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A non-restricted counterexample to the first Kac–Weisfeiler conjecture

Author: Lewis Topley
Journal: Proc. Amer. Math. Soc. 145 (2017), 1937-1942
MSC (2010): Primary 17B50
Published electronically: October 27, 2016
MathSciNet review: 3611310
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Abstract: In 1971 Kac and Weisfeiler made two important conjectures regarding the representation theory of restricted Lie algebras over fields of positive characteristic. The first of these predicts the maximal dimension of the simple modules, and can be stated without the hypothesis that the Lie algebra is restricted. In this short article we construct the first example of a non-restricted Lie algebra for which the prediction of the first Kac–Weisfeiler conjecture fails. Our method is to present pairs of Lie algebras which have isomorphic enveloping algebras but distinct indexes.

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Additional Information

Lewis Topley
Affiliation: Dipartimento di Matematica, Universita di Padova, via Trieste 63, Padova, Italy
MR Author ID: 1048635

Keywords: Modular Lie algebras, irreducible representations
Received by editor(s): May 23, 2016
Received by editor(s) in revised form: June 28, 2016, and July 4, 2016
Published electronically: October 27, 2016
Communicated by: Kailash C. Misra
Article copyright: © Copyright 2016 American Mathematical Society