Quantizations of D=3 Lorentz symmetry

Abstract

Using the isomorphism $\mathfrak{o}(3;\mathbb{C})\simeq \mathfrak{sl}(2;\mathbb{C})$ we develop a new simple algebraic technique for complete classification of quantum deformations (the classical $r$-matrices) for real forms $\mathfrak{o}(3)$ and $\mathfrak{o}(2,1)$ of the complex Lie algebra $\mathfrak{o}(3;\mathbb{C})$ in terms of real forms of $\mathfrak{sl}(2;\mathbb{C})$: $\mathfrak{su}(2)$, $\mathfrak{su}(1,1)$ and $\mathfrak{sl}(2;\mathbb{R})$. We prove that the $D=3$ Lorentz symmetry $\mathfrak{o}(2,1)\simeq \mathfrak{su}(1,1)\simeq \mathfrak{sl}(2;\mathbb{R})$ has three different Hopf-algebraic quantum deformations which are expressed in the simplest way by two standard $\mathfrak{su}(1,1)$ and $\mathfrak{sl}(2;\mathbb{R})$ $q$-analogs and by simple Jordanian $\mathfrak{sl}(2;\mathbb{R})$ twist deformations. These quantizations are presented in terms of the quantum Cartan-Weyl generators for the quantized algebras $\mathfrak{su}(1,1)$ and $\mathfrak{sl}(2;\mathbb{R})$ as well as in terms of quantum Cartesian generators for the quantized algebra $\mathfrak{o}(2,1)$. Finaly, some applications of the deformed $D=3$ Lorentz symmetry are mentioned.

22 pages, V2: First and final sections (Sect. 1, Sect. 6) has been partialy rewritten and extended, in Sect. 2-4 only minor corrections, in Sect. 5 notational changes and the clarifications of some formulas; 13 new references added

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• Basic quantizations of $D=4$ Euclidean, Lorentz, Kleinian and quaternionic ${\mathfrak{o}}^{\star }(4)$ symmetries
• Darboux families and the classification of real four-dimensional indecomposable coboundary Lie bialgebras
• Noncommutative Lightcones from Quantum SO(2,1) Conformal Groups
• On Carrollian and Galilean contractions of BMS algebra in 3 and 4 dimensions