简单排列的 Kac-Moody 算法的最大紧凑子代数的高自旋表征

Robin Lautenbacher, Ralf Köhl
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引用次数: 0

摘要

给定类型为 $A$ 的分裂实 Kac-Moody 代数 $/mathfrak{g}(A)$的最大紧凑子代数 $/mathfrak{k}(A)$,我们研究 $/mathfrak{k}(A)$的某些无限维表示、的最大紧凑子群 $K(A)$,而只是其自旋盖 $Spin(A)$。我们研究了它们的(非)还原性、半简约性以及提升到群层面的问题。我们利用这些表征与自旋扩展韦尔群的相互作用,通过$\mathfrak{g}(A)$的实根推导出了表征矩阵的部分参数化结果。
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Higher spin representations of maximal compact subalgebras of simply-laced Kac-Moody-algebras
Given the maximal compact subalgebra $\mathfrak{k}(A)$ of a split-real Kac-Moody algebra $\mathfrak{g}(A)$ of type $A$, we study certain finite-dimensional representations of $\mathfrak{k}(A)$, that do not lift to the maximal compact subgroup $K(A)$ of the minimal Kac-Moody group $G(A)$ associated to $\mathfrak{g}(A)$ but only to its spin cover $Spin(A)$. Currently, four elementary of these so-called spin representations are known. We study their (ir-)reducibility, semi-simplicity, and lift to the group level. The interaction of these representations with the spin-extended Weyl-group is used to derive a partial parametrization result of the representation matrices by the real roots of $\mathfrak{g}(A)$.
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