Sheaf realization of Bridgeland's Hall algebra of Dynkin type
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917120135135232 |
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| author | Fang, Jiepeng Lan, Yixin Xiao, Jie |
| author_facet | Fang, Jiepeng Lan, Yixin Xiao, Jie |
| contents | As one of results in [6], Bridgeland realized the quantum group $\mathbf{U}_v$ via the localization of Ringel-Hall algebra for the two-periodic projective complexes of quiver representations over a finite field. In the present paper, we generalize Lusztig's categorical construction for the nilpotent part $\mathbf{U}_v^+$ to Bridgeland's Hall algebra of Dynkin type. In particular, we obtain a basis of the Ringel-Hall algebra for the two-periodic projective complexes which has the positivity, and we categorify an integral form of the generic Bridgeland's Hall algebra which is isomorphic to the Poisson integral form of $\mathbf{U}_v$, and obtain a $\mathbb{Z}[v,v^{-1}]$-basis of this integral form. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_04993 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Sheaf realization of Bridgeland's Hall algebra of Dynkin type Fang, Jiepeng Lan, Yixin Xiao, Jie Representation Theory 16G20, 17B37, 18N25 As one of results in [6], Bridgeland realized the quantum group $\mathbf{U}_v$ via the localization of Ringel-Hall algebra for the two-periodic projective complexes of quiver representations over a finite field. In the present paper, we generalize Lusztig's categorical construction for the nilpotent part $\mathbf{U}_v^+$ to Bridgeland's Hall algebra of Dynkin type. In particular, we obtain a basis of the Ringel-Hall algebra for the two-periodic projective complexes which has the positivity, and we categorify an integral form of the generic Bridgeland's Hall algebra which is isomorphic to the Poisson integral form of $\mathbf{U}_v$, and obtain a $\mathbb{Z}[v,v^{-1}]$-basis of this integral form. |
| title | Sheaf realization of Bridgeland's Hall algebra of Dynkin type |
| topic | Representation Theory 16G20, 17B37, 18N25 |
| url | https://arxiv.org/abs/2303.04993 |