Classification and double commutant property for dual pairs in an orthosymplectic Lie supergroup
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arXiv
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2022
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| _version_ | 1866912921421873152 |
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| author | Merino, Allan Salmasian, Hadi |
| author_facet | Merino, Allan Salmasian, Hadi |
| contents | In this paper, we obtain a full classification of reductive dual pairs in a, real or complex, Lie superalgebra $\mathfrak{spo}(E)$ and Lie supergroup $\textbf{SpO}(E)$. Moreover, by looking at the natural action of the orthosymplectic Lie supergroup $\textbf{SpO}(E)$ on the Weyl-Clifford algebra $\textbf{WC}(E)$, we prove that for a reductive dual pair $(\mathscr{G}\,, \mathscr{G}') = ((G\,, \mathfrak{g})\,, (G'\,, \mathfrak{g}'))$ in $\textbf{SpO}(E)$, the superalgebra $\textbf{WC}(E)^{\mathscr{G}}$ consisting of $\mathscr{G}$-invariant elements in $\textbf{WC}(E)$ is generated by the Lie superalgebra $\mathfrak{g}'$.
We obtain a full classification of reductive dual pairs in the (real or complex) Lie superalgebra $\mathfrak{spo}(\mathrm E)$ and the Lie supergroup $\textbf{SpO}(\mathrm E)$. Using this classification we prove that for a reductive dual pair $(\mathscr{G}\,, \mathscr{G}') = ((\mathrm G\,, \mathfrak{g})\,, (\mathrm G'\,, \mathfrak{g}'))$ in $\textbf{SpO}(\mathrm E)$, the superalgebra $\textbf{WC}(\mathrm E)^{\mathscr{G}}$ consisting of $\mathscr{G}$-invariant elements in the Weyl-Clifford algebra $\textbf{WC}(\mathrm E)$, equipped with the natural action of the orthosymplectic Lie supergroup $\textbf{SpO}(\mathrm E)$, is generated by the Lie superalgebra $\mathfrak{g}'$.
As an application, we prove that Howe duality holds for the dual pairs $({\textbf{SpO}}(2n|1)\,, {\textbf{OSp}}(2k|2l)) \subseteq {\textbf{SpO}}(\mathbb{C}^{2k|2l} \otimes \mathbb{C}^{2n|1})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_09746 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Classification and double commutant property for dual pairs in an orthosymplectic Lie supergroup Merino, Allan Salmasian, Hadi Representation Theory Primary: 22E45, Secondary: 22E46, 22E30 In this paper, we obtain a full classification of reductive dual pairs in a, real or complex, Lie superalgebra $\mathfrak{spo}(E)$ and Lie supergroup $\textbf{SpO}(E)$. Moreover, by looking at the natural action of the orthosymplectic Lie supergroup $\textbf{SpO}(E)$ on the Weyl-Clifford algebra $\textbf{WC}(E)$, we prove that for a reductive dual pair $(\mathscr{G}\,, \mathscr{G}') = ((G\,, \mathfrak{g})\,, (G'\,, \mathfrak{g}'))$ in $\textbf{SpO}(E)$, the superalgebra $\textbf{WC}(E)^{\mathscr{G}}$ consisting of $\mathscr{G}$-invariant elements in $\textbf{WC}(E)$ is generated by the Lie superalgebra $\mathfrak{g}'$. We obtain a full classification of reductive dual pairs in the (real or complex) Lie superalgebra $\mathfrak{spo}(\mathrm E)$ and the Lie supergroup $\textbf{SpO}(\mathrm E)$. Using this classification we prove that for a reductive dual pair $(\mathscr{G}\,, \mathscr{G}') = ((\mathrm G\,, \mathfrak{g})\,, (\mathrm G'\,, \mathfrak{g}'))$ in $\textbf{SpO}(\mathrm E)$, the superalgebra $\textbf{WC}(\mathrm E)^{\mathscr{G}}$ consisting of $\mathscr{G}$-invariant elements in the Weyl-Clifford algebra $\textbf{WC}(\mathrm E)$, equipped with the natural action of the orthosymplectic Lie supergroup $\textbf{SpO}(\mathrm E)$, is generated by the Lie superalgebra $\mathfrak{g}'$. As an application, we prove that Howe duality holds for the dual pairs $({\textbf{SpO}}(2n|1)\,, {\textbf{OSp}}(2k|2l)) \subseteq {\textbf{SpO}}(\mathbb{C}^{2k|2l} \otimes \mathbb{C}^{2n|1})$. |
| title | Classification and double commutant property for dual pairs in an orthosymplectic Lie supergroup |
| topic | Representation Theory Primary: 22E45, Secondary: 22E46, 22E30 |
| url | https://arxiv.org/abs/2208.09746 |