The generalized Casimir operator and tensor representations of groups

TL;DR

This paper introduces a new method for constructing tensor representation functions of Lie groups using a generalized Casimir operator based on Lie derivatives, enabling better separation of irreducible components.

Contribution

It proposes a novel generalized Casimir operator utilizing Lie derivatives and invariance conditions to improve tensor representation analysis of Lie groups.

Findings

Constructed invariant projection operators for irreducible components.

Applied method to Bianchi type G^3 IX and G^3 II groups.

Demonstrated the effectiveness of the approach in specific group cases.

Abstract

There has been proposed a new method of the constructing of the basic functions for spaces of tensor representations of the Lie groups with the help of the generalized Casimir operator. In the definition of the operator there were used the Lie derivatives instead of the corresponding infenitisemal operators. When introducing the generalized Casimir operator we use the metric for which a group being considered will be isometry that follows from the invariance condition for the generalized Casimir operator. This allows us to formulate the eigenvalue and eigenfunction problems correctly. The invariant projection operators have been constructed in order to separate irreducible components. The cases of the Bianchi type G^3 IX and G^3 II groups are considered as examples.