Weight decomposition of $\mathfrak{sl}_d(\mathbb R)$ with respect to the adjoint representation of $\mathfrak{so}(p,q)$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909112476893184 |
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| author | Han, Jiyoung |
| author_facet | Han, Jiyoung |
| contents | In this concise article, we compute the weight decomposition of $\mathfrak{sl}_d(\mathbb R)$ with respect to the adjoint representation of $\mathfrak{so}(p,q)$, where $d=p+q$ and demonstrate in detail that $\mathfrak{sl}_d(\mathbb R)$ comprises two irreducible $\mathfrak{so}(p,q)$-invariant subspaces. This can be employed to establish the well-known fact that the identity component of $\mathrm{SO}(p,q)$ is a maximal connected subgroup of $\mathrm{SL}_d(\mathbb R)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_12929 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weight decomposition of $\mathfrak{sl}_d(\mathbb R)$ with respect to the adjoint representation of $\mathfrak{so}(p,q)$ Han, Jiyoung Representation Theory Rings and Algebras 17B10 In this concise article, we compute the weight decomposition of $\mathfrak{sl}_d(\mathbb R)$ with respect to the adjoint representation of $\mathfrak{so}(p,q)$, where $d=p+q$ and demonstrate in detail that $\mathfrak{sl}_d(\mathbb R)$ comprises two irreducible $\mathfrak{so}(p,q)$-invariant subspaces. This can be employed to establish the well-known fact that the identity component of $\mathrm{SO}(p,q)$ is a maximal connected subgroup of $\mathrm{SL}_d(\mathbb R)$. |
| title | Weight decomposition of $\mathfrak{sl}_d(\mathbb R)$ with respect to the adjoint representation of $\mathfrak{so}(p,q)$ |
| topic | Representation Theory Rings and Algebras 17B10 |
| url | https://arxiv.org/abs/2402.12929 |