Isomorphisms of Twisted Hilbert Loop Algebras
Isomorphisms of Twisted Hilbert Loop Algebras
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扭曲希尔伯特环代数的同构
DOI：
10.4153/cjm-2016-003-x
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
K. Neeb
中科院分区：
文献类型：
--
作者：
T. Marquis;K. Neeb
紧李代数最接近的无限维亲戚是Hilbert-Lie代数，即具有标量积不变的李代数结构的实数Hilbert空间。局部仿射李代数 $\left( \text{LALAs} \right)$ 对应于（扭曲）环代数在简单Hilbert-Lie代数上的双重扩展 $\mathfrak{k}$ ，也称为affinations of $\mathfrak{k}$ 。它们具有根空间分解，对应的根系为7科之一的局部仿射根系 $$A_{J}^{\left( 1 \right)},\,B_{J}^{\left( 1 \right)},\,C_{J}^{\left( 1 \right)},\,D_{J}^{\left( 1 \right)},\,B_{J}^{\left( 2 \right)},\,C_{J}^{\left( 2 \right)},\,\,\text{and}\,BC_{J}^{\left( 2 \right)}$$ 对于某个无限集 $J$ 。每一种类型都对应着一些简单的希尔伯特-李代数的“最小”亲和 $\mathfrak{k}$ ，我们称之为标准。在本文中，我们给出了每一种发音 $\mathfrak{g}$ 一个简单的希尔伯特-李代数 $\mathfrak{k}$ 的显式同构 $\mathfrak{g}$ 的标准词缀之一 $\mathfrak{k}$ 。这种同构的存在也可以从局部仿射根系的分类中推导出来，但为了表示理论的目的，将其明确地作为与根分解相容的两个扭曲之间的变形是至关重要的。我们通过将同构定理应用于正能量最高权值表示的研究来说明这一点 $\mathfrak{g}$ 。在后续的工作中，本文将用于获得一个完整的正能量最高权重表示的类型化 $\mathfrak{k}$ ．
Abstract The closest infinite-dimensional relatives of compact Lie algebras are Hilbert-Lie algebras, i.e., real Hilbert spaces with a Lie algebra structure for which the scalar product is invariant. Locally affine Lie algebras $\left( \text{LALAs} \right)$ correspond to double extensions of (twisted) loop algebras over simple Hilbert-Lie algebras $\mathfrak{k}$ , also called affinisations of $\mathfrak{k}$ . They possess a root space decomposition whose corresponding root system is a locally affine root system of one of the 7 families $$A_{J}^{\left( 1 \right)},\,B_{J}^{\left( 1 \right)},\,C_{J}^{\left( 1 \right)},\,D_{J}^{\left( 1 \right)},\,B_{J}^{\left( 2 \right)},\,C_{J}^{\left( 2 \right)},\,\,\text{and}\,BC_{J}^{\left( 2 \right)}$$ for some infinite set $J$ . To each of these types corresponds a “minimal ” affinisation of some simple Hilbert-Lie algebra $\mathfrak{k}$ , which we call standard. In this paper, we give for each affinisation $\mathfrak{g}$ of a simple Hilbert-Lie algebra $\mathfrak{k}$ an explicit isomorphism from $\mathfrak{g}$ to one of the standard affinisations of $\mathfrak{k}$ . The existence of such an isomorphism could also be derived from the classiffication of locally affine root systems, but for representation theoretic purposes it is crucial to obtain it explicitly as a deformation between two twists that is compatible with the root decompositions. We illustrate this by applying our isomorphism theorem to the study of positive energy highest weight representations of $\mathfrak{g}$ . In subsequent work, this paper will be used to obtain a complete classification of the positive energy highest weight representations of affinisations of $\mathfrak{k}$ .
局部扩展仿射李代数
DOI：
--
发表时间：
2006
期刊：
Journal of Algebra 301・1
影响因子：
--
作者：
山本顯治;H.Katsurada.;Jun Morita
通讯作者：
Jun Morita
局部环代数和局部仿射李代数
影响因子：
0.9
作者：
Jun Morita;Yoji Yoshii
通讯作者：
Yoji Yoshii