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

TL;DR

This paper classifies all quantum deformations of four-dimensional orthogonal Lie algebras, including complex and real forms, providing explicit structures and universal R-matrices, with new results for several real Lie algebra cases.

Contribution

It systematically constructs and classifies all quantum deformations of $rak{o}(4; ext{C})$ and its real forms, including explicit R-matrices, extending previous work to new algebraic cases.

Findings

Complete list of five quantum deformations of $rak{o}(4; ext{C})$

Sixteen Hopf-algebraic quantum deformations for real forms

Explicit universal R-matrices for each deformation

Abstract

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.