Laguerre polynomials of derivations

TL;DR

This paper introduces a novel grading switching method for nonassociative algebras in prime characteristic, using generalized Laguerre polynomials as analogues of exponentials to enhance algebra classification techniques.

Contribution

It develops a new approach to grading switching in nonassociative algebras by employing Laguerre polynomials, extending and clarifying the classical toral switching method.

Findings

Established a congruence for Laguerre polynomials analogous to the exponential functional equation.

Provided a more transparent framework for toral switching in modular Lie algebras.

Extended the scope of grading switching techniques in algebra classification.

Abstract

We introduce a 'grading switching' for arbitrary nonassociative 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 exp(x)*exp(y)=exp(x+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.