Quantum Groups, $q$-Oscillators and Covariant Algebras

Abstract

The physical interpretation of the main notions of the quantum group theory (coproduct, representations and corepresentations, action and coaction) is discussed using the simplest examples of $q$-deformed objects (quantum group $F_q(GL(2))$, quantum algebra $sl_q(2)$, $q$-oscillator and $F_q$-covariant algebra.) Appropriate reductions of the covariant algebra of second rank $q$-tensors give rise to the algebras of the $q$-oscillator and the $q$-sphere. A special covariant algebra related to the reflection equation corresponds to the braid group in a space with nontrivial topology.