Matrix structure of classical ${\mathbb Z}_2 \times {\mathbb Z}_2$ graded Lie algebras
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909525116715008 |
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| author | Stoilova, N. I. Van der Jeugt, J. |
| author_facet | Stoilova, N. I. Van der Jeugt, J. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_09274 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Matrix structure of classical ${\mathbb Z}_2 \times {\mathbb Z}_2$ graded Lie algebras Stoilova, N. I. Van der Jeugt, J. Mathematical Physics Group Theory Rings and Algebras Representation Theory 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. |
| title | Matrix structure of classical ${\mathbb Z}_2 \times {\mathbb Z}_2$ graded Lie algebras |
| topic | Mathematical Physics Group Theory Rings and Algebras Representation Theory |
| url | https://arxiv.org/abs/2408.09274 |