Geometry of bi-Lagrangian Grassmannian
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929500691890176 |
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| author | Kozlov, I. K. |
| author_facet | Kozlov, I. K. |
| contents | This paper explores the structure of bi-Lagrangian Grassmanians for pencils of $2$-forms on real or complex vector spaces. We reduce the analysis to the pencils whose Jordan-Kronecker Canonical Form consists of Jordan blocks with the same eigenvalue. We demonstrate that this is equivalent to studying Lagrangian subspaces invariant under a nilpotent self-adjoint operator. We calculate the dimension of bi-Lagrangian Grassmanians and describe their open orbit under the automorphism group. We completely describe the automorphism orbits in the following three cases: for one Jordan block, for sums of equal Jordan blocks and for a sum of two distinct Jordan blocks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_09855 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Geometry of bi-Lagrangian Grassmannian Kozlov, I. K. Rings and Algebras Symplectic Geometry This paper explores the structure of bi-Lagrangian Grassmanians for pencils of $2$-forms on real or complex vector spaces. We reduce the analysis to the pencils whose Jordan-Kronecker Canonical Form consists of Jordan blocks with the same eigenvalue. We demonstrate that this is equivalent to studying Lagrangian subspaces invariant under a nilpotent self-adjoint operator. We calculate the dimension of bi-Lagrangian Grassmanians and describe their open orbit under the automorphism group. We completely describe the automorphism orbits in the following three cases: for one Jordan block, for sums of equal Jordan blocks and for a sum of two distinct Jordan blocks. |
| title | Geometry of bi-Lagrangian Grassmannian |
| topic | Rings and Algebras Symplectic Geometry |
| url | https://arxiv.org/abs/2409.09855 |