Symplectic groups over Lie subgroups of involutive algebras
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911207287422976 |
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| author | Rogozinnikov, Eugen |
| author_facet | Rogozinnikov, Eugen |
| contents | We introduce the symplectic group $\mathrm{Sp}_2(G, σ)$ associated to a Lie subgroup $G$ of a (possibly noncommutative) associative algebra $A$ equipped with an anti-involution $σ$. Our construction recovers several classical Lie groups as special cases, and in particular provides new realizations of spin groups as instances of $\mathrm{Sp}_2(G, σ)$ for suitable subgroups $G$ of the Clifford algebra. This case is not covered by the framework, which focuses on the specific situation $G = A^\times$, and is thus of particular interest.
We construct and study geometric spaces on which $\mathrm{Sp}_2(G, σ)$ acts. In particular, we define the space of $G$-isotropic elements and the corresponding space of $G$-isotropic lines, which generalize the classical projective line. We analyze the group action on these spaces and introduce natural invariants, such as the notion of positive triples and quadruples of $G$-isotropic lines and a generalized cross-ratio of positive quadruples of $G$-isotropic lines. Finally, when the Lie algebra of $G$ is Hermitian, we define the associated Riemannian symmetric space of $\mathrm{Sp}_2(G,σ)$ and provide several models for it. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_11267 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Symplectic groups over Lie subgroups of involutive algebras Rogozinnikov, Eugen Differential Geometry Rings and Algebras 53C35, 32M15, 16W10 We introduce the symplectic group $\mathrm{Sp}_2(G, σ)$ associated to a Lie subgroup $G$ of a (possibly noncommutative) associative algebra $A$ equipped with an anti-involution $σ$. Our construction recovers several classical Lie groups as special cases, and in particular provides new realizations of spin groups as instances of $\mathrm{Sp}_2(G, σ)$ for suitable subgroups $G$ of the Clifford algebra. This case is not covered by the framework, which focuses on the specific situation $G = A^\times$, and is thus of particular interest. We construct and study geometric spaces on which $\mathrm{Sp}_2(G, σ)$ acts. In particular, we define the space of $G$-isotropic elements and the corresponding space of $G$-isotropic lines, which generalize the classical projective line. We analyze the group action on these spaces and introduce natural invariants, such as the notion of positive triples and quadruples of $G$-isotropic lines and a generalized cross-ratio of positive quadruples of $G$-isotropic lines. Finally, when the Lie algebra of $G$ is Hermitian, we define the associated Riemannian symmetric space of $\mathrm{Sp}_2(G,σ)$ and provide several models for it. |
| title | Symplectic groups over Lie subgroups of involutive algebras |
| topic | Differential Geometry Rings and Algebras 53C35, 32M15, 16W10 |
| url | https://arxiv.org/abs/2510.11267 |