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Simple Lie algebras of small characteristic VI. Completion of the classification, in preparation
"... Abstract. Let L be a finitedimensional simple Lie algebra over an algebraically closed field F of characteristic p> 3. We prove in this paper that if for every torus T of maximal dimension in the penvelope of ad L in Der L the centralizer of T in ad L acts triangulably on L, then L is either class ..."
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Cited by 17 (3 self)
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Abstract. Let L be a finitedimensional simple Lie algebra over an algebraically closed field F of characteristic p> 3. We prove in this paper that if for every torus T of maximal dimension in the penvelope of ad L in Der L the centralizer of T in ad L acts triangulably on L, then L is either classical or of Cartan type. As a consequence we obtain that any finitedimensional simple Lie algebra over an algebraically closed field of characteristic p> 5 is either classical or of Cartan type. This settles the last remaining case of the generalized KostrikinShafarevich conjecture (the case where p = 7). 1. Introduction and
Brauertype reciprocity for a class of graded associative algebras
 J. Algebra
, 1991
"... Classical Brauer reciprocity can be stated roughly as follows: Let G be a finite group and let k be an algebraically closed field of characteristic p> 0. If S is a simple kGmodule and P (S) is its projective cover, then the multiplicity of a simple module L as a composition factor of a characterist ..."
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Cited by 12 (5 self)
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Classical Brauer reciprocity can be stated roughly as follows: Let G be a finite group and let k be an algebraically closed field of characteristic p> 0. If S is a simple kGmodule and P (S) is its projective cover, then the multiplicity of a simple module L as a composition factor of a characteristic zero lift of P (S) is the same as the multiplicity of S as a composition factor of a modulo p reduction of L. (All modules in this paper are assumed to be finite dimensional.) One thinks of the module L as playing an intermediate role between P (S) and S. Reciprocities similar to Brauer reciprocity (here called“Brauertype reciprocities”) have subsequently been found to occur in many other settings. For instance, if g is a classical (modular) Lie algebra (definition given in §2), then there exists a set Z of gmodules with the following property: Given a simple gmodule S, the projective cover P (S) of S has a filtration with each successive quotient (isomorphic to a module) in Z and for each such filtration, the number of times Z ∈ Z occurs is the same as the multiplicity of S as a composition factor of Z. This was proved for g of type A1 by Pollack ([10]) and for arbitrary g by Humphreys ([3]). (The reciprocity in this setting is often called “Humphreys reciprocity.”) Inspired by Humphreys ’ result, Bernstein, Gelfand and Gelfand sought and found a Brauertype reciprocity
On Cocommutative Hopf Algebras Of Finite Representation Type
 Universitat Bielefeld, SFB Preprint
"... Let G be a finite algebraic group, defined over an algebraically closed field k of characteristic p ? 0. Such a group decomposes into a semidirect product G = G 0 \Theta G red with a constant group G red and a normal infinitesimal subgroup G 0 . If the principal block B 0 (G) of the group alge ..."
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Cited by 8 (6 self)
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Let G be a finite algebraic group, defined over an algebraically closed field k of characteristic p ? 0. Such a group decomposes into a semidirect product G = G 0 \Theta G red with a constant group G red and a normal infinitesimal subgroup G 0 . If the principal block B 0 (G) of the group algebra H(G) has finite representation type, then both constituents have the same property, with at least one of them being semisimple. We determine the structure of the infinitesimal constituent G 0 up to the classification of Vuniserial groups. 0. Introduction This paper is concerned with the representation theory of finitedimensional cocommutative Hopf algebras over algebraically closed fields of positive characteristic. As is wellknown, such an algebra can be viewed as the group algebra of a finite algebraic kgroup G. Special cases are the Hopf algebras associated to constant groups, i.e., the modular group algebras, as well as those of the infinitesimal groups of height 1, that is, ...
