Lusztig sheaves and integrable highest weight modules
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910876191162368 |
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| author | Fang, Jiepeng Lan, Yixin Xiao, Jie |
| author_facet | Fang, Jiepeng Lan, Yixin Xiao, Jie |
| contents | We consider the localization $\mathcal{Q}_{\mathbf{V},\mathbf{W}}/\mathcal{N}_{\mathbf{V}}$ of Lusztig's sheaves for framed quivers, and define functors $E^{(n)}_{i},F^{(n)}_{i},K^{\pm}_{i},n\in \mathbb{N},i \in I$ between the localizations. With these functors, the Grothendieck group of localizations realizes the irreducible integrable highest weight modules $L(Λ)$ of quantum groups. Moreover, the nonzero simple perverse sheaves in localizations form the canonical bases of $L(Λ)$. We also compare our realization (at $v \rightarrow 1$) with Nakajima's realization via quiver varieties and prove that the transition matrix between canonical bases and fundamental classes is upper triangular with diagonal entries all equal to $\pm 1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_16131 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Lusztig sheaves and integrable highest weight modules Fang, Jiepeng Lan, Yixin Xiao, Jie Representation Theory Algebraic Geometry Quantum Algebra 16G20, 17B37 We consider the localization $\mathcal{Q}_{\mathbf{V},\mathbf{W}}/\mathcal{N}_{\mathbf{V}}$ of Lusztig's sheaves for framed quivers, and define functors $E^{(n)}_{i},F^{(n)}_{i},K^{\pm}_{i},n\in \mathbb{N},i \in I$ between the localizations. With these functors, the Grothendieck group of localizations realizes the irreducible integrable highest weight modules $L(Λ)$ of quantum groups. Moreover, the nonzero simple perverse sheaves in localizations form the canonical bases of $L(Λ)$. We also compare our realization (at $v \rightarrow 1$) with Nakajima's realization via quiver varieties and prove that the transition matrix between canonical bases and fundamental classes is upper triangular with diagonal entries all equal to $\pm 1$. |
| title | Lusztig sheaves and integrable highest weight modules |
| topic | Representation Theory Algebraic Geometry Quantum Algebra 16G20, 17B37 |
| url | https://arxiv.org/abs/2307.16131 |