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 gmodules 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.
the dimensions of simple modules of a restricted Lie algebra g. The first predicts the maximal dimension of simple gmodules 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 (fromto)  278293 
Number of pages  16 
Journal  Representation Theory 
Volume  23 
Early online date  16 Sept 2019 
DOIs  
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
 LIEALGEBRAS
 REPRESENTATIONS
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Ben Martin
 School of Natural & Computing Sciences, Mathematical Science  Personal Chair
Person: Academic