考虑Dynkin型遗传代数的Hall代数，利用遗传代数表示范畴的结构和性质给出Ringel意义下canonical基的构造。进而用遗传代数表示范畴信息给出canonical基对合不变性的解释，由此推广上述结果到任意型。这样就能完全给出Lusztig意义下的canonical基的代数表示论解释，为研究量子群的表示理论提供全新的角度和有力的工具。

By using the Ringel-Hall algebra approach, we find a Lie algebra.arising in each 2-period triangulated category; as application, we realize all symmetrizable Kac-Moody Lie algebras from root categrories.of hereditary algebras. We also realize elliptic Lie algebras of type D_4 and E from root categrories of tubular algebras and get the Chevalley bases..A natural embedding is obtained from the Ringel-Hall algebra of a tilted.algebra into the Drinfeld double of the Ringel-Hall algebra of the.corresponding hereditary algebra; moreover, we get some isomorphisms and.automorphisms of the quantum group corresponding to isomorphisms and.automorphisms of the derived categories of hereditary algebras. We define.APR-transformations for the root base attached to a generalizedintersection matrix and explain them by using APR-tilting modules. And we give an inductive algorithm for Ringel-Hall polynomials of cyclic serial algebras..Interpreting Hurwitz numbers as the relative Gromov － Witten invariants and using a gluing formula we derive a recursive formula for Hurwitz numbers of any compact Riemann surface. Blaschke hypersurfaces with prescribed Gauss － Kronecker curvature are studied globally. We classify Moebius isotropic submanifolds in the n-dimensional sphere and Moebius isoparametric hypersurfaces in the n+1-dimensional sphere with two distinct principal curvatures.