New approach to representation theory of semisimple Lie algebras and quantum algebras

TL;DR

The paper introduces an explicit method for constructing generators of simple roots in finite-dimensional representations of semisimple Lie and quantum algebras, using finite difference equations and matrix forms.

Contribution

It provides a new explicit construction method for simple root generators applicable to any finite-dimensional representation of semisimple and quantum algebras.

Findings

Generators expressed as solutions to finite difference equations

Matrices of size N_l x N_l for each representation

Applied to rank two algebras like A_2, B_2, D_2, G_2

Abstract

A method to construct in explicit form the generators of the simple roots of an arbitrary finite-dimensional representation of a quantum or standard semisimple algebra is found. The method is based on general results from the global theory of representations of semisimple groups. The rank two algebras $A_2$, $B_2=C_2$, $D_2$ and $G_2$ are considered as examples. The generators of the simple roots are presented as solutions of a system of finite difference equations and given in the form of $N_l\times N_l$ matrices, where $N_l$ is the dimension of the representation.