Weights and characters over Borcherds-Kac-Moody algebras
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2025
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| author | Pal, Souvik Teja, G. Krishna |
| author_facet | Pal, Souvik Teja, G. Krishna |
| contents | Fix any Borcherds-Kac-Moody $\mathbb{C}$-Lie algebra (BKM LA) $\mathfrak{g}=\mathfrak{g}(A)$ of BKM-Cartan matrix $A$, and Cartan subalgebra $\mathfrak{h}\subset \mathfrak{g}$. In this paper, we obtain explicit weight formulas of any highest weight $\mathfrak{g}$-module $V$ with top weight $λ\in \mathfrak{h}^*$ : 1) Generalizing and extending those in one stroke from Kac-Moody (KM) case, of simples $V=L(λ)$ by Khare [Trans. Amer. Math. Soc. 2017] and Dhillon-Khare [Adv. Math. 2017 \& J. Algebra. 2022] and recently of all $V$ by Khare-Teja; via parabolic and higher order Verma $V$. 2) Uniform for all $(\mathfrak{g}, λ, V)$; seemingly novel even for integrable ($L(λ)$ and all intermediate) $V$ for dominant integral $λ\in P^+$. 3) As Weyl-orbit formulas (of finite-dim. $L(λ)$s) for several $V$; and for our candidates of parabolic Vermas over BKM LAs. 4) Using our concepts of holes (1-dim. weight-spaces lost) in $V$, and $P^{\pm}$-dominant weights to cover Chevalley-Serre relations in generic simple $V$s. We define $P^{\pm}$ to be the set of $μ\in \mathfrak{h}^*$ paired with simple co-roots for $A_{ii}\geq 0$ as usual, but notably by negative multiples of $\frac{|A_{ii}|}{2}$ if $A_{ii}<0$.
By-products of working with $P^{\pm}$: study of simples $L(λ)\ \forall\ λ\in P^{\pm}$, notably of $L(ρ)$ for Weyl vectors $ρ\in P^{\pm} \setminus P^+$, and their Verma covers; all novel to our best knowledge. For Weyl-Kac-Borcherds character type formulas of these $L(λ)$s over negative rank-2 $\mathfrak{g}$ and of $L(ρ)$ over negative $A_n$ type $\mathfrak{g}$ $\forall$ $n\in \mathbb{N}$, we explore : i) their presentations; ii) their Verma modules' structures; iii) problems on maximal vectors or Verma embeddings, from Kac-Kazhdan [Adv. Math. 1979], for our BKM $P^{\pm}$ setting. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_08102 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weights and characters over Borcherds-Kac-Moody algebras Pal, Souvik Teja, G. Krishna Representation Theory Primary: 17B10, Secondary: 17B20, 17B22, 17B67, 17B70, 52B20, 52B99 Fix any Borcherds-Kac-Moody $\mathbb{C}$-Lie algebra (BKM LA) $\mathfrak{g}=\mathfrak{g}(A)$ of BKM-Cartan matrix $A$, and Cartan subalgebra $\mathfrak{h}\subset \mathfrak{g}$. In this paper, we obtain explicit weight formulas of any highest weight $\mathfrak{g}$-module $V$ with top weight $λ\in \mathfrak{h}^*$ : 1) Generalizing and extending those in one stroke from Kac-Moody (KM) case, of simples $V=L(λ)$ by Khare [Trans. Amer. Math. Soc. 2017] and Dhillon-Khare [Adv. Math. 2017 \& J. Algebra. 2022] and recently of all $V$ by Khare-Teja; via parabolic and higher order Verma $V$. 2) Uniform for all $(\mathfrak{g}, λ, V)$; seemingly novel even for integrable ($L(λ)$ and all intermediate) $V$ for dominant integral $λ\in P^+$. 3) As Weyl-orbit formulas (of finite-dim. $L(λ)$s) for several $V$; and for our candidates of parabolic Vermas over BKM LAs. 4) Using our concepts of holes (1-dim. weight-spaces lost) in $V$, and $P^{\pm}$-dominant weights to cover Chevalley-Serre relations in generic simple $V$s. We define $P^{\pm}$ to be the set of $μ\in \mathfrak{h}^*$ paired with simple co-roots for $A_{ii}\geq 0$ as usual, but notably by negative multiples of $\frac{|A_{ii}|}{2}$ if $A_{ii}<0$. By-products of working with $P^{\pm}$: study of simples $L(λ)\ \forall\ λ\in P^{\pm}$, notably of $L(ρ)$ for Weyl vectors $ρ\in P^{\pm} \setminus P^+$, and their Verma covers; all novel to our best knowledge. For Weyl-Kac-Borcherds character type formulas of these $L(λ)$s over negative rank-2 $\mathfrak{g}$ and of $L(ρ)$ over negative $A_n$ type $\mathfrak{g}$ $\forall$ $n\in \mathbb{N}$, we explore : i) their presentations; ii) their Verma modules' structures; iii) problems on maximal vectors or Verma embeddings, from Kac-Kazhdan [Adv. Math. 1979], for our BKM $P^{\pm}$ setting. |
| title | Weights and characters over Borcherds-Kac-Moody algebras |
| topic | Representation Theory Primary: 17B10, Secondary: 17B20, 17B22, 17B67, 17B70, 52B20, 52B99 |
| url | https://arxiv.org/abs/2505.08102 |