Geometric representation theory of restricted Lie algebras of classical type
 BEZRUKAVNIKOV, IVAN MIRKOVIĆ, AND DMITRIY RUMYNIN
"... Abstract. We modify the Hochschild ϕmap to construct central extensions of a restricted Lie algebra. Such central extension gives rise to a group scheme which leads to a geometric construction of unrestricted representations. For a classical semisimple Lie algebra, we construct equivariant line bun ..."
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Abstract. We modify the Hochschild ϕmap to construct central extensions of a restricted Lie algebra. Such central extension gives rise to a group scheme which leads to a geometric construction of unrestricted representations. For a classical semisimple Lie algebra, we construct equivariant line bundles whose global sections afford representations with a nilpotent pcharacter. Let G be a connected simply connected semisimple algebraic group over an algebraically closed field K of characteristic p and g be its Lie algebra. The representation theory of g is connected with the coadjoint orbits through the notion of a pcharacter [27, 3, 14, 10]. An irreducible representation ρ is finitedimensional and determines a pcharacter χ ∈ g ∗ by χ(x) p Id = ρ(x) p −ρ(x [p] ) for each x ∈ g [27]. There are indications that a geometry stands behind this representation theory, for instance, the KacWeisfeiler conjecture proved by Premet [21]. This work has been motivated by an idea of Humphreys that the representations affording χ should be related to the Springer fiber B χ. Some of our intuition comes from algebraic calculations of Jantzen [12, 13]. The most interesting evidence for the relation between Springer fibers and representations of g is now given by Lusztig [17]. The main goal of this paper is to introduce a method for constructing unrestricted representations of g by taking global sections of line bundles on infinitesimal neighborhoods of certain subvarieties of B χ. A more general approach implementing twisted sheaves of crystalline differential operators will be explained elsewhere. An attempt to study representations of g with a single pcharacter χ has led to the notion of a reduced enveloping algebra. We modify this approach by considering a set of p different pcharacters {0, χ, 2χ,...,
Quantizations of generalizedWitt algebra and of JacobsonWitt algebra in modular case
, 2006
"... Xiuling Wang ..."
DEFORMATIONS OF RESTRICTED SIMPLE LIE ALGEBRAS II
, 2007
"... Abstract. We compute the infinitesimal deformations of two families of restricted simple modular Lie algebras of Cartantype: the Contact and the Hamiltonian Lie algebras. 1. ..."
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Cited by 6 (5 self)
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Abstract. We compute the infinitesimal deformations of two families of restricted simple modular Lie algebras of Cartantype: the Contact and the Hamiltonian Lie algebras. 1.
HopfGalois extensions with central invariants and their geometric properties
 Algebr. Represent. Theory
, 1998
"... We study a class of algebra extensions which usually appear in the study of restricted Lie algebras or various quantum objects at roots of unity. The present paper was inspired by the theory of nonrestricted representations of restricted Lie algebras and the theory of quantum groups at roots of unit ..."
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Cited by 6 (2 self)
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We study a class of algebra extensions which usually appear in the study of restricted Lie algebras or various quantum objects at roots of unity. The present paper was inspired by the theory of nonrestricted representations of restricted Lie algebras and the theory of quantum groups at roots of unity where algebras are usually finitely generated modules over their centers. Our objective is to demonstrate that all these theories admit a unifying approach. We use the concept of a HopfGalois extension with central invariants to treat these phenomena from a general point of view. We discuss extensions of algebras in Section 1. We define HopfGalois extensions with central invariants early in Section 2. Then we study them locally: i.e. we consider localizations at points of the prime spectrum of the subalgebra of invariants. The localizations form a vector bundle of not necessarily commutative algebras on a scheme. The fibers of the bundle are finitedimensional Frobenius algebras and their irreducible representations coincide with irreducible representations of the HopfGalois extension we study. We describe some known examples in which the developed formalism takes place in Section 3. It seems that further attempts to study these extensions should be based on exploring the inherent geometry of the situation. The author is grateful to J. Humphreys for fruitful discussions and the proofreading of the manuscript.
