Quantizations of D=3 Lorentz symmetry

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.Comment: 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 adde

On a general analytical formula for U_q(su(3))-Clebsch-Gordan coefficients

We present the projection operator method in combination with the Wigner-Racah calculus of the subalgebra U_q(su(2)) for calculation of Clebsch-Gordan coefficients (CGCs) of the quantum algebra U_q(su(3)). The key formulas of the method are couplings of the tensor and projection operators and also a tensor form for the projection operator of U_q(su(3)). We obtain a very compact general analytical formula for the U_q(su(3)) CGCs in terms of the U_q(su(2)) Wigner 3nj-symbols.Comment: 9 pages, LaTeX; to be published in Yad. Fiz. (Phys. Atomic Nuclei), (2001