We introduce a grading switching for arbitrary non-associative algebras of prime characteristic p, aimed at producing a new grading of an algebra from a given one. We take inspiration from a fundamental tool in the classification theory of modular Lie algebras known as toral switching, which relies on a delicate adaptation of the exponential of a derivation. Our grading switching is achieved by evaluating certain generalized Laguerre polynomials of degree p − 1, which play the role of generalized exponentials, on a derivation of the algebra. A crucial part of our argument is establishing a congruence for them which is an appropriate analogue of the functional equation e<sup>x</sup> · e<sup>y</sup> = e<sup>x+y</sup> for the classical exponential. Besides having a wider scope, our treatment provides a more transparent explanation of some aspects of the original toral switching, which can be recovered as a special case.

AVITABILE, M., & Mattarei, S. (2015). Laguerre polynomials of derivations. ISRAEL JOURNAL OF MATHEMATICS, 205(1), 109-126 [10.1007/s11856-014-1116-x].

Laguerre polynomials of derivations

AVITABILE, MARINA
Primo
;
2015

Abstract

We introduce a grading switching for arbitrary non-associative algebras of prime characteristic p, aimed at producing a new grading of an algebra from a given one. We take inspiration from a fundamental tool in the classification theory of modular Lie algebras known as toral switching, which relies on a delicate adaptation of the exponential of a derivation. Our grading switching is achieved by evaluating certain generalized Laguerre polynomials of degree p − 1, which play the role of generalized exponentials, on a derivation of the algebra. A crucial part of our argument is establishing a congruence for them which is an appropriate analogue of the functional equation ex · ey = ex+y for the classical exponential. Besides having a wider scope, our treatment provides a more transparent explanation of some aspects of the original toral switching, which can be recovered as a special case.
Articolo in rivista - Articolo scientifico
Nonassociative algebra; grading; derivation; Laguerre polynomial; restricted Lie algebra; toral switching
English
109
126
18
Pubblicato online first nel luglio 2014
AVITABILE, M., & Mattarei, S. (2015). Laguerre polynomials of derivations. ISRAEL JOURNAL OF MATHEMATICS, 205(1), 109-126 [10.1007/s11856-014-1116-x].
Avitabile, M; Mattarei, S
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/100882
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