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

TL;DR

This work addresses realizing $\mathbb{Z}_2^2$-graded Lie algebras associated with classical Lie algebras via explicit matrix realizations. It introduces the graded transpose and embeds the classical types as $\mathbb{Z}_2^2$-GLA subalgebras inside $\mathfrak{sl}_{p,q,r,s}(n)$, with concrete block forms for types $C$, $D$, and $B$. For type $A$, the approach agrees with a known class, while for types $B$, $C$, and $D$ it yields new defining matrices that closely resemble the classical counterparts but differ in essential signs and block structure. An explicit parafermion-inspired example illustrates the applicability to parastatistics and suggests potential physical realizations of graded symmetry.

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.