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

Abstract

Abstract We construct firstly the complete list of five quantum deformations of D = 4 complex homogeneous orthogonal Lie algebra $$ \mathfrak{o}\left(4;\mathbb{C}\right)\cong \mathfrak{o}\left(3;\mathbb{C}\right)\oplus \mathfrak{o}\left(3;\mathbb{C}\right) $$ 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 $$ \mathfrak{o}\left(4;\mathbb{C}\right) $$ o 4 ℂ : Euclidean $$ \mathfrak{o}(4) $$ o 4 , Lorentz $$ \mathfrak{o}\left(3,\ 1\right) $$ o 3 , 1 , Kleinian $$ \mathfrak{o}\left(2,\ 2\right) $$ o 2 , 2 and quaternionic $$ {\mathfrak{o}}^{\star }(4) $$ o ⋆ 4 . For $$ \mathfrak{o}\left(3,\ 1\right) $$ 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 $$ \mathfrak{o}\left(4;\mathbb{C}\right) $$ o 4 ℂ we present new results.