On induced completely prime primitive ideals in enveloping algebras of classical Lie algebras
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866918145119223808 |
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| author | Goodwin, Simon M. Topley, Lewis Westaway, Matthew |
| author_facet | Goodwin, Simon M. Topley, Lewis Westaway, Matthew |
| contents | A distinguished family of completely prime primitive ideals in the universal enveloping algebra of a reductive Lie algebra ${\mathfrak g}$ over ${\mathbb C}$ are those ideals constructed from one-dimensional representations of finite $W$-algebras. We refer to these ideals as Losev--Premet ideals. For ${\mathfrak g}$ simple of classical type, we prove that for a Losev-Premet ideal $I$ in $U({\mathfrak g})$, there exists a Losev-Premet ideal $I_0$ for a certain Levi subalgebra ${\mathfrak g}_0$ of ${\mathfrak g}$ such that associated variety of $I_0$ is the closure of a rigid nilpotent orbit in ${\mathfrak g}_0$ and $I$ is obtained from $I_0$ by parabolic induction; in turn, this gives a classification of rigid Losev-Premet ideals in $U({\mathfrak g})$. This is deduced from the corresponding statement about one-dimensional representations of finite $W$-algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_10603 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On induced completely prime primitive ideals in enveloping algebras of classical Lie algebras Goodwin, Simon M. Topley, Lewis Westaway, Matthew Representation Theory Rings and Algebras 17B35 (Primary), 16D60, 17B08, 17B10 (Secondary) A distinguished family of completely prime primitive ideals in the universal enveloping algebra of a reductive Lie algebra ${\mathfrak g}$ over ${\mathbb C}$ are those ideals constructed from one-dimensional representations of finite $W$-algebras. We refer to these ideals as Losev--Premet ideals. For ${\mathfrak g}$ simple of classical type, we prove that for a Losev-Premet ideal $I$ in $U({\mathfrak g})$, there exists a Losev-Premet ideal $I_0$ for a certain Levi subalgebra ${\mathfrak g}_0$ of ${\mathfrak g}$ such that associated variety of $I_0$ is the closure of a rigid nilpotent orbit in ${\mathfrak g}_0$ and $I$ is obtained from $I_0$ by parabolic induction; in turn, this gives a classification of rigid Losev-Premet ideals in $U({\mathfrak g})$. This is deduced from the corresponding statement about one-dimensional representations of finite $W$-algebras. |
| title | On induced completely prime primitive ideals in enveloping algebras of classical Lie algebras |
| topic | Representation Theory Rings and Algebras 17B35 (Primary), 16D60, 17B08, 17B10 (Secondary) |
| url | https://arxiv.org/abs/2311.10603 |