Multiparametric quantum gl(2): Lie bialgebras, quantum R-matrices and non-relativistic limits

Abstract

Multiparametric quantum deformations of $gl(2)$ are studied through a complete classification of $gl(2)$ Lie bialgebra structures. From them, the non-relativistic limit leading to harmonic oscillator Lie bialgebras is implemented by means of a contraction procedure. New quantum deformations of $gl(2)$ together with their associated quantum $R$-matrices are obtained and other known quantizations are recovered and classified. Several connections with integrable models are outlined.

21 pages, LaTeX. To appear in J. Phys. A. New contents added

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• On Combined Standard-Nonstandard or Hybrid (q,h)-Deformations
• Quantization of Drinfel'd doubles
• Duality for Exotic Bialgebras
• Drinfeld twist for two-parametric deformation of gl(2) and sl(1/2)
• Yang-Baxter deformations of the $GL(2,\mathbb{R})$ WZW model and non-Abelian T-duality
• Generalized rotational hamiltonians from nonlinear angular momentum algebras
• Quantum Jordanian twist
• Darboux families and the classification of real four-dimensional indecomposable coboundary Lie bialgebras
• On a "New" Deformation of GL(2)
• On the biparametric quantum deformation of GL(2) x GL(1)