Quantum Algebras and Cyclic Quiver Varieties
Quantum Algebras and Cyclic Quiver Varieties
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量子代数和循环箭袋簇
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发表时间：
2015
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通讯作者：
Andrei Neguct
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作者：
Andrei Neguct
量子代数与循环箭图簇
安德烈·内古特
本论文的目的是提出关于中岛循环箭图簇的几何表示理论的某些观点，这些观点与毛里克 - 奥昆科夫稳定基相关。我们的主要技术工具是洗牌代数，它作为双循环箭图的K - 理论霍尔代数出现。我们证明了洗牌代数与量子环面代数\(U_{q,t}(\hat{\mathfrak{s}\mathfrak{l}_n})\)之间的同构，并将洗牌代数的 Verma 模的商与中岛和瓦拉尼奥洛 - 瓦塞罗所研究的中岛循环箭图簇的K - 理论群等同起来。洗牌代数的观点使我们能够构造量子环面代数\(U_{q,t}(\hat{\mathfrak{s}\mathfrak{l}_n})\)的通用\(R\) - 矩阵，并根据来自与各种\(m\)的量子仿射群\(U_q(\hat{\mathfrak{g}\mathfrak{l}_m})\)同构的子代数的部分对其进行分解。这种分解将霍罗什金 - 托尔斯泰的构造推广到环面情形，并与毛里克 - 奥昆科夫通过中岛箭图簇的K - 理论中的稳定基所产生的分解相匹配。我们通过计算\(U_{q,t}(\hat{\mathfrak{s}\mathfrak{l}_n})\)的根生成元作用于稳定基的公式来连接这两种情形，这从组合学角度对默纳汉 - 中岛和皮埃里型规则进行了广泛的扩展。
Quantum Algebras and Cyclic Quiver Varieties Andrei Negut, The purpose of this thesis is to present certain viewpoints on the geometric representation theory of Nakajima cyclic quiver varieties, in relation to the Maulik-Okounkov stable basis. Our main technical tool is the shuffle algebra, which arises as the K−theoretic Hall algebra of the double cyclic quiver. We prove the isomorphism between the shuffle algebra and the quantum toroidal algebra Uq,t(s̈ln), and identify the quotients of Verma modules for the shuffle algebra with the K−theory groups of Nakajima cyclic quiver varieties, which were studied by Nakajima and VaragnoloVasserot. The shuffle algebra viewpoint allows us to construct the universal R−matrix of the quantum toroidal algebra Uq,t(s̈ln), and to factor it in terms of pieces that arise from subalgebras isomorphic to quantum affine groups Uq(ġlm), for various m. This factorization generalizes constructions of Khoroshkin-Tolstoy to the toroidal case, and matches the factorization that Maulik-Okounkov produce via the stable basis in the K−theory of Nakajima quiver varieties. We connect the two pictures by computing formulas for the root generators of Uq,t(s̈ln) acting on the stable basis, which provide a wide extension of Murnaghan-Nakayama and Pieri type rules from combinatorics.