Algorithm for computing canonical bases and foldings of quantum groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918042694320128 |
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| author | Shoji, Toshiaki Zhou, Zhiping |
| author_facet | Shoji, Toshiaki Zhou, Zhiping |
| contents | Let ${\mathbf U}_q^-$ be the negative half of a quantum group of finite type. Let $P$ be the transition matrix between the canonical basis and a PBW basis of ${\mathbf U}_q^-$. In the case ${\mathbf U}_q^-$ is symmetric, Antor gave a simple algorithm of computing $P$ by making use of monomial bases. By the folding theory, ${\mathbf U}_q^-$ (symmetric, with a certain automorphism) is related to a quantum group $\underline{\mathbf U}_q^-$ of non-symmetric type. In this paper, we extend the results of Antor to the non-symmetric case, and discuss the relationship between the algorithms for ${\mathbf U}_q^-$ and for $\underline{\mathbf U}_q^-$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_00793 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Algorithm for computing canonical bases and foldings of quantum groups Shoji, Toshiaki Zhou, Zhiping Quantum Algebra 17B37, 81R50 Let ${\mathbf U}_q^-$ be the negative half of a quantum group of finite type. Let $P$ be the transition matrix between the canonical basis and a PBW basis of ${\mathbf U}_q^-$. In the case ${\mathbf U}_q^-$ is symmetric, Antor gave a simple algorithm of computing $P$ by making use of monomial bases. By the folding theory, ${\mathbf U}_q^-$ (symmetric, with a certain automorphism) is related to a quantum group $\underline{\mathbf U}_q^-$ of non-symmetric type. In this paper, we extend the results of Antor to the non-symmetric case, and discuss the relationship between the algorithms for ${\mathbf U}_q^-$ and for $\underline{\mathbf U}_q^-$. |
| title | Algorithm for computing canonical bases and foldings of quantum groups |
| topic | Quantum Algebra 17B37, 81R50 |
| url | https://arxiv.org/abs/2506.00793 |