Wide Regular Subalgebras of Symmetrizable Kac-Moody Algebras and an Extension of Schur's Lemma
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918534388383744 |
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| author | Douglas, Andrew Ali, Abid |
| author_facet | Douglas, Andrew Ali, Abid |
| contents | The behavior of representations under restriction is a central theme in Lie theory. We study wide regular subalgebras of symmetrizable Kac-Moody algebras, extending work of Douglas and Repka on semisimple Lie algebras. A subalgebra is wide if every irreducible integrable highest weight module remains indecomposable upon restriction. Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra with Cartan subalgebra $\mathfrak{h}$, root system $Φ$, simple roots $Π$, and root space decomposition $\mathfrak{g}=\mathfrak{h}\oplus\bigoplus_{α\inΦ}\mathfrak{g}_α$. Denote by $Φ_{\operatorname{re}}$ the set of real roots. To a regular subalgebra $\mathfrak{s}$ normalized by $\mathfrak{h}$, we associate a closed subset $T\subseteq Φ$ by declaring $α\in T$ if $\mathfrak{s}\cap \mathfrak{g}_α\ne \{0\}$. Our main result is an extension of Schur's lemma: if $\mathfrak{h}\subseteq \mathfrak{s}$ and the real closure of $(T\cup(-T))\cap Φ_{\operatorname{re}}$ contains $Π$, then $(\operatorname{End} V)^{\mathfrak{s}}=\mathbb{C}\operatorname{Id}_V$ for every irreducible integrable highest weight module $V$. As a consequence, this real-root closure condition yields a sufficient condition for wideness. In the affine case, we establish a converse: if $\mathfrak{s}$ is wide, then the closure of $T\cup(-T)$ in $Φ$ is all of $Φ$, and this implication holds without assuming that $\mathfrak{h}\subseteq \mathfrak{s}$. A key ingredient is a structural result showing that closed subsets of affine root systems are closed under arbitrary finite root sums that remain roots. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2606_01371 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Wide Regular Subalgebras of Symmetrizable Kac-Moody Algebras and an Extension of Schur's Lemma Douglas, Andrew Ali, Abid Representation Theory 17B05, 17B10, 17B20, 17B22, 17B67 The behavior of representations under restriction is a central theme in Lie theory. We study wide regular subalgebras of symmetrizable Kac-Moody algebras, extending work of Douglas and Repka on semisimple Lie algebras. A subalgebra is wide if every irreducible integrable highest weight module remains indecomposable upon restriction. Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra with Cartan subalgebra $\mathfrak{h}$, root system $Φ$, simple roots $Π$, and root space decomposition $\mathfrak{g}=\mathfrak{h}\oplus\bigoplus_{α\inΦ}\mathfrak{g}_α$. Denote by $Φ_{\operatorname{re}}$ the set of real roots. To a regular subalgebra $\mathfrak{s}$ normalized by $\mathfrak{h}$, we associate a closed subset $T\subseteq Φ$ by declaring $α\in T$ if $\mathfrak{s}\cap \mathfrak{g}_α\ne \{0\}$. Our main result is an extension of Schur's lemma: if $\mathfrak{h}\subseteq \mathfrak{s}$ and the real closure of $(T\cup(-T))\cap Φ_{\operatorname{re}}$ contains $Π$, then $(\operatorname{End} V)^{\mathfrak{s}}=\mathbb{C}\operatorname{Id}_V$ for every irreducible integrable highest weight module $V$. As a consequence, this real-root closure condition yields a sufficient condition for wideness. In the affine case, we establish a converse: if $\mathfrak{s}$ is wide, then the closure of $T\cup(-T)$ in $Φ$ is all of $Φ$, and this implication holds without assuming that $\mathfrak{h}\subseteq \mathfrak{s}$. A key ingredient is a structural result showing that closed subsets of affine root systems are closed under arbitrary finite root sums that remain roots. |
| title | Wide Regular Subalgebras of Symmetrizable Kac-Moody Algebras and an Extension of Schur's Lemma |
| topic | Representation Theory 17B05, 17B10, 17B20, 17B22, 17B67 |
| url | https://arxiv.org/abs/2606.01371 |