Characters of modules over negative rank-2 Borcherds-Kac-Moody Lie algebras

Mathematics > Representation Theory [Submitted on 18 Jun 2026] Title:Characters of modules over negative rank-2 Borcherds-Kac-Moody Lie algebras View PDF HTML (experimental)Abstract:Let $\mathfrak{g}=\mathfrak{g}(A)$ be the Borcherds-Kac-Moody Lie algebra (BKM LA), corresponding to a BKM Cartan matrix $A$ filled by negative integers. Let $P^+\subset \mathfrak{h}^*$ the classical dominant integral cone (wherein pairings are non-negative). The non-integrable simple highest weight modules $L(\mu)$'s widely studied were broadly those by Naito ([Trans. Amer. Soc., 1995]), for $\mu$'s dot-linked to $P^+$-translates of sums $- \sum_{j\in J}\alpha_j$ of mutually orthogonal and imaginary simple roots $\alpha_j$'s. Recently, we computed weights of all highest weight $\mathfrak{g}$-modules $V$'s, and characters of $L(\rho)$ for Weyl vector $\rho$ in negative type-$A$. These needed a family of ``integrable'' $L(\mu)$'s for $\mu$'s inside our novel signed-dominant-integral cone $P^{\pm}$ (which generalizes $P^+$). Pairings $\mu(\alpha_i^{\vee})\leq 0$ therein are multiples of $\frac{A_{ii}}{2}$ for all $i$. Nevertheless, $L(\mu)$ contain ``Chevalley-Serre relations'' $f_i^{\frac{2}{A_{ii}}{\mu(\alpha_i^{\vee})}+1}L(\mu)_{\mu}=0$; which differ from relations in $L(\lambda)$ for all $\lambda\in P^+$, and are seemingly unstudied earlier (also by Naito). This paper initiates the study in rank-2, of the module structures and maximal vectors (or Verma embeddings) in the Verma covers $M(\mu)$ of $L(\mu)$'s for $\mu\in P^{\pm}$. In this, our goal is to explore in weight spaces of those Verma covers, the strictness (or otherwise, an 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 the unique solution case in Lemma 3.1 from that paper. References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.