Narrow, wide, and $λ$-wide regular subalgebras of semisimple Lie algebras
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913287854096384 |
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| author | Douglas, Andrew Repka, Joe |
| author_facet | Douglas, Andrew Repka, Joe |
| contents | A subalgebra of a semisimple Lie algebra is wide if every simple module of the semisimple Lie algebra remains indecomposable when restricted to the subalgebra. From a finer viewpoint, a subalgebra is $λ$-wide if the simple module of a semisimple Lie algebra of highest weight $λ$ remains indecomposable when restricted to the subalgebra. A subalgebra is narrow if the restriction of all non-trivial simple modules to the subalgebra have proper decompositions. We determine necessary and sufficient conditions for regular subalgebras of semisimple Lie algebras to be $λ$-wide. As a natural consequence, we establish necessary and sufficient conditions for regular subalgebras to be wide, a result which has already been established by Panyushev for essentially all regular solvable subalgebras. Next, we show that establishing whether or not a regular subalgebra of a simple Lie algebra is wide does not require consideration of all simple modules. It is necessary and sufficient to only consider the adjoint representation. Finally, we show that a regular subalgebra of the special linear algebra $\mathfrak{sl}_{n+1}$ is either narrow or wide; this property does not hold for non-regular subalgebras of $\mathfrak{sl}_{n+1}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_18847 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Narrow, wide, and $λ$-wide regular subalgebras of semisimple Lie algebras Douglas, Andrew Repka, Joe Representation Theory 17B05, 17B10, 17B20, 17B22, 17B30 A subalgebra of a semisimple Lie algebra is wide if every simple module of the semisimple Lie algebra remains indecomposable when restricted to the subalgebra. From a finer viewpoint, a subalgebra is $λ$-wide if the simple module of a semisimple Lie algebra of highest weight $λ$ remains indecomposable when restricted to the subalgebra. A subalgebra is narrow if the restriction of all non-trivial simple modules to the subalgebra have proper decompositions. We determine necessary and sufficient conditions for regular subalgebras of semisimple Lie algebras to be $λ$-wide. As a natural consequence, we establish necessary and sufficient conditions for regular subalgebras to be wide, a result which has already been established by Panyushev for essentially all regular solvable subalgebras. Next, we show that establishing whether or not a regular subalgebra of a simple Lie algebra is wide does not require consideration of all simple modules. It is necessary and sufficient to only consider the adjoint representation. Finally, we show that a regular subalgebra of the special linear algebra $\mathfrak{sl}_{n+1}$ is either narrow or wide; this property does not hold for non-regular subalgebras of $\mathfrak{sl}_{n+1}$. |
| title | Narrow, wide, and $λ$-wide regular subalgebras of semisimple Lie algebras |
| topic | Representation Theory 17B05, 17B10, 17B20, 17B22, 17B30 |
| url | https://arxiv.org/abs/2403.18847 |