The Schur-Weyl duality and Invariants for classical Lie superalgebras
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Luo, Yang Wang, Yongjie |
| author_facet | Luo, Yang Wang, Yongjie |
| contents | In this article, we provide a comprehensive characterization of invariants of classical Lie superalgebras from the super-analog of the Schur-Weyl duality in a unified way. We establish $\mathfrak{g}$-invariants of the tensor algebra $T(\mathfrak{g})$, the supersymmetric algebra $S(\mathfrak{g})$, and the universal enveloping algebra $\mathrm{U}(\mathfrak{g})$ of a classical Lie superalgebra $\mathfrak{g}$ corresponding to every element in centralizer algebras and their relationship under supersymmetrization. As a byproduct, we prove that the restriction on $T(\mathfrak{g})^{\mathfrak{g}}$ of the projection from $T(\mathfrak{g})$ to $\mathrm{U}(\mathfrak{g})$ is surjective, which enables us to determine the generators of the center $\mathcal{Z}(\mathfrak{g})$ except for $\mathfrak{g}=\mathfrak{osp}_{2m|2n}$. Additionally, we present an alternative algebraic proof of the triviality of $\mathcal{Z}(\mathfrak{p}_n)$. The key ingredient involves a technique lemma related to the symmetric group and Brauer diagrams. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17093 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Schur-Weyl duality and Invariants for classical Lie superalgebras Luo, Yang Wang, Yongjie Representation Theory (2020): 17B35, 17B10, 20C30 In this article, we provide a comprehensive characterization of invariants of classical Lie superalgebras from the super-analog of the Schur-Weyl duality in a unified way. We establish $\mathfrak{g}$-invariants of the tensor algebra $T(\mathfrak{g})$, the supersymmetric algebra $S(\mathfrak{g})$, and the universal enveloping algebra $\mathrm{U}(\mathfrak{g})$ of a classical Lie superalgebra $\mathfrak{g}$ corresponding to every element in centralizer algebras and their relationship under supersymmetrization. As a byproduct, we prove that the restriction on $T(\mathfrak{g})^{\mathfrak{g}}$ of the projection from $T(\mathfrak{g})$ to $\mathrm{U}(\mathfrak{g})$ is surjective, which enables us to determine the generators of the center $\mathcal{Z}(\mathfrak{g})$ except for $\mathfrak{g}=\mathfrak{osp}_{2m|2n}$. Additionally, we present an alternative algebraic proof of the triviality of $\mathcal{Z}(\mathfrak{p}_n)$. The key ingredient involves a technique lemma related to the symmetric group and Brauer diagrams. |
| title | The Schur-Weyl duality and Invariants for classical Lie superalgebras |
| topic | Representation Theory (2020): 17B35, 17B10, 20C30 |
| url | https://arxiv.org/abs/2411.17093 |