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Register a new userQuantum deformations of D=4 Euclidean, Lorentz, Kleinian and quaternionic o^*(4) symmetries in unified o(4;C) setting
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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 applying 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.
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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 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.
We present the class of deformations of simple Euclidean superalgebra, which describe the supersymmetrization of some Lie algebraic noncommutativity of D=4 Euclidean space-time. The presented deformations are generated by the supertwists. We provide new explicit formulae for a chosen twisted D=4 Euclidean Hopf superalgebra and describe the corresponding quantum covariant deformation of chiral Euclidean superspace.
This paper together with the previous one (arXiv:hep-th/0604146) presents the detailed description of all quantum deformations of D=4 Lorentz algebra as Hopf algebra in terms of complex and real generators. We describe here in detail two quantum deformations of the D=4 Lorentz algebra o(3,1) obtained by twisting of the standard q-deformation U_{q}(o(3,1)). For the first twisted q-deformation an Abelian twist depending on Cartan generators of o(3,1) is used. The second example of twisting provides a quantum deformation of Cremmer-Gervais type for the Lorentz algebra. For completeness we describe also twisting of the Lorentz algebra by standard Jordanian twist. By twist quantization techniques we obtain for these deformations new explicit formulae for the deformed coproducts and antipodes of the o(3,1)-generators.
We show how some classical r-matrices for the D=4 Poincare algebra can be supersymmetrized by an addition of part depending on odd supercharges. These r-matrices for D=4 super-Poincare algebra can be presented as a sum of the so-called subordinated r-matrices of super-Abelian and super-Jordanian type. Corresponding twists describing quantum deformations are obtained in an explicit form. These twists are the super-extensions of twists obtained in the paper arXiv:0712.3962.
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