Remarks on simple interpolation between Jordanian twists

Abstract

In this paper, we propose a simple generalization of the locally r-symmetric Jordanian twist, resulting in the one-parameter family of Jordanian twists. All the proposed twists differ by the coboundary twists and produce the same Jordanian deformation of the corresponding Lie algebra. They all provide the $\kappa$-Minkowski spacetime commutation relations. Constructions from noncommutative coordinates to the star product and coproduct, and from the star product to the coproduct and the twist are presented. The corresponding twist in the Hopf algebroid approach is given. Our results are presented symbolically by a diagram relating all of the possible constructions.

12 pages

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• $\kappa$-deformed phase spaces, Jordanian twists, Lorentz-Weyl algebra and dispersion relations
• Dispersion Relations in $\kappa$-Noncommutative Cosmology
• Associative realizations of $\kappa$-deformed extended Snyder model
• Generalized Heisenberg algebra applied to realizations of the orthogonal, Lorentz and Poincare algebras and their dual extensions
• Interpolations between Jordanian Twists Induced by Coboundary Twists
• Symmetric ordering and Weyl realizations for quantum Minkowski spaces
• Twist for Snyder space
• Twisted submanifolds of R^n
• Exponential Formulas, Normal Ordering and the Weyl-Heisenberg Algebra
• Generalized Heisenberg algebra, realizations of the $\mathfrak{gl}(n)$ algebra and applications
• Families of vector-like deformed relativistic quantum phase spaces, twists and symmetries
• One Parameter Family of Jordanian Twists
• On Light-like Deformations of the Poincaré Algebra
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• Braided Commutative Geometry and Drinfel'd Twist Deformations
• Jordanian spin chains for twisted strings in $Ad{S}_5\times S^5$
• Interpolations between Jordanian twists, the Poincaré-Weyl algebra and dispersion relations