Higher spin representations of maximal compact subalgebras of simply-laced Kac-Moody-algebras
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916566092742656 |
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| author | Lautenbacher, Robin Köhl, Ralf |
| author_facet | Lautenbacher, Robin Köhl, Ralf |
| contents | 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 $\mathfrak{g}(A)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_07247 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Higher spin representations of maximal compact subalgebras of simply-laced Kac-Moody-algebras Lautenbacher, Robin Köhl, Ralf Representation Theory 17B67, 81R10 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 $\mathfrak{g}(A)$. |
| title | Higher spin representations of maximal compact subalgebras of simply-laced Kac-Moody-algebras |
| topic | Representation Theory 17B67, 81R10 |
| url | https://arxiv.org/abs/2409.07247 |