Ortogonal rotation in theory of finite-dimensional representations of quantum semisimple algebras. The case of $A_2$ algebra

TL;DR

This paper applies orthogonal rotation methods to explicitly construct generators of simple roots in finite-dimensional representations of quantum and classical semisimple algebras, focusing on the $A_2$ algebra case.

Contribution

It introduces a detailed method for constructing simple root generators using orthogonal rotations specifically for the $A_2$ quantum algebra.

Findings

Explicit formulas for generators of simple roots in $A_2$ algebra

Detailed calculations for finite-dimensional representations $(p,q)$

Method applicable to both quantum and classical semisimple algebras

Abstract

The metohod of ortogonal rotations introduced in the previous papers of the author is used for construction of the explicit form the generators of the simple roots for quantum (and ussual) semisimple algebras. All calculations are presented in details for finite-dimensional representation $(p,q)$ of $A^q_2$ algebra.