q-Deformed quantum Lie algebras

TL;DR

This paper explores q-deformed quantum Lie algebras such as $U_q(su_2)$, $U_q(so_4)$, and q-deformed Lorentz algebra, establishing a consistent framework for their representation theory in physics.

Contribution

It provides explicit formulas for coproducts, antipodes, and representations of these q-deformed algebras, serving as a foundation for further physical applications.

Findings

Derived explicit Hopf algebra structures for q-deformed symmetries

Calculated spinor and vector representations of the generators

Presented quantum Lie algebra relations and Casimir operators

Abstract

Attention is focused on q-deformed quantum algebras with physical importance, i.e. $U_{q}(su_{2})$, $U_{q}(so_{4})$ and q-deformed Lorentz algebra. The main concern of this article is to assemble important ideas about these symmetry algebras in a consistent framework which shall serve as starting point for representation theoretic investigations in physics, especially quantum field theory. In each case considerations start from a realization of symmetry generators within the differential algebra. Formulae for coproducts and antipodes on symmetry generators are listed. The action of symmetry generators in terms of their Hopf structure is taken as q-analog of classical commutators and written out explicitly. Spinor and vector representations of symmetry generators are calculated. A review of the commutation relations between symmetry generators and components of a spinor or vector operator is given. Relations for the corresponding quantum Lie algebras are computed. Their Casimir operators are written down in a form similar to the undeformed case.