Characters of modules over negative rank-2 Borcherds-Kac-Moody Lie algebras
Abstract
Let g=g(A) be the Borcherds-Kac-Moody Lie algebra (BKM LA) for a BKM Cartan matrix A that is filled by negative integers. Fix a Cartan subalgebra h of g and the classical cone of dominant integral weights P+⊂ h*. The non-integrable simple highest weight g-modules L(μ)'s widely studied were those by Naito ([Trans. Amer. Soc., 1995]), for μ's dot-linked to P+-translates of sums − ∑j∈ Jαj of mutually orthogonal and imaginary simple roots αj's. Recently, we computed weights of all highest weight g-modules V's (over all BKM LA's), and character of L(ρ) for Weyl vector ρ. These needed a family of ``integrable'' L(μ)'s for μ's inside our novel signed-dominant-integral cone P± (which generalizes P+). Pairings μ(αi∨)≤ 0 for μ∈ P± are multiples of Aii/2 for all i. Nevertheless, L(μ) contains ``Chevalley-Serre relations'' fi2/Aiiμ(αi∨)+1L(μ)μ=0, which seem to be previously unstudied and even in Naito's works. This paper initiates in rank-2, the study of module structures and maximal vectors (or Verma embeddings) in Verma covers M(μ)'s of L(μ)'s for μ∈ P±. Our goal in this is to explore in weight spaces of those Vermas, the strictness, or else a uniform equality, of lower bounds by Kac and Kazhdan ([Adv. Math., 1979]) for count of linearly independent maximal vectors. We obtain presentations and characters of all V's when Kac-Kazhdan equation has unique solution in the interior of root-cone. This builds on results of Kac and Kazhdan in crucial unique solution case.
Related
- Chromatic characters and root multiplicities for Borcherds--Kac--Moody algebras
- Irreducibility of Verma modules for symmetrizable Kac-Moody superalgebras
- On characters of irreducible modules for Lie superalgebra q(n)
- Ramified convolution and monodromy bases for symmetrizable Kac--Moody algebras
- Finite Root Sums in Symmetrizable Kac-Moody Algebras
BibTeX
@misc{pal2026characters,
title = {{Characters of modules over negative rank-2 Borcherds-Kac-Moody Lie algebras}},
author = {Souvik Pal and Supravat Sarkar and G. Krishna Teja},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.20386},
eprint = {2606.20386},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.20386},
}
RIS
TY - UNPB
TI - Characters of modules over negative rank-2 Borcherds-Kac-Moody Lie algebras
AU - Souvik Pal
AU - Supravat Sarkar
AU - G. Krishna Teja
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2606.20386
UR - https://arxiv.org/abs/2606.20386
AB - Let g=g(A) be the Borcherds-Kac-Moody Lie algebra (BKM LA) for a BKM Cartan matrix A that is filled by negative integers. Fix a Cartan subalgebra h of g and the classical cone of dominant integral weights P^+⊂ h^*. The non-integrable simple highest weight g-modules L(μ)'s widely studied were those by Naito ([Trans. Amer. Soc., 1995]), for μ's dot-linked to P^+-translates of sums − ∑_j∈ Jα_j of mutually orthogonal and imaginary simple roots α_j's. Recently, we computed weights of all highest weight g-modules V's (over all BKM LA's), and character of L(ρ) for Weyl vector ρ. These needed a family of ``integrable'' L(μ)'s for μ's inside our novel signed-dominant-integral cone P^± (which generalizes P^+). Pairings μ(α_i^∨)≤ 0 for μ∈ P^± are multiples of A_ii/2 for all i. Nevertheless, L(μ) contains ``Chevalley-Serre relations'' f_i^2/A_iiμ(α_i^∨)+1L(μ)_μ=0, which seem to be previously unstudied and even in Naito's works. This paper initiates in rank-2, the study of module structures and maximal vectors (or Verma embeddings) in Verma covers M(μ)'s of L(μ)'s for μ∈ P^±. Our goal in this is to explore in weight spaces of those Vermas, the strictness, or else a uniform equality, of lower bounds by Kac and Kazhdan ([Adv. Math., 1979]) for count of linearly independent maximal vectors. We obtain presentations and characters of all V's when Kac-Kazhdan equation has unique solution in the interior of root-cone. This builds on results of Kac and Kazhdan in crucial unique solution case.
ER -