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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2506.11562 |
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| _version_ | 1866912428313280512 |
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| author | Mathevet, Hugo |
| author_facet | Mathevet, Hugo |
| contents | Given a natural number m and a Lie algebra g, the m th generalized Takiff Lie algebra of g is the Lie algebra gm\,:= g $\otimes$ C[T ]/T m+1 . For n $\ge$ m, we define the (m, n)-modality of an adjoint orbit $Ω$m in gm to be the minimum codimension of an adjoint orbit in the pullback of $Ω$m in gn. In this paper, we study this family of invariants in generalized Takiff Lie algebras associated to a quadratic Lie algebra g. We show that this family of invariants satisfies some concavity and hereditary properties. From which we deduce that (n -m)$χ$(g) is a lower bound, where $χ$(g) is the index of g. We prove that this lower bound is in fact an equality for a dense set of orbits, and that if g is reductive, it is always an equality when m = 0 (and also some special orbits). We conjecture that equality holds for all m when g is reductive. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_11562 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Relative modality of elements in generalized Takiff Lie algebras Mathevet, Hugo Rings and Algebras Given a natural number m and a Lie algebra g, the m th generalized Takiff Lie algebra of g is the Lie algebra gm\,:= g $\otimes$ C[T ]/T m+1 . For n $\ge$ m, we define the (m, n)-modality of an adjoint orbit $Ω$m in gm to be the minimum codimension of an adjoint orbit in the pullback of $Ω$m in gn. In this paper, we study this family of invariants in generalized Takiff Lie algebras associated to a quadratic Lie algebra g. We show that this family of invariants satisfies some concavity and hereditary properties. From which we deduce that (n -m)$χ$(g) is a lower bound, where $χ$(g) is the index of g. We prove that this lower bound is in fact an equality for a dense set of orbits, and that if g is reductive, it is always an equality when m = 0 (and also some special orbits). We conjecture that equality holds for all m when g is reductive. |
| title | Relative modality of elements in generalized Takiff Lie algebras |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2506.11562 |