Basic quantizations of D = 4 Euclidean, Lorentz, Kleinian and quaternionic $$ {\mathfrak{o}}^{\star }(4) $$ symmetries

Abstract

We construct firstly the complete list of five quantum deformations of D = 4 complex homogeneous orthogonal Lie algebra o(4; C) ∼ = o(3; C) ⊕ o(3; C), describing quantum rotational symmetries 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 o(4; C): Euclidean o(4), Lorentz o(3, 1), Kleinian o(2, 2) and quaternionic o ⋆ (4). For 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 o(4; C) we present new results.

Quantizations of $$D=3$$ D = 3 Lorentz symmetry

“…We first recall different most popular bases of the complex D = 3 Euclidean Lie algebra o(3; C): metric, Cartesian and Cartan-Weyl bases (see [8]). …”

“…There are two types of explicit sl(2; C)-automorphisms which were presented in [8]. First type connecting the classical r -matrices with zero γ -characteristic is given by the formulas (see (3.15) in [8]) 4 :…”

“…First type connecting the classical r -matrices with zero γ -characteristic is given by the formulas (see (3.15) in [8]) 4 :…”

“…15 The next task could be to describe all deformations of D = 4 space-times with constant curvature and arbitrary signature, which would classify all possible D = 4 quantum d S and Ad S algebras as well as the quantum-deformed D = 5 Euclidean o(5) symmetries. For such a purpose one can look for the extension of algebraic methods used to classify the deformations of o(4; C) and its real forms (see [8]) to the case of o(5; C) and the real forms o (5), o(4, 1) and o(3, 2). Finally the systematic study of deformations of o(6; C) is another important challenge, in particular because the deformations of its real form o(4, 2) su(2, 2) will provide the list of quantum D = 4 conformal algebras.…”

“…Firstly, following [8], we obtain the complete classifications of the non-equivalent (nonisomorphic) classical r -matrices for complex Lie algebra sl(2; C) and its real forms su (2), su(1, 1) and sl(2; R) with the help of explicit formulas for the automorphisms of these Lie algebras in terms of the Cartan-Weyl bases. In the case of sl(2; C) there are two non-equivalent classical r -matrices -standard and Jordanian ones.…”