Maximally extended $\boldsymbol {\mathfrak{sl}(2\vert 2)}$ , q-deformed $\boldsymbol{\mathfrak{d}{(2, 1;\epsilon)}}$ and 3D kappa-Poincaré

Abstract

We show that the maximal extension $ \newcommand{\Complex}{\mathbb{C}} \newcommand{\alg}[1]{\mathfrak{#1}} \alg{sl}(2)\ltimes\alg{psl}(2\vert 2)\ltimes \Complex^3$ of the $ \newcommand{\alg}[1]{\mathfrak{#1}} \alg{sl}(2\vert 2)$ superalgebra can be obtained as a contraction limit of the semi-simple superalgebra $ \newcommand{\alg}[1]{\mathfrak{#1}} \newcommand{\directsum}{\times} \alg{d}(2, 1;\epsilon) \directsum \alg{sl}(2)$ . We reproduce earlier results on the corresponding q-deformed Hopf algebra and its universal R-matrix by means of contraction. We make the curious observation that the above algebra is related to kappa-Poincaré symmetry. When dropping the graded part $ \newcommand{\alg}[1]{\mathfrak{#1}} \alg{psl}(2\vert 2)$ we find a novel one-parameter deformation of the 3D kappa-Poincaré algebra. Our construction also provides a concise exact expression for its universal R-matrix.