Quantizations of D=3 Lorentz symmetry

Using the isomorphism $mathfrak{o}(3;mathbb{C})simeqmathfrak{sl}(2;mathbb{C})$ 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 $mathfrak{o}(2,1)simeqmathfrak{su}(1,1)simeqmathfrak{sl}(2;mathbb{R})$ 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.