximate homogeneity is not a local property
- Gradient regularity for minimizers under general growth conditions
- On the mock Peter-Weyl theorem and the Drinfeld double of a double
- Simple inductive limit C*-algebras: Spectra and approximations by interval algebras
- On the K-theory of elliptic curves
- Polar representations and symmetric spaces
- On the divisor-sum problem for binary forms
- Block-compatible metaplectic cocycles
- Converse theorems for GLn, II
- Representations of restricted Lie algebras and families of associative ℒ-algebras
- A limit theorem for Bohr-Jessen's probability measures of the Riemann zeta-function
- Erratum to the paper: Terme constant des fonctions tempérées sur un espace symétrique réductif
Articles in the same Issue
- Approximate homogeneity is not a local property
- Gradient regularity for minimizers under general growth conditions
- On the mock Peter-Weyl theorem and the Drinfeld double of a double
- Simple inductive limit C*-algebras: Spectra and approximations by interval algebras
- On the K-theory of elliptic curves
- Polar representations and symmetric spaces
- On the divisor-sum problem for binary forms
- Block-compatible metaplectic cocycles
- Converse theorems for GLn, II
- Representations of restricted Lie algebras and families of associative ℒ-algebras
- A limit theorem for Bohr-Jessen's probability measures of the Riemann zeta-function
- Erratum to the paper: Terme constant des fonctions tempérées sur un espace symétrique réductif
