Notes on Chevalley Groups and Root Category II: Compact Lie Groups and Representations
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915816537063424 |
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| author | Li, Buyan Xiao, Jie |
| author_facet | Li, Buyan Xiao, Jie |
| contents | This paper is a continuation of [5]. Using the root categories, we define the compact real forms of the complex semisimple Lie algebras, and maximal compact subgroups of the Chevalley groups over $\mathbb{C}$. In [7], Lusztig used the modified quantum group $\dot{\mathbf{U}}$ and its canonical basis to obtain the reductive group and its coordinate ring $\mathbf{O}_A$, in particular the tensor product decomposition of $\mathbf{O}_A$. By combining these two kinds of structures, we explore in this paper how the classical theory of the compact Lie groups, such as Peter-Weyl theorem and Plancherel theorem, can be recovered completely. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_05752 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Notes on Chevalley Groups and Root Category II: Compact Lie Groups and Representations Li, Buyan Xiao, Jie Representation Theory 16G20, 17B20, 22E46 This paper is a continuation of [5]. Using the root categories, we define the compact real forms of the complex semisimple Lie algebras, and maximal compact subgroups of the Chevalley groups over $\mathbb{C}$. In [7], Lusztig used the modified quantum group $\dot{\mathbf{U}}$ and its canonical basis to obtain the reductive group and its coordinate ring $\mathbf{O}_A$, in particular the tensor product decomposition of $\mathbf{O}_A$. By combining these two kinds of structures, we explore in this paper how the classical theory of the compact Lie groups, such as Peter-Weyl theorem and Plancherel theorem, can be recovered completely. |
| title | Notes on Chevalley Groups and Root Category II: Compact Lie Groups and Representations |
| topic | Representation Theory 16G20, 17B20, 22E46 |
| url | https://arxiv.org/abs/2512.05752 |