Wide Regular Subalgebras of Symmetrizable Kac-Moody Algebras and an Extension of Schur's Lemma
Abstract
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 \emph{wide} if every irreducible integrable highest weight module remains indecomposable upon restriction. Let g be a symmetrizable Kac--Moody algebra with Cartan subalgebra h, root system Φ, simple roots Π, and root space decomposition g=h⊕⨁α∈Φgα. Denote by Φre the set of real roots. To a regular subalgebra s normalized by h, we associate a subset T⊆ Φ by declaring α∈ T if s∩ gα≠ {0}. One of our main results is an extension of Schur's lemma: if h⊆ s and the real closure of (T∪(−T))∩ Φre contains Π, then (End V)s=ℂIdV 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 necessary condition for wideness: if s is wide, then the closure of T∪(−T) in Φ is all of Φ, and this implication holds without assuming that h⊆ 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.
Related
- Ramified convolution and monodromy bases for symmetrizable Kac--Moody algebras
- Finite Root Sums in Symmetrizable Kac-Moody Algebras
- Irreducibility of Verma modules for symmetrizable Kac-Moody superalgebras
- A decomposition of Weyl group multiple Dirichlet series for symmetrizable Kac-Moody root systems
- Slice regular functions on alternative *-algebras : prescribing zeroes and values on discrete sets and related extension problems
BibTeX
@misc{douglas2026wide,
title = {{Wide Regular Subalgebras of Symmetrizable Kac-Moody Algebras and an Extension of Schur's Lemma}},
author = {Andrew Douglas and Abid Ali},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.01371},
eprint = {2606.01371},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.01371},
}
RIS
TY - UNPB
TI - Wide Regular Subalgebras of Symmetrizable Kac-Moody Algebras and an Extension of Schur's Lemma
AU - Andrew Douglas
AU - Abid Ali
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2606.01371
UR - https://arxiv.org/abs/2606.01371
AB - 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 \emph{wide} if every irreducible integrable highest weight module remains indecomposable upon restriction. Let g be a symmetrizable Kac--Moody algebra with Cartan subalgebra h, root system Φ, simple roots Π, and root space decomposition g=h⊕⨁_α∈Φg_α. Denote by Φ_re the set of real roots. To a regular subalgebra s normalized by h, we associate a subset T⊆ Φ by declaring α∈ T if s∩ g_α≠ {0}. One of our main results is an extension of Schur's lemma: if h⊆ s and the real closure of (T∪(−T))∩ Φ_re contains Π, then (End V)^s=ℂId_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 necessary condition for wideness: if s is wide, then the closure of T∪(−T) in Φ is all of Φ, and this implication holds without assuming that h⊆ 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.
ER -