Boson representations, non-standard quantum algebras and contractions

Abstract

A Gelfan'd--Dyson mapping is used to generate a one-boson realization for the non-standard quantum deformation of $sl(2,\mathbb{R})$ which directly provides its infinite and finite dimensional irreducible representations. Tensor product decompositions are worked out for some examples. Relations between contraction methods and boson realizations are also explored in several contexts. So, a class of two-boson representations for the non-standard deformation of $sl(2,\mathbb{R})$ is introduced and contracted to the non-standard quantum (1+1) Poincaré representations. Likewise, a quantum extended Hopf $sl(2,\mathbb{R})$ algebra is constructed and the Jordanian $q$-oscillator algebra representations are obtained from it by means of another contraction procedure.

21 pages, LaTeX; two new references added

References in corpus (5)

• Universal $R$-matrix for non-standard quantum $sl(2,\mathbb{R})$
• Lie bialgebra quantizations of the oscillator algebra and their universal $R$--matrices
• Irreducible Representations of Jordanian Quantum Algebra ${\mathcal{U}}_h(sl(2))$ Via a Nonlinear Map
• The universal R-matrix for the Jordanian deformation of sl(2), and the contracted forms of so(4)
• On ${\mathcal{U}}_h(sl(2))$, ${\mathcal{U}}_h(e(3))$ and their Representations

Cited by in corpus (11)

• A systematic construction of completely integrable Hamiltonians from coalgebras
• (1+1) Schrodinger Lie bialgebras and their Poisson-Lie groups
• Multiparametric quantum gl(2): Lie bialgebras, quantum R-matrices and non-relativistic limits
• A Jordanian quantum two-photon/Schrodinger algebra
• Twist maps for non-standard quantum algebras and discrete Schrodinger symmetries
• Kinematical superalgebras and Lie algebras of order 3
• Poisson-Hopf deformations of Lie-Hamilton systems revisited: deformed superposition rules and applications to the oscillator algebra
• Quantum two-photon algebra from non-standard U_z(sl(2,R)) and a discrete time Schrödinger equation
• Non-standard quantum algebras and infinite-dimensional PT-symmetric systems
• Quantum harmonic oscillator algebras as non-relativistic limits of multiparametric $gl(2)$ quantizations
• Generalized Buchdahl equations as Lie-Hamilton systems from the 'book' and oscillator algebras: Quantum deformations and their general solution