A proof of the first Kac-Weisfeiler conjecture in large characteristics

Benjamin Martin, David I. Stewart (Corresponding Author), Lewis Topley, Akaki Tikaradze (Collaborator)

Research output: Contribution to journalArticle

3 Downloads (Pure)

Abstract

In 1971, Kac and Weisfeiler made two influential conjectures describing
the dimensions of simple modules of a restricted Lie algebra g. The first predicts the maximal dimension of simple g-modules and in this paper we apply the Lefschetz Principle and classical techniques from Lie theory to prove this conjecture for all restricted Lie subalgebras of glnpkq whenever k is an algebraically closed field of sufficiently large characteristic p (depending on n). As a consequence we deduce that the conjecture holds for the Lie algebra of an affine algebraic group scheme over any commutative ring, after specialising to an algebraically closed field of almost any characteristic.
In the appendix to this paper, written by Akaki Tikaradze, an alternative, short proof of the first Kac–Weisfeiler conjecture is given for the Lie algebra of group scheme over a finitely generated ring R Ď C, after base change to a field of large positive characteristic.
Original languageEnglish
Pages (from-to)278-293
Number of pages16
JournalRepresentation Theory
Volume23
Early online date16 Sep 2019
DOIs
Publication statusPublished - 2019

Fingerprint

Lie Algebra
Group Scheme
Algebraically closed
Base Change
Affine Group
Simple Module
Positive Characteristic
Algebraic Groups
Commutative Ring
Finitely Generated
Subalgebra
Deduce
Ring
Predict
Module
Alternatives

Keywords

  • LIE-ALGEBRAS
  • REPRESENTATIONS

Cite this

A proof of the first Kac-Weisfeiler conjecture in large characteristics. / Martin, Benjamin; Stewart, David I. (Corresponding Author); Topley, Lewis; Tikaradze, Akaki (Collaborator).

In: Representation Theory, Vol. 23, 2019, p. 278-293.

Research output: Contribution to journalArticle

Martin, Benjamin ; Stewart, David I. ; Topley, Lewis ; Tikaradze, Akaki. / A proof of the first Kac-Weisfeiler conjecture in large characteristics. In: Representation Theory. 2019 ; Vol. 23. pp. 278-293.
@article{584f55a131a843068d87dc3dc3b6d611,
title = "A proof of the first Kac-Weisfeiler conjecture in large characteristics",
abstract = "In 1971, Kac and Weisfeiler made two influential conjectures describingthe dimensions of simple modules of a restricted Lie algebra g. The first predicts the maximal dimension of simple g-modules and in this paper we apply the Lefschetz Principle and classical techniques from Lie theory to prove this conjecture for all restricted Lie subalgebras of glnpkq whenever k is an algebraically closed field of sufficiently large characteristic p (depending on n). As a consequence we deduce that the conjecture holds for the Lie algebra of an affine algebraic group scheme over any commutative ring, after specialising to an algebraically closed field of almost any characteristic. In the appendix to this paper, written by Akaki Tikaradze, an alternative, short proof of the first Kac–Weisfeiler conjecture is given for the Lie algebra of group scheme over a finitely generated ring R Ď C, after base change to a field of large positive characteristic.",
keywords = "LIE-ALGEBRAS, REPRESENTATIONS",
author = "Benjamin Martin and Stewart, {David I.} and Lewis Topley and Akaki Tikaradze",
note = "The authors would like to thank Akaki Tikaradze for useful correspondence and for contributing the appendix to this paper, as well as James Waldron for useful remarks on the first draft. We also thank both of the referees for numerous helpful suggestions, including the alternative proof of Proposition 3.8 which we use here. The third author also gratefully acknowledges the support of EPSRC grant number EP/N034449/1.",
year = "2019",
doi = "10.1090/ert/529",
language = "English",
volume = "23",
pages = "278--293",
journal = "Representation Theory",
issn = "1088-4165",
publisher = "American Mathematical Society",

}

TY - JOUR

T1 - A proof of the first Kac-Weisfeiler conjecture in large characteristics

AU - Martin, Benjamin

AU - Stewart, David I.

AU - Topley, Lewis

A2 - Tikaradze, Akaki

N1 - The authors would like to thank Akaki Tikaradze for useful correspondence and for contributing the appendix to this paper, as well as James Waldron for useful remarks on the first draft. We also thank both of the referees for numerous helpful suggestions, including the alternative proof of Proposition 3.8 which we use here. The third author also gratefully acknowledges the support of EPSRC grant number EP/N034449/1.

PY - 2019

Y1 - 2019

N2 - In 1971, Kac and Weisfeiler made two influential conjectures describingthe dimensions of simple modules of a restricted Lie algebra g. The first predicts the maximal dimension of simple g-modules and in this paper we apply the Lefschetz Principle and classical techniques from Lie theory to prove this conjecture for all restricted Lie subalgebras of glnpkq whenever k is an algebraically closed field of sufficiently large characteristic p (depending on n). As a consequence we deduce that the conjecture holds for the Lie algebra of an affine algebraic group scheme over any commutative ring, after specialising to an algebraically closed field of almost any characteristic. In the appendix to this paper, written by Akaki Tikaradze, an alternative, short proof of the first Kac–Weisfeiler conjecture is given for the Lie algebra of group scheme over a finitely generated ring R Ď C, after base change to a field of large positive characteristic.

AB - In 1971, Kac and Weisfeiler made two influential conjectures describingthe dimensions of simple modules of a restricted Lie algebra g. The first predicts the maximal dimension of simple g-modules and in this paper we apply the Lefschetz Principle and classical techniques from Lie theory to prove this conjecture for all restricted Lie subalgebras of glnpkq whenever k is an algebraically closed field of sufficiently large characteristic p (depending on n). As a consequence we deduce that the conjecture holds for the Lie algebra of an affine algebraic group scheme over any commutative ring, after specialising to an algebraically closed field of almost any characteristic. In the appendix to this paper, written by Akaki Tikaradze, an alternative, short proof of the first Kac–Weisfeiler conjecture is given for the Lie algebra of group scheme over a finitely generated ring R Ď C, after base change to a field of large positive characteristic.

KW - LIE-ALGEBRAS

KW - REPRESENTATIONS

UR - http://www.scopus.com/inward/record.url?scp=85073961670&partnerID=8YFLogxK

U2 - 10.1090/ert/529

DO - 10.1090/ert/529

M3 - Article

VL - 23

SP - 278

EP - 293

JO - Representation Theory

JF - Representation Theory

SN - 1088-4165

ER -