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