Affine Brauer category and category O in types B,C,D
Affine Brauer category and category O in types B,C,D
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B、C、D 类型中的仿射布劳尔类别和 O 类别
DOI：
10.1007/s00209-018-2207-x
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发表时间：
2018
影响因子：
0.8
通讯作者：
Linliang Song
中科院分区：
文献类型：
--
作者：
Hebing Rui;Linliang Song
在包含乘法单位1和可逆元2的交换环κ上，引入了一个严格的一元范畴，即仿射Brauer范畴AB。证明了AB中的态射空间在κ上是自由的。环切（或水平k） Brauer范畴CBf（ω）是AB的商范畴。我们证明了CBf（ω）中的任何态射空间在κ上自由且秩极大，当且仅当u可容许条件在（1.32）意义下成立。仿射的Nazarov - wenzl代数（Nazarov in J Algebra 182(3): 664-693, 1996）和环切的Nazarov - wenzl代数（Ariki et al. in Nagoya mathematics 182:47 - 134, 2006）将分别在AB和CBf（ω）中被实现为一定的自同态代数。我们将在复域c上建立环切Nazarov-Wenzl代数和抛物型BGG类O之间更高的Schur-Weyl对偶性。这使我们能够在(Anderson et al. in Pac J Math 292(1): 21-59, 2018；数学学报，2015 (3):669-689；Rui and Song, J Algebra, 444:246-271, 2015)，计算分环Nazarov-Wenzl代数的分解矩阵。二级案例由Ehrig和Stroppel在（Adv. Math. 331:58-142, 2018）中考虑。
A strict monoidal category referred to as affine Brauer category AB is introduced over a commutative ring κ containing multiplicative identity 1 and invertible element 2. We prove that morphism spaces in AB are free over κ. The cyclotomic (or level k) Brauer category CBf(ω) is a quotient category of AB. We prove that any morphism space in CBf(ω) is free over κ with maximal rank if and only if the u-admissible condition holds in the sense of (1.32). Affine Nazarov–Wenzl algebras (Nazarov in J Algebra 182(3):664–693, 1996) and cyclotomic Nazarov–Wenzl algebras (Ariki et al. in Nagoya Math J 182:47–134, 2006) will be realized as certain endomorphism algebras in AB and CBf(ω), respectively. We will establish higher Schur–Weyl duality between cyclotomic Nazarov–Wenzl algebras and parabolic BGG categories O associated to symplectic and orthogonal Lie algebras over the complex field C. This enables us to use standard arguments in (Anderson et al. in Pac J Math 292(1):21–59, 2018; Rui and Song in Math Zeit 280(3–4):669–689, 2015; Rui and Song in J Algebra 444:246–271, 2015), to compute decomposition matrices of cyclotomic Nazarov–Wenzl algebras. The level two case was considered by Ehrig and Stroppel in (Adv. Math. 331:58–142, 2018).