Crystal bases and canonical bases for quantum Borcherds-Bozec algebras
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866914734723301376 |
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| author | Fan, Zhaobing Han, Shaolong Kang, Seok-Jin Kim, Young Rock |
| author_facet | Fan, Zhaobing Han, Shaolong Kang, Seok-Jin Kim, Young Rock |
| contents | Let $U_{q}^{-}(\mathfrak g)$ be the negative half of a quantum Borcherds-Bozec algebra $U_{q}(\mathfrak g)$ and $V(λ)$ be the irreducible highest weight module with $λ\in P^{+}$. In this paper, we investigate the structures, properties and their close connections between crystal bases and canonical bases of $U_{q}^{-}(\mathfrak g)$ and $V(λ)$. We first re-construct crystal basis theory with modified Kashiwara operators. While going through Kashiwara's grand-loop argument, we prove several important lemmas, which play crucial roles in the later developments of the paper. Next, based on the theory of canonical bases on quantum Bocherds-Bozec algebras, we introduce the notion of primitive canonical bases and prove that primitive canonical bases coincide with lower global bases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_02859 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Crystal bases and canonical bases for quantum Borcherds-Bozec algebras Fan, Zhaobing Han, Shaolong Kang, Seok-Jin Kim, Young Rock Representation Theory Quantum Algebra Let $U_{q}^{-}(\mathfrak g)$ be the negative half of a quantum Borcherds-Bozec algebra $U_{q}(\mathfrak g)$ and $V(λ)$ be the irreducible highest weight module with $λ\in P^{+}$. In this paper, we investigate the structures, properties and their close connections between crystal bases and canonical bases of $U_{q}^{-}(\mathfrak g)$ and $V(λ)$. We first re-construct crystal basis theory with modified Kashiwara operators. While going through Kashiwara's grand-loop argument, we prove several important lemmas, which play crucial roles in the later developments of the paper. Next, based on the theory of canonical bases on quantum Bocherds-Bozec algebras, we introduce the notion of primitive canonical bases and prove that primitive canonical bases coincide with lower global bases. |
| title | Crystal bases and canonical bases for quantum Borcherds-Bozec algebras |
| topic | Representation Theory Quantum Algebra |
| url | https://arxiv.org/abs/2211.02859 |