Representations of restricted Lie algebras and families of associative ℒ-algebras

1999 ◽  
Vol 1999 (507) ◽  
pp. 189-218 ◽  
Author(s):  
Alexander Premet ◽  
Serge Skryabin
Keyword(s):  
Linear Function ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Maximal Dimension ◽  
Closed Field ◽  
Enveloping Algebra ◽  
Restricted Lie Algebra ◽  
Restricted Lie Algebras ◽  
Support Varieties

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.

THE FOX PROBLEM FOR FREE RESTRICTED LIE ALGEBRAS

2008 ◽  
Vol 18 (02) ◽  
pp. 271-283 ◽  
Author(s):  
HAMID USEFI
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Free Groups ◽  
Enveloping Algebra ◽  
Restricted Lie Algebra ◽  
Restricted Lie Algebras

Let L be a free restricted Lie algebra and R a restricted ideal of L. Denote by u(L) the restricted enveloping algebra of L and by ω(L) the associative ideal of u(L) generated by L. The purpose of this paper is to identify the subalgebra R ∩ ωn(L)ω(R) in terms of R only. This problem is the analogue of the Fox problem for free groups.

SCHEMES OF TORI AND THE STRUCTURE OF TAME RESTRICTED LIE ALGEBRAS

2001 ◽  
Vol 63 (3) ◽  
pp. 553-570 ◽  
Author(s):  
ROLF FARNSTEINER ◽  
DETLEF VOIGT
Keyword(s):  
Representation Theory ◽  
Lie Algebras ◽  
Classical Case ◽  
Forthcoming Paper ◽  
Structure Theory ◽  
Group Schemes ◽  
Restricted Lie Algebra ◽  
Restricted Lie Algebras ◽  
Geometric Techniques ◽  
Support Varieties

Much of the recent progress in the representation theory of infinitesimal group schemes rests on the application of algebro-geometric techniques related to the notion of cohomological support varieties (cf. [6, 8–10]). The noncohomological characterization of these varieties via the so-called rank varieties (see [21, 22]) involves schemes of additive subgroups that are the infinitesimal counterparts of the elementary abelian groups. In this note we introduce another geometric tool by considering schemes of tori of restricted Lie algebras. Our interest in these derives from the study of infinitesimal groups of tame representation type, whose determination [12] necessitates the results to be presented in §4 and §5 as well as techniques from abstract representation theory.In contrast to the classical case of complex Lie algebras, the information on the structure of a restricted Lie algebra that can be extracted from its root systems is highly sensitive to the choice of the underlying maximal torus. Schemes of tori obviate this defect by allowing us to study algebraic families of root spaces. Accordingly, these schemes may also shed new light on various aspects of the structure theory of restricted Lie algebras. We intend to pursue these questions in a forthcoming paper [13], and focus here on first applications within representation theory.

Restricted Lie Algebras and Their Envelopes

1995 ◽  
Vol 47 (1) ◽  
pp. 146-164 ◽  
Author(s):  
D. M. Riley ◽  
A. Shalev
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Codimension ◽  
Augmentation Ideal ◽  
Nilpotent Ideal ◽  
Enveloping Algebra ◽  
Group Of Units ◽  
Restricted Lie Algebra ◽  
Restricted Lie Algebras ◽  
Algebra Over A Field

AbstractLet L be a restricted Lie algebra over a field of characteristic p. Denote by u(L) its restricted enveloping algebra and by ωu(L) the augmentation ideal of u(L). We give an explicit description for the dimension subalgebras of L, namely those ideals of L defined by Dn(L) - L∩ωu(L)n for each n ≥ 1. Using this expression we describe the nilpotence index of ωU(L). We also give a precise characterisation of those L for which ωu(L) is a residually nilpotent ideal. In this case we show that the minimal number of elements required to generate an arbitrary ideal of u(L) is finitely bounded if and only if L contains a 1-generated restricted subalgebra of finite codimension. Subsequently we examine certain analogous aspects of the Lie structure of u(L). In particular we characterise L for which u(L) is residually nilpotent when considered as a Lie algebra, and give a formula for the Lie nilpotence index of u(L). This formula is then used to describe the nilpotence class of the group of units of u(L).

scholarly journals The nilpotent variety of W(1;n)p is irreducible

2019 ◽  
Vol 18 (03) ◽  
pp. 1950056
Author(s):  
Cong Chen
Keyword(s):  
Large Class ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Algebraic Groups ◽  
Closed Field ◽  
Nilpotent Variety ◽  
Restricted Lie Algebra ◽  
Finite Dimensional ◽  
Restricted Lie Algebras ◽  
Minimal Formula

In the late 1980s, Premet conjectured that the nilpotent variety of any finite dimensional restricted Lie algebra over an algebraically closed field of characteristic [Formula: see text] is irreducible. This conjecture remains open, but it is known to hold for a large class of simple restricted Lie algebras, e.g. for Lie algebras of connected algebraic groups, and for Cartan series [Formula: see text] and [Formula: see text]. In this paper, with the assumption that [Formula: see text], we confirm this conjecture for the minimal [Formula: see text]-envelope [Formula: see text] of the Zassenhaus algebra [Formula: see text] for all [Formula: see text].

scholarly journals Nilpotent orbits of exceptional Lie algebras over algebraically closed fields of bad characteristic

1985 ◽  
Vol 38 (3) ◽  
pp. 330-350 ◽  
Author(s):  
D. F. Holt ◽  
N. Spaltenstein
Keyword(s):  
Finite Field ◽  
Lie Algebras ◽  
Closed Field ◽  
Nilpotent Orbits ◽  
Reductive Algebraic Group ◽  
Exceptional Lie Algebras ◽  
Closed Fields ◽  
Characteristic 3 ◽  
Algebraically Closed Fields

AbstractThe classification of the nilpotent orbits in the Lie algebra of a reductive algebraic group (over an algebraically closed field) is given in all the cases where it was not previously known (E7 and E8 in bad characteristic, F4 in characteristic 3). The paper exploits the tight relation with the corresponding situation over a finite field. A computer is used to study this case for suitable choices of the finite field.

