Quantum Groups,Deformed Oscillators and their Interrelations

TL;DR

This paper explores fundamental concepts of quantum groups, including coproducts and representations, using simple examples like $GL_q(2)$ and $sl_q(2)$, and examines their decompositions and realizations in terms of deformed oscillators.

Contribution

It provides a detailed discussion of quantum group notions and their realizations through $q$-oscillator algebra, with explicit decompositions and examples.

Findings

Gauss decompositions of quantum groups are derived.

Realizations of quantum groups in terms of $q$-oscillator algebra are presented.

Key notions like coproduct and coaction are illustrated with simple examples.

Abstract

The main notions of the quantum groups: coproduct, action and coaction, representation and corepresentation are discussed using simplest examples: $GL_q(2)$, $sl_q(2)$, $q$-oscillator algebra ${\cal A}(q)$, and reflection equation algebra. The Gauss decompositions of quantum groups and their realizations in terms of\, ${\cal A}(q)$ are given.