Branching problem on winding subalgebras of affine Kac-Moody algebras A^{(1)}_1 and A^{(2)}_2
| Autor: | Duc, Khanh Nguyen |
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| Rok vydání: | 2019 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We consider an affine Kac-Moody algebra $\mathfrak{g}$ with Cartan subalgebra $\mathfrak{h}$. Let $\mathfrak{g}[u]$ be a winding subalgebra of $\mathfrak{g}$. Given $\Lambda$ (resp. $\lambda$) in the set $P_+$ (resp. $\dot{P}_+$) of dominant integral weights of $\mathfrak{g}$ (resp. $\mathfrak{g}[u]$), we denote by $L(\Lambda)$ (resp. $\dot{L}(\lambda)$) the integrable highest weight $\mathfrak{g}$-module (resp. $\mathfrak{g}[u]$-module) with highest weight $\Lambda$ (resp. $\lambda$). The $\mathfrak{g}$-module $L(\Lambda)$ decomposes as a direct sum of weight spaces $L(\Lambda)_\mu$ ($\mu \in \mathfrak{h}^*$). It also decomposes as a direct sum of $\mathfrak{g}[u]$-modules $\dot{L}(\lambda)$. We are interested in the supports of these decompositions $\Gamma(\mathfrak{g},\mathfrak{h})$ and $\Gamma(\mathfrak{g},\mathfrak{g}[u])$. We show that they are semigroups. Let $P(\Lambda)$ (resp. $P_{\mathfrak{g},u}(\Lambda)$) be the set of all $\lambda \in \mathfrak{h}^*$ (resp. $\dot{P}_+$) such that $L(\Lambda)_\lambda \ne 0$ (resp. $\dot{L}(\lambda) \subset L(\Lambda)$). Let $\delta$ be the basis imaginary root of $\mathfrak{g}$. For each $\lambda \in P(\Lambda)+\mathbb{C}\delta$ (resp. $P_{\mathfrak{g},u}(\Lambda)+\mathbb{C}\delta$), we set $b_{\Lambda,\lambda}$ (resp. $b_{\Lambda,\lambda,u}$) the complex number $b$ such that $\lambda+b\delta \in P(A)$ (resp. $P_{\mathfrak{g},u}(\Lambda)$) and $\lambda+(b+n)\delta \not\in P(\Lambda)$ (resp. $P_{\mathfrak{g},u}(\Lambda)$) for any $n\in \mathbb{Z}_{>0}$. For the cases $A^{(1)}_1$ and $A^{(2)}_2$, we determine explicitly the number $b_{\Lambda,\lambda}$ and a set $\mathcal{A}_u(\Lambda)$ of $\lambda$ satisfying $b_{\Lambda,\lambda,u}=b_{\Lambda,\lambda}$. This help us realize the relation between $\Gamma(\mathfrak{g},\mathfrak{g}[u])$ and its satured setting. Comment: 28 pages |
| Databáze: | arXiv |
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