On explicit realization of algebra of complex powers of generators of $U_{q}(\mathfrak{sl}(3))$

TL;DR

This paper establishes an integral identity involving complex powers of generators in the quantum group $U_q(sl(3))$, extending Lusztig's relations and enhancing understanding of positive principal series representations.

Contribution

It introduces a continuous analog of Lusztig's relations for complex powers of quantum group generators and provides new definitions and proofs related to positive principal series representations.

Findings

Proved an integral identity for complex powers of generators.

Defined functions of quantum group generators in positive representations.

Provided alternative proofs for known results in the representation theory.

Abstract

In this note we prove an integral identity involving complex powers of generators of quantum group $U_{q}(\mathfrak{sl}(3))$ considered as certain positive operators in the setting of positive principal series representations. This identity represents a continuous analog of one of the Lusztig's relations between divided powers of generators of quantum groups, which play an important role in the study of irreducible modules \cite{Lu 1}. We also give definitions of arbitrary functions of $U_{q}(\mathfrak{sl}(3))$ generators and give another proofs for some of the known results concerning positive principal series representations of $U_{q}(\mathfrak{sl}(3))$.