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 $s{l}_q(2)$, $q$-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.

11 pages. (Figures are not included.) Dedicated to the memory of M. C. Polivanov

References in corpus (2)

• Correlation Functions of the XXZ model for $\Delta <-1$
• Algebraic Structures Related to Reflection Equations

Cited by in corpus (9)

• Towards a full solution of the large N double-scaled SYK model
• Deformed Dolan-Grady relations in quantum integrable models
• Comodule algebras and integrable systems
• Quantum groups and deformed special relativity
• $\U_q(sl(n))$-invariant quantization of symmetric coadjoint orbits via reflection equation algebra
• Reflection Equation, Twist, and Equivariant Quantization
• Characters of $\U_q\bigl(gl(n)\bigr)$-reflection equation algebra
• Some Remarks on Producing Hopf Algebras
• Quantum Integrability and Quantum Groups: a special issue in memory of Petr P. Kulish