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abstract: ‘It has been shown that the Lie superalgebra $\mathfrak{osp}(2\vert2,\mathbb{C})$ admits a realisation as a three-grading Lie superalgebra, while most known realisations of the Lie superalgebra $\mathfrak{osp}(4\vert2,\mathbb{C})$ consist of two even and one odd part. In this paper we give two realisations of $\mathfrak{osp}(4\vert2,\mathbb{C})$, which describe a super Poincaré and a super Poincaré-Bergmann algebra (in one of which the Lorentz condition is included).’
author:
– Marc Zirnbauer
title: ‘Realisations of the Lie superalgebra $\mathfrak{osp}(4\vert2,\mathbb{C})$’
—
For $\mathbb{R} = \mathbb{C}$, $x\to -\mathrm{i}x$ is an inner automorphism and therefore $\mathfrak{osp}(2\vert2,\mathbb{C})$ can be written as the subalgebra of $\mathfrak{gl}(2\vert2,\mathbb{C})$ generated by $E_{12},E_{21},F_{12},F_{21}$ $$\begin{aligned}
[E_{12}, E_{21}]&=&E_{23},\\
[F_{12}, F_{21}]&=&F_{13},\\
[E_{12}, F_{21}]&=&F_{22},\\
[F_{12}, E_{21}]&=&E_{22},\end{aligned}$$ that is, as the Lie superalgebra of formal series in even and odd variables $\mathbb{C}\langle x, y\rangle$ with values in the supervector space $\mathbb{C}^{2\vert2}$. We shall denote by $$\label{strange}
\begin{split}
\psi & = \pmatrix{1&
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