Quasi-hereditary Algebras And Generalized Path Algebras

Quasi-hereditary algebras and their generalizations such as standardly strat-ified algebras were first introduced by Cline, Parshall and Scott in order to study highest weight categories in the representation theory of complex semi-simple Lie algebras and algebraic groups. Since then, these algebras, which possess many good properties, have been studied by many authors (for exam-ple:Dlab,Ringel,C.C.Xi). Chapter Two introduces Quasi-hereditary algebras’ definition, basic properties, construction methods and applications, as well as gives a concrete proof for hereditary algebras being quasi-hereditary. In Chapter Three, we introduce two kinds of stratified algebras, and then compare them with cellular algebras respectively.Last chapter gives a generalization of the concept of path algebras, i.e. gen-eralized path algebras. We prove isomorphism problem for normal generalized path algebras.It is known that the generalized path algebras over quasi-hereditary algebras are quasi-hereditary. Similarity, we show that the tensor algebras over quasi-hereditary algebras are quasi-hereditary.