Strongly real adjoint orbits of complex symplectic Lie group
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917837707149312 |
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| author | Lohan, Tejbir Maity, Chandan |
| author_facet | Lohan, Tejbir Maity, Chandan |
| contents | We consider the adjoint action of the symplectic Lie group $\mathrm{Sp}(2n,\mathbb{C})$ on its Lie algebra $\mathfrak{sp}(2n,\mathbb{C})$. An element $X \in \mathfrak{sp}(2n,\mathbb{C})$ is called $\mathrm{Ad}_{\mathrm{Sp}(2n,\mathbb{C})}$-real if $ -X = \mathrm{Ad}(g)X$ for some $g \in \mathrm{Sp}(2n,\mathbb{C})$. Moreover, if $ -X = \mathrm{Ad}(h)X $ for some involution $h \in \mathrm{Sp}(2n,\mathbb{C})$, then $X \in \mathfrak{sp}(2n,\mathbb{C})$ is called strongly $\mathrm{Ad}_{\mathrm{Sp}(2n,\mathbb{C})}$-real. In this paper, we prove that for every element $X \in \mathfrak{sp}(2n,\mathbb{C})$, there exists a skew-involution $g \in \mathrm{Sp}(2n,\mathbb{C})$ such that $-X =\mathrm{Ad}(g)X$. Furthermore, we classify the strongly $\mathrm{Ad}_{\mathrm{Sp}(2n,\mathbb{C})}$-real elements in $\mathfrak{sp}(2n,\mathbb{C})$. We also classify skew-Hamiltonian matrices that are similar to their negatives via a symplectic involution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_09575 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Strongly real adjoint orbits of complex symplectic Lie group Lohan, Tejbir Maity, Chandan Group Theory Representation Theory Symplectic Geometry Primary: 15A21, 15B30, Secondary: 22E60, 20E45 [2020] We consider the adjoint action of the symplectic Lie group $\mathrm{Sp}(2n,\mathbb{C})$ on its Lie algebra $\mathfrak{sp}(2n,\mathbb{C})$. An element $X \in \mathfrak{sp}(2n,\mathbb{C})$ is called $\mathrm{Ad}_{\mathrm{Sp}(2n,\mathbb{C})}$-real if $ -X = \mathrm{Ad}(g)X$ for some $g \in \mathrm{Sp}(2n,\mathbb{C})$. Moreover, if $ -X = \mathrm{Ad}(h)X $ for some involution $h \in \mathrm{Sp}(2n,\mathbb{C})$, then $X \in \mathfrak{sp}(2n,\mathbb{C})$ is called strongly $\mathrm{Ad}_{\mathrm{Sp}(2n,\mathbb{C})}$-real. In this paper, we prove that for every element $X \in \mathfrak{sp}(2n,\mathbb{C})$, there exists a skew-involution $g \in \mathrm{Sp}(2n,\mathbb{C})$ such that $-X =\mathrm{Ad}(g)X$. Furthermore, we classify the strongly $\mathrm{Ad}_{\mathrm{Sp}(2n,\mathbb{C})}$-real elements in $\mathfrak{sp}(2n,\mathbb{C})$. We also classify skew-Hamiltonian matrices that are similar to their negatives via a symplectic involution. |
| title | Strongly real adjoint orbits of complex symplectic Lie group |
| topic | Group Theory Representation Theory Symplectic Geometry Primary: 15A21, 15B30, Secondary: 22E60, 20E45 [2020] |
| url | https://arxiv.org/abs/2411.09575 |