Quantum deformations of $D=4$ Euclidean, Lorentz, Kleinian and quaternionic ${\mathfrak{o}}^{\star }(4)$ symmetries in unified $\mathfrak{o}(4;\mathbb{C})$ setting -- Addendum

Abstract

In our previous paper we obtained a full classification of nonequivalent quasitriangular quantum deformations for the complex $D=4$ Euclidean Lie symmetry $\mathfrak{o}(4;\mathbb{C})$. The result was presented in the form of a list consisting of three three-parameter, one two-parameter and one one-parameter nonisomorphic classical $r$-matrices which provide 'directions' of the nonequivalent quantizations of $\mathfrak{o}(4;\mathbb{C})$. Applying reality conditions to the complex $\mathfrak{o}(4;\mathbb{C})$ $r$-matrices we obtained the nonisomorphic classical $r$-matrices for all possible real forms of $\mathfrak{o}(4;\mathbb{C})$: Euclidean $\mathfrak{o}(4)$, Lorentz $\mathfrak{o}(3,1)$, Kleinian $\mathfrak{o}(2,2)$ and quaternionic ${\mathfrak{o}}^{\star }(4)$ Lie algebras. In the case of $\mathfrak{o}(4)$ and $\mathfrak{o}(3,1)$ real symmetries these $r$-matrices give the full classifications of the inequivalent quasitriangular quantum deformations, however for $\mathfrak{o}(2,2)$ and ${\mathfrak{o}}^{\star }(4)$ the classifications are not full. In this paper we complete these classifications by adding three new three-parameter $\mathfrak{o}(2,2)$-real $r$-matrices and one new three-parameter ${\mathfrak{o}}^{\star }(4)$-real $r$-matrix. All nonisomorphic classical $r$-matrices for all real forms of $\mathfrak{o}(4;\mathbb{C})$ are presented in the explicite form what is convenient for providing the quantizations. We will mention also some applications of our results to the deformations of space-time symmetries and string $\sigma$-models.

10 pages. We supplement results of our previous paper by adding new $\mathfrak{o}(2,2)$ and ${\mathfrak{o}}^{\star }(4)$ $r$-matrices needed for the complete classification of real classical $r$-matrices for all four real forms of $\mathfrak{o}(4;\mathbb{C})$

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• Drinfel'd double structures for Poincaré and Euclidean groups
• Darboux families and the classification of real four-dimensional indecomposable coboundary Lie bialgebras
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• On Carrollian and Galilean contractions of BMS algebra in 3 and 4 dimensions