Affine cellularity of quantum affine algebras
Affine cellularity of quantum affine algebras
复制标题
量子仿射代数的仿射元胞性
DOI：
10.1016/j.jalgebra.2015.07.017
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发表时间：
2015
影响因子：
0.9
通讯作者：
Hiraku Nakajima
中科院分区：
文献类型：
--
作者：
小峰昇一;秋山健太郎;蕨 栄治;正田純一;Hiraku Nakajima
这是 Cui 论文 [3] 的附录，表明修改后的 0 级量子仿射代数 U= Uq (g)（更准确地说是它的商，BLN 代数）是 Koenig 和 Xi [5] 意义上的仿射元胞。该证明基于 U 的单元结构，之前在作者与 Beck 合作的[1]中研究过。我们在这里给出基于[1，Lem. 6.17]，以及[11]中引入的双线性形式的属性。请注意[1，莱姆。 6.17]和双线性形式是[1]中细胞结构研究的重要组成部分。从这个意义上说，下面的证明比[3]中的证明更直接和基本。我们还证明了单元理想是幂等的，因此[5，Th。 4.4]适用。因此BLN代数具有有限的全局维数，并且其派生范畴允许分层，其截面等价于一般线性群乘积的表示环的派生范畴。一旦我们记住 U 具有最高权重理论，证明或多或少就是一个简单的观察。
This is an appendix to Cui's paper [3] showing that the modified quantum affine algebra U= Uq (g) of level 0 (more precisely its quotients, BLN algebras) is affine cellular in the sense of Koenig and Xi [5]. The proof is based on the structure of cells of U, studied previously in [1], the author's joint work with Beck. We here give a proof based on [1, Lem. 6.17], together with a property of the bilinear form introduced in [11]. Note that [1, Lem. 6.17] and the bilinear form are crucial ingredients for the study of the structure of cells in [1]. In this sense the following proof is more direct and fundamental than one in [3].We also prove that cell ideals are idempotent, and hence [5, Th. 4.4] is applicable. Therefore BLN algebras are of finite global dimension, and its derived category admits a stratification whose sections are equivalent to derived categories of representation rings of products of general linear groups. The proof is, more or less, a simple observation once we remember that U has the highest weight theory.