Ideals in enveloping algebras of affine Kac-Moody algebras
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| Format: | Preprint |
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2021
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| author | Biswal, Rekha Sierra, Susan J. |
| author_facet | Biswal, Rekha Sierra, Susan J. |
| contents | Let $L$ be an affine Kac-Moody algebra, with central element $c$, and let $λ\in \mathbb C$. We study two-sided ideals in the central quotient $U_λ(L):= U(L)/(c-λ)$ of the universal enveloping algebra of $L$, and prove:
Theorem 1. If $λ\neq 0$ then $U_λ(L)$ is simple.
Theorem 2. The algebra $U_0(L)$ has just-infinite growth, in the sense that any proper quotient has polynomial growth.
As an immediate corollary, we show that the annihilator of any nontrivial integrable highest weight representation of $L$ is centrally generated, extending a result of Chari for Verma modules.
We also show that universal enveloping algebras of loop algebras and current algebras of finite-dimensional simple Lie algebras have just-infinite growth, and prove similar results to Theorems 1 and 2 for quotients of symmetric algebras of these Lie algebras by Poisson ideals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_08334 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Ideals in enveloping algebras of affine Kac-Moody algebras Biswal, Rekha Sierra, Susan J. Rings and Algebras Representation Theory Primary: 16S30, 17B67, 16P90, 17B10, Secondary 17B65, 16D30 Let $L$ be an affine Kac-Moody algebra, with central element $c$, and let $λ\in \mathbb C$. We study two-sided ideals in the central quotient $U_λ(L):= U(L)/(c-λ)$ of the universal enveloping algebra of $L$, and prove: Theorem 1. If $λ\neq 0$ then $U_λ(L)$ is simple. Theorem 2. The algebra $U_0(L)$ has just-infinite growth, in the sense that any proper quotient has polynomial growth. As an immediate corollary, we show that the annihilator of any nontrivial integrable highest weight representation of $L$ is centrally generated, extending a result of Chari for Verma modules. We also show that universal enveloping algebras of loop algebras and current algebras of finite-dimensional simple Lie algebras have just-infinite growth, and prove similar results to Theorems 1 and 2 for quotients of symmetric algebras of these Lie algebras by Poisson ideals. |
| title | Ideals in enveloping algebras of affine Kac-Moody algebras |
| topic | Rings and Algebras Representation Theory Primary: 16S30, 17B67, 16P90, 17B10, Secondary 17B65, 16D30 |
| url | https://arxiv.org/abs/2112.08334 |