Matrix structure of classical ${\mathbb Z}_2 \times {\mathbb Z}_2$ graded Lie algebras

Abstract

A ${\mathbb Z}_2\times{\mathbb Z}_2$-graded Lie algebra $\mathfrak g$ is a ${\mathbb Z}_2\times{\mathbb Z}_2$-graded algebra $\mathfrak g$ with a bracket $[|. , . |]$ that satisfies certain graded versions of the symmetry and Jacobi identity. In particular, despite the common terminology, $\mathfrak g$ is not a Lie algebra. We construct classes of ${\mathbb Z}_2\times{\mathbb Z}_2$-graded Lie algebras corresponding to the classical Lie algebras, in terms of their defining matrices. For the ${\mathbb Z}_2\times{\mathbb Z}_2$-graded Lie algebra of type $A$, the construction coincides with the previously known class. For the ${\mathbb Z}_2\times{\mathbb Z}_2$-graded Lie algebra of type $B$, $C$ and $D$ our construction is new and gives rise to interesting defining matrices closely related to the classical ones but undoubtedly different. We also give some examples and possible applications to parastatistics.