Graded Casimir elements and central extensions of color Lie algebras
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866915929738182656 |
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| author | Aizawa, N. Fujii, I. Segar, J. Van der Jeugt, J. |
| author_facet | Aizawa, N. Fujii, I. Segar, J. Van der Jeugt, J. |
| contents | A color Lie algebra is a generalization of a Lie (super)algebra by an Abelian group $Γ$. The underlying vector space and defining relations of the algebra are graded by $Γ$, and the color Lie algebra admits graded Casimir elements. Furthermore, its loop algebra admits graded central extensions. We present a general method for constructing 2nd order graded Casimir elements and graded central extensions for a given color Lie algebra and its loop algebra, respectively. We also show that there exists a large class of color Lie algebras admitting such graded Casimir elements or central extensions by providing three examples, namely, $\mathfrak{sl}(2)$ for $Γ= \mathbb{Z}_3^2$, and $\mathfrak{q}(n)$ and $\mathfrak{osp}(m|2n)$ for $Γ= \mathbb{Z}_2^2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_08900 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Graded Casimir elements and central extensions of color Lie algebras Aizawa, N. Fujii, I. Segar, J. Van der Jeugt, J. Representation Theory Mathematical Physics A color Lie algebra is a generalization of a Lie (super)algebra by an Abelian group $Γ$. The underlying vector space and defining relations of the algebra are graded by $Γ$, and the color Lie algebra admits graded Casimir elements. Furthermore, its loop algebra admits graded central extensions. We present a general method for constructing 2nd order graded Casimir elements and graded central extensions for a given color Lie algebra and its loop algebra, respectively. We also show that there exists a large class of color Lie algebras admitting such graded Casimir elements or central extensions by providing three examples, namely, $\mathfrak{sl}(2)$ for $Γ= \mathbb{Z}_3^2$, and $\mathfrak{q}(n)$ and $\mathfrak{osp}(m|2n)$ for $Γ= \mathbb{Z}_2^2$. |
| title | Graded Casimir elements and central extensions of color Lie algebras |
| topic | Representation Theory Mathematical Physics |
| url | https://arxiv.org/abs/2604.08900 |