Finite Root Sums in Symmetrizable Kac-Moody Algebras
Abstract
Let g be a symmetrizable Kac-Moody algebra with root system Φ. A subset S⊆Φ is \emph{closed} if α,β∈ S and α+β∈Φ imply α+β∈ S. For finite and affine Kac-Moody algebras, this closure property is known to extend from pairs to arbitrary finite sums that are roots. We prove that every closed subset of the root system of any symmetrizable Kac-Moody algebra is closed under arbitrary finite sums that are roots. The proof uses real-root strings and simultaneous separation of positive imaginary roots, together with an addition lemma for fundamental imaginary roots. As an application, we extend a necessary condition for wideness of regular subalgebras from finite and affine types to all symmetrizable Kac-Moody algebras: the symmetric closure of the roots associated to a wide regular subalgebra is the entire root system.
Related
- Ramified convolution and monodromy bases for symmetrizable Kac--Moody algebras
- A decomposition of Weyl group multiple Dirichlet series for symmetrizable Kac-Moody root systems
- Wide Regular Subalgebras of Symmetrizable Kac-Moody Algebras and an Extension of Schur's Lemma
- Irreducibility of Verma modules for symmetrizable Kac-Moody superalgebras
- Chromatic characters and root multiplicities for Borcherds--Kac--Moody algebras
BibTeX
@misc{douglas2026finite,
title = {{Finite Root Sums in Symmetrizable Kac-Moody Algebras}},
author = {Andrew Douglas},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15923},
eprint = {2610.15923},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15923},
}
RIS
TY - UNPB
TI - Finite Root Sums in Symmetrizable Kac-Moody Algebras
AU - Andrew Douglas
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.15923
UR - https://arxiv.org/abs/2610.15923
AB - Let g be a symmetrizable Kac-Moody algebra with root system Φ. A subset S⊆Φ is \emph{closed} if α,β∈ S and α+β∈Φ imply α+β∈ S. For finite and affine Kac-Moody algebras, this closure property is known to extend from pairs to arbitrary finite sums that are roots. We prove that every closed subset of the root system of any symmetrizable Kac-Moody algebra is closed under arbitrary finite sums that are roots. The proof uses real-root strings and simultaneous separation of positive imaginary roots, together with an addition lemma for fundamental imaginary roots. As an application, we extend a necessary condition for wideness of regular subalgebras from finite and affine types to all symmetrizable Kac-Moody algebras: the symmetric closure of the roots associated to a wide regular subalgebra is the entire root system.
ER -