The Square-Zero Basis of Matrix Lie Algebras

Raúl Durán Díaz,Jaime Muñoz Masqué,Luis Hernández Encinas,Víctor Gayoso Martínez

doi:10.3390/math8061032

Abstract

A method is presented that allows one to compute the maximum number of functionally-independent invariant functions under the action of a linear algebraic group as long as its Lie algebra admits a basis of square-zero matrices even on a field of positive characteristic. The class of such Lie algebras is studied in the framework of the classical Lie algebras of arbitrary characteristic. Some examples and applications are also given.

Highlights

The modern theory of invariants of linear representations was formulated in the fundamental book [1], which currently represents an essential chapter in the theory of group representations
Is a linear representation of a linear algebraic group G and its Lie algebra g is in the SQZ-LA class, every G-invariant function I ∈ F[V ∗ ] is a common first-integral of the system of derivations ρ∗ ( X ), ∀ X ∈ g, and the number of algebraically independent G-invariant functions in F[V ∗ ] is upper-bounded by the difference n2 − r, where r is the generic rank of the F[V ∗ ]-module M spanned by all the derivations ρ∗ ( X )
It would be interesting to adapt the algorithms given in [6] to the linear representations of a linear algebraic group whose Lie algebra is in the SQZ-LA class of positive characteristic

Summary

Introduction

The modern theory of invariants of linear representations was formulated in the fundamental book [1], which currently represents an essential chapter in the theory of group representations (cf. MSC2020: 20Cxx). If ρ : G → GL(V ) is a linear representation of a connected affine group defined over a ground field F, ρ induces a Lie-algebra homomorphism ρ∗ : g → gl(V ), and in the case F = C, the invariant functions are the first integrals of the vector space defined by the image of ρ∗. Square-zero matrices have been dealt with in several settings and with different purposes; for example, see [8,9,10,11,12,13], among other papers and authors We will consider such matrices in connection with the aforementioned problem of linearizing the calculation of invariants of a representation of an affine algebraic group defined on a field of positive characteristic.

Terminology and Notation

A Class of Lie Algebras

The SQZ-LA Class Studied

Conclusions