论文标题
$ \ imath $ Quantum组的相对编织组对称性的固有方法
An intrinsic approach to relative braid group symmetries on $\imath$quantum groups
论文作者
论文摘要
我们对（通用）$ \ imath $量子组的相对编织组对称性启动了一种通用方法，该组是由量子对称对对称的任意有限类型及其模块引起的。我们的方法建立在我们开发的新的$ k $ - amatrices的新相互交织属性的基础上，并在（德林菲尔德双重）量子组上进行了编织组对称性。获得了这些新对称性的明确公式，以$ \ imath $量子组获得。我们在$ \ imath $量子组上为这些对称性建立了许多基本属性，与其众所周知的量子组同行平行。我们应用这些对称性来完全建立排名的准$ k $ - amatrices，而这种分解属性反过来有助于表明新的对称性满足相对辫子关系。结果，肯定地解决了科尔布 - 佩尔利格里尼和多布森 - 科尔布的猜想。最后，上述方法使我们能够首次在模块上构建兼容的相对辫子组动作。
We initiate a general approach to the relative braid group symmetries on (universal) $\imath$quantum groups, arising from quantum symmetric pairs of arbitrary finite types, and their modules. Our approach is built on new intertwining properties of quasi $K$-matrices which we develop and braid group symmetries on (Drinfeld double) quantum groups. Explicit formulas for these new symmetries on $\imath$quantum groups are obtained. We establish a number of fundamental properties for these symmetries on $\imath$quantum groups, strikingly parallel to their well-known quantum group counterparts. We apply these symmetries to fully establish rank one factorizations of quasi $K$-matrices, and this factorization property in turn helps to show that the new symmetries satisfy relative braid relations. As a consequence, conjectures of Kolb-Pellegrini and Dobson-Kolb are settled affirmatively. Finally, the above approach allows us to construct compatible relative braid group actions on modules over quantum groups for the first time.
