Quantized function algebras at $q=0$: type $A_{n}$ case
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866909316970184704 |
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| author | Giri, Manabendra Pal, Arup Kumar |
| author_facet | Giri, Manabendra Pal, Arup Kumar |
| contents | We define the notion of quantized function algebras at $q=0$ or crystallization of the $q$ deformations of the type $A_{n}$ compact Lie groups at the $C^*$-algebra level. The $C^{*}$-algebra $A_{n}(0)$ is defined as a universal $C^*$-algebra given by a finite set of generators and relations. We obtain these relations by looking at the irreducible representations of the quantized function algebras for $q>0$ and taking limit as $q\to 0+$ after rescaling the generating elements appropriately. We then prove that in the $n=2$ case the irreducible representations $A_{2}(0)$ are precisely the $q\to 0+$ limits of the irreducible representations of the $C^*$-algebras $A_{2}(q)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_14665 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Quantized function algebras at $q=0$: type $A_{n}$ case Giri, Manabendra Pal, Arup Kumar Quantum Algebra Operator Algebras 17B37, 20G42, 46L67 We define the notion of quantized function algebras at $q=0$ or crystallization of the $q$ deformations of the type $A_{n}$ compact Lie groups at the $C^*$-algebra level. The $C^{*}$-algebra $A_{n}(0)$ is defined as a universal $C^*$-algebra given by a finite set of generators and relations. We obtain these relations by looking at the irreducible representations of the quantized function algebras for $q>0$ and taking limit as $q\to 0+$ after rescaling the generating elements appropriately. We then prove that in the $n=2$ case the irreducible representations $A_{2}(0)$ are precisely the $q\to 0+$ limits of the irreducible representations of the $C^*$-algebras $A_{2}(q)$. |
| title | Quantized function algebras at $q=0$: type $A_{n}$ case |
| topic | Quantum Algebra Operator Algebras 17B37, 20G42, 46L67 |
| url | https://arxiv.org/abs/2203.14665 |