Engellike characterization of radicals in finite dimensional Lie algebras and finite groups
 Manuscripta Math
"... Abstract. A classical theorem of R. Baer describes the nilpotent radical of a finite group G as the set of all Engel elements, i.e. elements y ∈ G such that for any x ∈ G the nth commutator [x, y,...,y] equals 1 for n big enough. We obtain a characterization of the solvable radical of a finite dimen ..."
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Cited by 5 (4 self)
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Abstract. A classical theorem of R. Baer describes the nilpotent radical of a finite group G as the set of all Engel elements, i.e. elements y ∈ G such that for any x ∈ G the nth commutator [x, y,...,y] equals 1 for n big enough. We obtain a characterization of the solvable radical of a finite dimensional Lie algebra defined over a field of characteristic zero in similar terms. We suggest a conjectural description of the solvable radical of a finite group as the set of Engellike elements and reduce this conjecture to the case of a finite simple group. Contents
Representation Theory Of Noetherian Hopf Algebras Satisfying A Polynomial Identity
, 1997
"... . A class of Noetherian Hopf algebras satisfying a polynomial identity is axiomatised and studied. This class includes group algebras of abelianbyfinite groups, finite dimensional restricted Lie algebras, and quantised enveloping algebras and quantised function algebras at roots of unity. Some ..."
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Cited by 5 (3 self)
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. A class of Noetherian Hopf algebras satisfying a polynomial identity is axiomatised and studied. This class includes group algebras of abelianbyfinite groups, finite dimensional restricted Lie algebras, and quantised enveloping algebras and quantised function algebras at roots of unity. Some common homological and representationtheoretic features of these algebras are described, with some indications of recent and current developments in research on each of the exemplar classes. It is shown that the finite dimensional representation theory of each of these algebras H reduces to the study of a collection Alg(H) of (finite dimensional) Frobenius algebras. The properties of this family of finite dimensional algebras are shown to be intimately connected with geometrical features of central subHopf algebras of H. A number of open questions are listed throughout. 1. Introduction My aim in this paper is to review some common properties exhibited by four large and important cla...
Cartan invariants for the restricted toral rank two contact Lie algebra
 Indag. Math. (N.S
, 1994
"... Abstract. Restricted modules for the restricted toral rank two contact Lie algebra are considered. Contragredients of the simple modules, Cartan invariants, and dimensions of the simple modules and their projective covers are determined. Let L be a finite dimensional restricted Lie algebra. All Lmo ..."
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Cited by 5 (3 self)
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Abstract. Restricted modules for the restricted toral rank two contact Lie algebra are considered. Contragredients of the simple modules, Cartan invariants, and dimensions of the simple modules and their projective covers are determined. Let L be a finite dimensional restricted Lie algebra. All Lmodules in this paper are assumed to be left, restricted and finite dimensional over the defining field. Each simple Lmodule has a projective cover; the multiplicities of the composition factors of the various projective covers are called Cartan invariants. Here, we use the method of [3] to compute the Cartan invariants for the restricted toral rank two contact Lie algebra K(3, 1). To carry out the computation, one needs to know the simple modules and their multiplicities as composition factors of certain induced modules. In [4]–which considered restricted contact Lie algebras of arbitrary toral rank–it was shown that these multiplicities are generically one, that is, the induced modules are, with a few exceptions, simple (see 1.1 below). Although it is not known at the time of this writing, it is expected that (for arbitrary toral rank) the few exceptional induced modules will not be simple. At least this is the case for the algebra K(3, 1) as will be shown in this paper (see 6.1). In addition to the Cartan invariants for K(3, 1) we will compute the dimensions of the simple modules which will give in turn the dimensions of their projective covers. Also, we determine the contragredient of each simple module. I thank the referee for the improvement in 2.3(2) and its proof as well as for other useful comments. 1. Statement of Main Results Let F be an algebraically closed field of characteristic p> 2 and let n = 2r + 1 with r ∈ N. For 1 ≤ k ≤ n let εk be the ntuple with jth component δjk (Kronecker delta). Set A = {a =