论文标题
$ d^\ mathrm b（\ mathcal o_0）$ chaffling函数和球形曲折
Shuffling functors and spherical twists on $D^\mathrm b(\mathcal O_0)$
论文作者
论文摘要
对于半圆形复合物lie代数$ \ mathfrak g $，bgg类别$ \ mathcal {o} $在表示理论中特别感兴趣。众所周知，Irving的改组功能元素$ \ MATHRM {SH} _ {W} $，由Weyl of w $ in W $ in Weyl of weyl of weyl of的元素索引，诱导了与派生的类别上$ w $相关的编织组$ b_w $ $ d $ d^$ d^$ d^\ d^\ mathrm {b}（ $ \ MATHCAL {O} $。 我们表明，对于最大抛物线副词，$ \ mathfrak {p} $ of $ \ mathfrak {sl} _n $对应于抛物线子群$ w_ \ mathfrak {p} = s_ {n-1} $ \ mathbf {l} \ mathrm {sh} {s_i} $是Seidel和Thomas的Spherical Twist函数的实例。也就是说，我们表明某些抛物面不可塑性的预测$ p^\ mathfrak {p}（w）$是球形对象，并且相关的扭曲函数自然是同构对$ \ mathbf {l} \ mathrm {sh} {sh} $ d^\ mathrm {b}（\ mathcal {o}^\ mathfrak {p}）$。 我们概述了BGG类别$ \ Mathcal {O} $的主要属性，改组和球形扭曲函数的构造，并给出一些示例，如何确定两者的图像。为此，我们采用了$ \ Mathcal {O} $的块和某些路径代数的模块类别的等效性。
For a semisimple complex Lie algebra $\mathfrak g$, the BGG category $\mathcal{O}$ is of particular interest in representation theory. It is known that Irving's shuffling functors $\mathrm{Sh}_{w}$, indexed by elements $w\in W$ of the Weyl group, induce an action of the braid group $B_W$ associated to $W$ on the derived categories $D^\mathrm{b}(\mathcal{O}_λ)$ of blocks of $\mathcal{O}$. We show that for maximal parabolic subalgebras $\mathfrak{p}$ of $\mathfrak{sl}_n$ corresponding to the parabolic subgroup $W_\mathfrak{p}=S_{n-1}\times S_1$ of $S_n$, the derived shuffling functors $\mathbf{L}\mathrm{Sh}{s_i}$ are instances of Seidel and Thomas' spherical twist functors. Namely, we show that certain parabolic indecomposable projectives $P^\mathfrak{p}(w)$ are spherical objects, and the associated twist functors are naturally isomorphic to $\mathbf{L}\mathrm{Sh}{w}[1]$ as auto-equivalences of $D^\mathrm{b}(\mathcal{O}^\mathfrak{p})$. We give an overview of the main properties of the BGG category $\mathcal{O}$, the construction of shuffling and spherical twist functors, and give some examples how to determine images of both. To this end, we employ the equivalence of blocks of $\mathcal{O}$ and the module categories of certain path algebras.
