q-Deformed quantum Lie algebras

Abstract

Attention is focused on q-deformed quantum algebras with physical importance, i.e. $U_q(s{u}_2)$, $U_q(s{o}_4)$ 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.

Latex, 49 pages, no figures

References in corpus (16)

• On a Lorentz-Invariant Interpretation of Noncommutative Space-Time and Its Implications on Noncommutative QFT
• Algebras and Hopf Algebras in Braided Categories
• Construction of $\theta$-Poincaré Algebras and their Invariants on ${\mathcal{M}}_{\theta }$
• Quantum and Braided Lie Algebras
• Differential Calculus on Iso_q (N), Quantum Poincare' Algebra and q - Gravity
• Braided Quantum Field Theory
• Bicovariant Quantum Algebras and Quantum Lie Algebras
• q-deformed Phase Space and its Lattice Structure
• Towards Finite Quantum Field Theory in Non-Commutative Geometry
• q-Integration on Quantum Spaces
• Realization of $U_q(so(N))$ within the differntial algebra on ${\mathbf{R}}_q^N$
• Free q-Deformed Relativistic Wave Equations by Representation Theory
• *-Products on Quantum Spaces
• Operator Representations on Quantum Spaces
• q-Exponentials on quantum spaces
• q-Translations on quantum spaces

Cited by in corpus (7)

• Probing quantum scars and weak ergodicity-breaking through quantum complexity
• q-Deformation of W(2,2) Lie algebra associated with quantum groups
• Spinor calculus for q-deformed quantum spaces II
• q-Deformed Superalgebras
• Quantum group structure of the q-deformed $W$ algebra $\WW_q$
• Conservation laws for a $q$-deformed nonrelativistic particle
• Quantum kinematics on q-deformed quantum spaces II, Wave functions on position and momentum space