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

Benjamin Martin, David Stewart, Lewis Topley

Research output: Working paper

### Abstract

In 1971, Kac and Weisfeiler made two influential conjectures describing the dimensions of simple modules of a restricted Lie algebra \$\mathfrak{g}\$. The first predicts the maximal dimension of simple \$\mathfrak{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 \$\mathfrak{gl}_n(k)\$ whenever \$k\$ is an algebraically closed field of characteristic \$p \gg n\$. As a consequence we deduce that the conjecture holds for the the Lie algebra of a group scheme when specialised to an algebraically closed field of almost any characteristic.
Original language English ArXiv Submitted - 30 Oct 2018

### Fingerprint

Algebraically closed
Lie Algebra
Simple Module
Group Scheme
Subalgebra
Deduce
Predict
Module

• math.RT
• math.RA

### Cite this

A proof of the first Kac-Weisfeiler conjecture in large characteristics. / Martin, Benjamin; Stewart, David; Topley, Lewis.

ArXiv, 2018.

Research output: Working paper

@techreport{584f55a131a843068d87dc3dc3b6d611,
title = "A proof of the first Kac-Weisfeiler conjecture in large characteristics",
abstract = "In 1971, Kac and Weisfeiler made two influential conjectures describing the dimensions of simple modules of a restricted Lie algebra \$\mathfrak{g}\$. The first predicts the maximal dimension of simple \$\mathfrak{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 \$\mathfrak{gl}_n(k)\$ whenever \$k\$ is an algebraically closed field of characteristic \$p \gg n\$. As a consequence we deduce that the conjecture holds for the the Lie algebra of a group scheme when specialised to an algebraically closed field of almost any characteristic.",
keywords = "math.RT, math.RA",
author = "Benjamin Martin and David Stewart and Lewis Topley",
note = "The third author is grateful for the support of EPSRC grant number EP/N034449/1.",
year = "2018",
month = "10",
day = "30",
language = "English",
publisher = "ArXiv",
type = "WorkingPaper",
institution = "ArXiv",

}

TY - UNPB

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

AU - Martin, Benjamin

AU - Stewart, David

AU - Topley, Lewis

N1 - The third author is grateful for the support of EPSRC grant number EP/N034449/1.

PY - 2018/10/30

Y1 - 2018/10/30

N2 - In 1971, Kac and Weisfeiler made two influential conjectures describing the dimensions of simple modules of a restricted Lie algebra \$\mathfrak{g}\$. The first predicts the maximal dimension of simple \$\mathfrak{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 \$\mathfrak{gl}_n(k)\$ whenever \$k\$ is an algebraically closed field of characteristic \$p \gg n\$. As a consequence we deduce that the conjecture holds for the the Lie algebra of a group scheme when specialised to an algebraically closed field of almost any characteristic.

AB - In 1971, Kac and Weisfeiler made two influential conjectures describing the dimensions of simple modules of a restricted Lie algebra \$\mathfrak{g}\$. The first predicts the maximal dimension of simple \$\mathfrak{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 \$\mathfrak{gl}_n(k)\$ whenever \$k\$ is an algebraically closed field of characteristic \$p \gg n\$. As a consequence we deduce that the conjecture holds for the the Lie algebra of a group scheme when specialised to an algebraically closed field of almost any characteristic.

KW - math.RT

KW - math.RA

M3 - Working paper

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

PB - ArXiv

ER -