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In our previous paper we obtained a full classification of nonequivalentquasitriangular quantum deformations for the complex $D=4$ Euclidean Liesymmetry $\mathfrak{o}(4;\mathbb{C})$. The result was presented in the form ofa list consisting of three three-parameter, one two-parameter and oneone-parameter nonisomorphic classical $r$-matrices which provide 'directions'of the nonequivalent quantizations of $\mathfrak{o}(4;\mathbb{C})$. Applyingreality conditions to the complex $\mathfrak{o}(4;\mathbb{C})$ $r$-matrices weobtained the nonisomorphic classical $r$-matrices for all possible real formsof $\mathfrak{o}(4;\mathbb{C})$: 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 fullclassifications of the inequivalent quasitriangular quantum deformations,however for $\mathfrak{o}(2,2)$ and $\mathfrak{o}^{\star}(4)$ theclassifications are not full. In this paper we complete these classificationsby adding three new three-parameter $\mathfrak{o}(2,2)$-real $r$-matrices andone new three-parameter $\mathfrak{o}^{\star}(4)$-real $r$-matrix. Allnonisomorphic classical $r$-matrices for all real forms of$\mathfrak{o}(4;\mathbb{C})$ are presented in the explicite form what isconvenient for providing the quantizations. We will mention also someapplications of our results to the deformations of space-time symmetries andstring $\sigma$-models.

physics letters b

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