导出Hall代数，丛代数和范畴化

We study the structures of Hall algebras over periodic derived categories and cluster categories by applying Bridgeland-Hall's approaches and derived Hall algebras. Then we deduce the algebra homomorphisms from derived Hall algebras to (quantum) cluster algebras, which reveal the interactive relations between two kinds of algebras. Furthermore, under the framework of the geometric constructions of canonical bases and KLR algebras, we study the categorification of Hall algebras and relate it to the categorification of (quantum) cluster algebras via the multiplication properties of dual canonical bases. We connect the above constructions to Nakajima's quiver variety through comparing Hall algebras over periodic derived categories with the geometric realizations of finite dimensional quotients of enveloping algebras via quiver varieties, quantum groups via cyclic quiver varieties and Hall algebras via graded quiver varieties.

应用Bridgeland-Hall代数方法和导出Hall代数，我们研究周期导出范畴和丛范畴上Hall代数的结构，推导出从导出Hall代数到（量子）丛代数的代数同态，从而建立导出Hall代数到丛代数的交互关系。进一步，基于典范基和KLR代数的几何构造，我们研究Hall代数范畴化。基于对偶典范基的乘法性质，研究它和（量子）丛代数范畴化的关系。我们关联以上的构造和Nakajima的quiver variety。这包括比较周期导出范畴上Hall代数和Nakajima使用quiver variety给出的关于包络代数的有限维商的几何实现以及cyclic quiver variety上Hall代数构造，研究graded quiver variety上Hall代数构造。

在本次项目中，我们扩展导出Hall代数的构造方法，实现了轨道三角范畴上Hall代数的定义， 包括根范畴上Hall代数和丛范畴上Hall代数。我们扩展了Ringel-Hall代数到量子丛代数的代数同态，构造了导出Hall代数到量子丛代数的满代数同态，加强了Hall代数和丛代数的关联。我们推广Lusztig关于量子群的范畴化构造到Ringel-Hall代数，这扩展了典范基的定义。我们定义了广义丛代数的量子版本， 刻画相应的Laurent现象。