Real and pseudoreal forms of D=4 complex Euclidean (super)algebras and super-Poincare / super-Euclidean r-matrices

TL;DR

This paper classifies real and pseudoreal forms of D=4 Euclidean (super)algebras and constructs super-Poincare and super-Euclidean r-matrices, extending Zakrzewski's work to supersymmetric cases with N=1,2.

Contribution

It provides a comprehensive classification of real forms of complex D=4 Euclidean superalgebras and derives new supersymmetric r-matrices for N=1,2 cases.

Findings

Classified real and pseudoreal forms of D=4 Euclidean superalgebras.

Constructed N=1 and N=2 supersymmetric r-matrices.

Extended Zakrzewski's Poincare r-matrices to supersymmetric cases.

Abstract

We provide the classification of real forms of complex D=4 Euclidean algebra $\mathcal{\epsilon}(4; \mathbb{C}) = \mathfrak{o}(4;\mathbb{C})) \ltimes \mathbf{T}_{\mathbb{C}}^4$ as well as (pseudo)real forms of complex D=4 Euclidean superalgebras $\mathcal{\epsilon}(4|N; \mathbb{C})$ for N=1,2. Further we present our results: N=1 and N=2 supersymmetric D=4 Poincare and Euclidean r-matrices obtained by using D= 4 Poincare r-matrices provided by Zakrzewski [1]. For N=2 we shall consider the general superalgebras with two central charges.