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تسجيل مستخدم جديدBasic quantizations of $D=4$ Euclidean, Lorentz, Kleinian and quaternionic $mathfrak{o}^{star}(4)$ symmetries
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We construct firstly the complete list of five quantum deformations of $D=4$ complex homogeneous orthogonal Lie algebra $mathfrak{o}(4;mathbb{C})cong mathfrak{o}(3;mathbb{C})oplus mathfrak{o}(3;mathbb{C})$, describing quantum rotational symmetry of four-dimensional complex space-time, in particular we provide the corresponding universal quantum $R$-matrices. Further applying four possible reality conditions we obtain all sixteen Hopf-algebraic quantum deformations for the real forms of $mathfrak{o}(4;mathbb{C})$: Euclidean $mathfrak{o}(4)$, Lorentz $mathfrak{o}(3,1)$, Kleinian $mathfrak{o}(2,2)$ and quaternionic $mathfrak{o}^{star}(4)$. For $mathfrak{o}(3,1)$ we only recall well-known results obtained previously by the authors, but for other real Lie algebras (Euclidean, Kleinian, quaternionic) as well as for the complex Lie algebra $mathfrak{o}(4;mathbb{C})$ we present new results.
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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.
We employ new calculational technique and present complete list of classical $r$-matrices for $D=4$ complex homogeneous orthogonal Lie algebra $mathfrak{o}(4;mathbb{C})$, the rotational symmetry of four-dimensional complex space-time. Further applyin
g reality conditions we obtain the 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. For $mathfrak{o}(3,1)$ we get known four classical $D=4$ Lorentz $r$-matrices, but for other real Lie algebras (Euclidean, Kleinian, quaternionic) we provide new results and mention some applications.
We use the decomposition of o(3,1)=sl(2;C)_1oplus sl(2;C)_2 in order to describe nonstandard quantum deformation of o(3,1) linked with Jordanian deformation of sl(2;C}. Using twist quantization technique we obtain the deformed coproducts and antipode
s which can be expressed in terms of real physical Lorentz generators. We describe the extension of the considered deformation of D=4 Lorentz algebra to the twist deformation of D=4 Poincare algebra with dimensionless deformation parameter.
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