Algebraic method for construction of infinitesimal invariants of Lie groups representations

TL;DR

This paper introduces an algebraic method that simplifies finding invariants of Lie group representations by reducing the problem to linear algebra and coadjoint invariants, providing a systematic approach.

Contribution

The paper presents a novel algebraic technique that extends the representation space, enabling the construction of invariants through linear algebra methods.

Findings

The method reduces the problem to known linear algebra invariants.

It connects the invariants of a representation to those of the coadjoint representation.

The approach simplifies the process of finding invariants for arbitrary Lie group representations.

Abstract

We propose the method for obtaining invariants of arbitrary representations of Lie groups that reduces this problem to known problems of linear algebra. The basis of this method is the idea of a special extension of the representation space, which allows us to regard it as a coalgebra of some Lie algebra. In its turn, this allows us to reduce the problem of constructing invariants of a given representation to the problem of constructing invariants of the coadjoint representation of the corresponding Lie group. In our previous paper it was shown that this problem can always be solved in a natural way by the methods of linear algebra.