Lusztig sheaves and integrable highest weight modules in symmetrizable cases
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911040147554304 |
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| author | Lan, Yixin Wu, Yumeng Xiao, Jie |
| author_facet | Lan, Yixin Wu, Yumeng Xiao, Jie |
| contents | The present paper continues the work of [10] and [6]. For any symmetrizable generalized Cartan Matrix $C$ and the corresponding quantum group $\mathbf{U}$, we consider the associated quiver $Q$ with an admissible automorphism $a$. We construct the category $\widetilde{\mathcal{Q}/\mathcal{N}}$ of the localization of Lusztig sheaves for the quiver with the automorphism of corresponding framed quiver and 2-framed quiver. Their Grothendieck groups give realizations of integrable highest weight module $L(λ)$ and the tensor product of integrable highest weights $\mathbf{U}-$module $L(λ_1)\otimes L(λ_2)$, and modulo the traceless ones Lusztig sheaves provide the (signed) canonical basis of $L(λ)$ and $L(λ_1)\otimes L(λ_2)$. As an application, the symmetrizable crystal structures on Nakajima's quiver/tensor product varieties and Lusztig's nilpotent varieties of preprojective algebras are deduced. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_09188 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lusztig sheaves and integrable highest weight modules in symmetrizable cases Lan, Yixin Wu, Yumeng Xiao, Jie Representation Theory Quantum Algebra Rings and Algebras 16G20, 17B37 The present paper continues the work of [10] and [6]. For any symmetrizable generalized Cartan Matrix $C$ and the corresponding quantum group $\mathbf{U}$, we consider the associated quiver $Q$ with an admissible automorphism $a$. We construct the category $\widetilde{\mathcal{Q}/\mathcal{N}}$ of the localization of Lusztig sheaves for the quiver with the automorphism of corresponding framed quiver and 2-framed quiver. Their Grothendieck groups give realizations of integrable highest weight module $L(λ)$ and the tensor product of integrable highest weights $\mathbf{U}-$module $L(λ_1)\otimes L(λ_2)$, and modulo the traceless ones Lusztig sheaves provide the (signed) canonical basis of $L(λ)$ and $L(λ_1)\otimes L(λ_2)$. As an application, the symmetrizable crystal structures on Nakajima's quiver/tensor product varieties and Lusztig's nilpotent varieties of preprojective algebras are deduced. |
| title | Lusztig sheaves and integrable highest weight modules in symmetrizable cases |
| topic | Representation Theory Quantum Algebra Rings and Algebras 16G20, 17B37 |
| url | https://arxiv.org/abs/2411.09188 |