The Projective Class Rings of Drinfeld doubles of pointed rank one Hopf algebras
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866917127059931136 |
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| author | Sun, Hua Chen, Hui-Xiang Li, Libin Zhang, Yinhuo |
| author_facet | Sun, Hua Chen, Hui-Xiang Li, Libin Zhang, Yinhuo |
| contents | Let $\Bbbk$ be an algebraically closed field of characteristic $0$. In this paper, we study the Grothendieck ring $G_0(D(H_\mathcal{D}))$ and the projective class ring $r_p(D(H_\mathcal{D}))$ of the Drinfeld double $D(H_{\mathcal{D}})$ of the rank one pointed Hopf algebra $H_{\mathcal{D}}$. We analyze the tensor products of simple modules with simple modules, simple modules with indecomposable projective modules, and indecomposable projective modules with indecomposable projective modules, providing explicit decomposition rules in each case. Finally, we compute both the Grothendieck ring $G_0(D(H_\mathcal{D}))$ and the projective class ring $r_p(D(H_\mathcal{D}))$, and present these two rings in terms of generators and defining relations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_21163 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Projective Class Rings of Drinfeld doubles of pointed rank one Hopf algebras Sun, Hua Chen, Hui-Xiang Li, Libin Zhang, Yinhuo Quantum Algebra Commutative Algebra Let $\Bbbk$ be an algebraically closed field of characteristic $0$. In this paper, we study the Grothendieck ring $G_0(D(H_\mathcal{D}))$ and the projective class ring $r_p(D(H_\mathcal{D}))$ of the Drinfeld double $D(H_{\mathcal{D}})$ of the rank one pointed Hopf algebra $H_{\mathcal{D}}$. We analyze the tensor products of simple modules with simple modules, simple modules with indecomposable projective modules, and indecomposable projective modules with indecomposable projective modules, providing explicit decomposition rules in each case. Finally, we compute both the Grothendieck ring $G_0(D(H_\mathcal{D}))$ and the projective class ring $r_p(D(H_\mathcal{D}))$, and present these two rings in terms of generators and defining relations. |
| title | The Projective Class Rings of Drinfeld doubles of pointed rank one Hopf algebras |
| topic | Quantum Algebra Commutative Algebra |
| url | https://arxiv.org/abs/2511.21163 |