Finite group schemes of p-rank ≤ 1

2017 ◽  
Vol 166 (2) ◽  
pp. 297-323
Author(s):  
HAO CHANG ◽  
ROLF FARNSTEINER
Keyword(s):  
Finite Group ◽  
Lie Algebras ◽  
Group Scheme ◽  
Closed Field ◽  
Difficult Case ◽  
Modular Representation Theory ◽  
Group Schemes ◽  
Restricted Lie Algebras ◽  
A Finite Group ◽  
Finite Group Schemes

AbstractLet be a finite group scheme over an algebraically closed field k of characteristic char(k) = p ≥ 3. In generalisation of the familiar notion from the modular representation theory of finite groups, we define the p-rank rkp() of and determine the structure of those group schemes of p-rank 1, whose linearly reductive radical is trivial. The most difficult case concerns infinitesimal groups of height 1, which correspond to restricted Lie algebras. Our results show that group schemes of p-rank ≤ 1 are closely related to those being of finite or domestic representation type.

Note on Primitive Ideals of Enveloping Algebras in Prime Characteristic

2011 ◽  
Vol 18 (04) ◽  
pp. 701-708
Author(s):  
Yufeng Yao
Keyword(s):  
Maximal Dimension ◽  
Closed Field ◽  
Central Character ◽  
Characteristic P ◽  
Reductive Algebraic Group ◽  
Restricted Lie Algebra ◽  
Primitive Ideals ◽  
Alternative Description ◽  
Stable Ideal ◽  
The Ideal

Let [Formula: see text] be a restricted Lie algebra over an algebraically closed field F of characteristic p > 0, [Formula: see text] the center of the universal enveloping algebra [Formula: see text] of [Formula: see text]. In this note, we study primitive ideals of [Formula: see text]. The following results are included: (1) The ideal of [Formula: see text] generated by the central character ideal associated with any irreducible [Formula: see text]-module has finite co-dimension in [Formula: see text]. Furthermore, the co-dimension is no less than [Formula: see text], where [Formula: see text] is the maximal dimension of irreducible [Formula: see text]-modules. (2) Each annihilator ideal of irreducible [Formula: see text]-modules of maximal dimension is generated by the corresponding central character ideal in [Formula: see text]. (3) Each G-stable ideal in [Formula: see text] for [Formula: see text] contains nonzero fixed points under the action of G, where G is a connected reductive algebraic group. Additionally, the arguments on ideals help us to give an alternative description of the Azumaya locus in the Zassenhaus variety without using the normality of the Zassenhaus variety.

scholarly journals Subalgebras of free restricted Lie algebras

2005 ◽  
Vol 72 (1) ◽  
pp. 147-156 ◽  
Author(s):  
R.M. Bryant ◽  
L.G. Kovács ◽  
Ralph Stöhr
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Main Step ◽  
Free Lie Algebra ◽  
Generating Set ◽  
Restricted Lie Algebra ◽  
Corrected Proof ◽  
Restricted Lie Algebras ◽  
Remarkable Result ◽  
Algebra Over A Field

A theorem independently due to A.I. Shirshov and E. Witt asserts that every subalgebra of a free Lie algebra (over a field) is free. The main step in Shirshov's proof is a little known but rather remarkable result: if a set of homogeneous elements in a free Lie algebra has the property that no element of it is contained in the subalgebra generated by the other elements, then this subset is a free generating set for the subalgebra it generates. Witt also proved that every subalgebra of a free restricted Lie algebra is free. Later G.P. Kukin gave a proof of this theorem in which he adapted Shirshov's argument. The main step is similar, but it has come to light that its proof contains substantial gaps. Here we give a corrected proof of this main step in order to justify its applications elsewhere.

scholarly journals The Existence of Affine Structures on the Borel Subalgebra of Dimension 6

2021 ◽  
Vol 12 (1) ◽  
pp. 45-52
Author(s):  
Edi Kurniadi ◽  
Ema Carnia ◽  
Herlina Napitupulu
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Borel Subalgebra ◽  
Future Research ◽  
Axiomatic Method ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Isomorphism Classes ◽  
Affine Structures

The notion of affine structures arises in many fields of mathematics, including convex homogeneous cones, vertex algebras, and affine manifolds. On the other hand, it is well known that Frobenius Lie algebras correspond to the research of homogeneous domains. Moreover, there are 16 isomorphism classes of 6-dimensional Frobenius Lie algebras over an algebraically closed field. The research studied the affine structures for the 6-dimensional Borel subalgebra of a simple Lie algebra. The Borel subalgebra was isomorphic to the first class of Csikós and Verhóczki’s classification of the Frobenius Lie algebras of dimension 6 over an algebraically closed field. The main purpose was to prove that the Borel subalgebra of dimension 6 was equipped with incomplete affine structures. To achieve the purpose, the axiomatic method was considered by studying some important notions corresponding to affine structures and their completeness, Borel subalgebras, and Frobenius Lie algebras. A chosen Frobenius functional of the Borel subalgebra helped to determine the affine structure formulas well. The result shows that the Borel subalgebra of dimension 6 has affine structures which are not complete. Furthermore, the research also gives explicit formulas of affine structures. For future research, another isomorphism class of 6-dimensional Frobenius Lie algebra still needs to be investigated whether it has complete affine structures or not.

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