# Quantizations of $D=3$ Lorentz symmetry

The European Physical Journal C, Apr 2017

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 deformation. 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 ${\mathfrak {o}}(2,1)$. Finally, some applications of the deformed $D=3$ Lorentz symmetry are mentioned.

This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1140%2Fepjc%2Fs10052-017-4786-9.pdf

J. Lukierski, V. N. Tolstoy. Quantizations of $D=3$ Lorentz symmetry, The European Physical Journal C, 2017, 226, DOI: 10.1140/epjc/s10052-017-4786-9