The invariant form of the generators of semisimple Lie and quantum algebras in their arbitrary finite-dimensional representation

TL;DR

This paper derives explicit matrix forms for generators of semisimple Lie and quantum algebras in any finite-dimensional representation, using solutions to matrix equations and the Weyl formula.

Contribution

It provides a novel explicit matrix representation of algebra generators applicable to all finite-dimensional cases, bridging Lie and quantum algebra representations.

Findings

Explicit matrix forms for generators are derived.

Generators are expressed via solutions to matrix equations.

Results are applicable to both Lie and quantum algebras.

Abstract

An explicit form of the generators of quantum and ordinary semisimple algebras for an arbitrary finite-dimensional representation is found. The generators corresponding to the simple roots are obtained in terms of a solution of a system of matrix equations. The result is presented in the form of $N_l\times N_l$ matrices, where $N_l$ is the dimension of the corresponding representation, determined by the invariant Weyl formula.