论文标题
$ \ imath $ hall代数和$ \ imath $量子组
$\imath$Hall algebras and $\imath$quantum groups
论文作者
论文摘要
我们调查了有关$ \ imath $ hall代数理论的一些最新发展。从$ \ imath $ Quivers（又名带有互动的Quivers）开始，我们构建了一类1-Gorenstein代数，称为$ \ imath $ Quiver代数，其半衍生的霍尔代数为我们提供了$ \ imath $ Hall algebras。然后，我们使用这些$ \ imath $ hall代数来实现由量子对称对产生的准拼写$ \ imath $量子组。 $ \ imath $量子组的相对编织组对称性通过反射函数实现。如果Jordan $ \ imath $ Quiver，则$ \ imath $ hall代数是可交换的，并且与$ \ imath $ hall $ hall-littlewood对称功能的连接得以开发。如果$ \ imath $的对角线类型震颤，我们的构造等于对德林菲尔德双重量子组的Bridgeland-hall代数实现（反过来概述了Ringel-hall代数实现量子组的一半））。提供许多等级1和等级2计算以说明一般构造。我们还简要介绍了加权线条的$ \ imath $ halter代数，并使用它们来实现$ \ imath $ Quantum Loop代数的Drinfeld类型演示文稿。
We survey some recent development on the theory of $\imath$Hall algebras. Starting from $\imath$quivers (aka quivers with involutions), we construct a class of 1-Gorenstein algebras called $\imath$quiver algebras, whose semi-derived Hall algebras give us $\imath$Hall algebras. We then use these $\imath$Hall algebras to realize quasi-split $\imath$quantum groups arising from quantum symmetric pairs. Relative braid group symmetries on $\imath$quantum groups are realized via reflection functors. In case of Jordan $\imath$quiver, the $\imath$Hall algebra is commutative and connections to $\imath$Hall-Littlewood symmetric functions are developed. In case of $\imath$quivers of diagonal type, our construction amounts to a reformulation of Bridgeland-Hall algebra realization of the Drinfeld double quantum groups (which in turn generalizes Ringel-Hall algebra realization of halves of quantum groups). Many rank 1 and rank 2 computations are supplied to illustrate the general constructions. We also briefly review $\imath$Hall algebras of weighted projective lines, and use them to realize Drinfeld type presentations of $\imath$quantum loop algebras.
