Quantum deformations of D=4 Euclidean, Lorentz, Kleinian and quaternionic o^*(4) symmetries in unified o(4;C) setting

TL;DR

This paper classifies all classical r-matrices for four-dimensional complex orthogonal Lie algebras and their real forms, providing new results for Euclidean, Kleinian, and quaternionic symmetries relevant to quantum deformations.

Contribution

It presents a complete list of classical r-matrices for D=4 complex orthogonal Lie algebra and its real forms, including new results for Euclidean, Kleinian, and quaternionic cases.

Findings

Complete classification of classical r-matrices for all real forms.

New classical r-matrices for Euclidean, Kleinian, and quaternionic symmetries.

Reaffirmation of known r-matrices for Lorentz symmetry.

Abstract

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.