Affine flag varieties and quantum symmetric pairs
Affine flag varieties and quantum symmetric pairs
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DOI：
10.1090/memo/1285
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发表时间：
2016-02
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影响因子：
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通讯作者：
Zhaobing Fan;C. Lai;Yiqiang Li;Lipeng Luo;Weiqiang Wang
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文献类型：
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作者：
Zhaobing Fan;C. Lai;Yiqiang Li;Lipeng Luo;Weiqiang Wang
有限型和仿射型\(A\)的量子群可以通过有限型和仿射型\(A\)的部分旗簇给出几何实现。最近，与有限型\(B/C\)的部分旗簇相关联的量子群被证明是有限型\(A\)量子群的一个余理想子代数。在本文中，我们研究与仿射型\(C\)的（四个变体的）部分旗簇相关联的舒尔代数和卢斯蒂格代数的结构。我们表明，通过稳定化过程从卢斯蒂格代数和舒尔代数产生的量子群分别是仿射\(\mathfrak{sl}\)型和\(\mathfrak{gl}\)型量子群的（幂等化的）余理想子代数。通过这种方式，我们给出了仿射型的八个量子对称对的几何实现。我们构造了所有这些量子（舒尔、卢斯蒂格和余理想）代数的单项式基和典范基。对于仿射\(\mathfrak{sl}\)型的幂等化余理想代数，我们建立了典范基关于乘法、余乘法和一个双线性配对的正性性质。特别地，我们得到了幂等化量子仿射\(\mathfrak{gl}\)及其典范基的一种新的几何构造。
The quantum groups of finite and affine type $A$ admit geometric realizations in terms of partial flag varieties of finite and affine type $A$. Recently, the quantum group associated to partial flag varieties of finite type $B/C$ is shown to be a coideal subalgebra of the quantum group of finite type $A$. In this paper we study the structures of Schur algebras and Lusztig algebras associated to (four variants of) partial flag varieties of affine type $C$. We show that the quantum groups arising from Lusztig algebras and Schur algebras via stabilization procedures are (idempotented) coideal subalgebras of quantum groups of affine $\mathfrak{sl}$ and $\mathfrak{gl}$ types, respectively. In this way, we provide geometric realizations of eight quantum symmetric pairs of affine types. We construct monomial and canonical bases of all these quantum (Schur, Lusztig, and coideal) algebras. For the idempotented coideal algebras of affine $\mathfrak{sl}$ type, we establish the positivity properties of the canonical basis with respect to multiplication, comultiplication and a bilinear pairing. In particular, we obtain a new and geometric construction of the idempotented quantum affine $\mathfrak{gl}$ and its canonical basis.