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

Benjamin Martin; David I. Stewart; Lewis Topley; Akaki Tikaradze

doi:10.1090/ert/529

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

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

Mathematical Science

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3 Citations (Scopus)

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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 language	English
Pages (from-to)	278-293
Number of pages	16
Journal	Representation Theory
Volume	23
Early online date	16 Sept 2019
DOIs	https://doi.org/10.1090/ert/529
Publication status	Published - 2019

Bibliographical 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.

Keywords

LIE-ALGEBRAS
REPRESENTATIONS

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10.1090/ert/529Licence: Unspecified

A proof of the first Kac-Weisfeiler conjecture in large characteristicsSubmitted manuscript, 195 KBLicence: Unspecified
Accepted Manuscript
This is the author's accepted manuscript of an article published in Representation Theory. The final published version can be found at: Martin, B, Stewart, DI, Topley, L & Tikaradze, A 2019, 'A proof of the first Kac-Weisfeiler conjecture in large characteristics' Representation Theory, vol. 23, pp. 278-293. https://doi.org/10.1090/ert/529 https://creativecommons.org/licenses/by-nc-nd/4.0/
Accepted author manuscript, 343 KBLicence: CC BY-NC-ND

http://arxiv.org/abs/1810.12632v1Licence: Unspecified

Cite this

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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 {\v D} C, after base change to a field of large positive characteristic.",

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AU - Stewart, David I.

AU - Topley, Lewis

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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

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JO - Representation Theory

JF - Representation Theory

ER -

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

Abstract

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Ben Martin

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

Abstract

Bibliographical note

Keywords

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Ben Martin

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