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

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

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3 Citations (Scopus)
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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
Early online date16 Sept 2019
Publication statusPublished - 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.




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