Several Types Of Lean Structural Characterizations Of Quasi-hereditary Algebra,

The Constructive Description of Some Classes ofLean Quasi-hereditary AlgebrasThou ZhongmeiAbstract The notion of the quasi-hereditary algebra has been introduced by E. dine, B. Parshall and L. Scott in order to study the highest weight categories arising in the representation theory of complex semi-simple Lie algebra and algebraic group. It has been proved that these algebras appear quite common, such as hereditary algebras, Schur algebras and Auslander algebras. I. Agoston, V. Dlab and E. Lukacs introduced a class of quasi-hereditary algebras with additional properties--lean quasi-hereditary algebra. In terms of the so-called top filtration, they characterized this class of the quasi-hereditary algebra and gave lower and upper bounds for the dimension of lean quasi-hereditary algebras in the same species. They proved that the dimension of a lean quasi--hereditary algebra is equal to the lower (upper) bound if and only if the algebra is shallow (replete, respectively).We recalled the basic concepts and properties of lean quasihereditary algebras in § 1. In § 2, first we recall the concepts of the left (right) medial algebra, then we give a characterization of the left (right) medial algebra.The trivially twisted extension algebra (esp. the dual extension algebra) was introduced by Xi Changchang. Deng Bangmeng , XiChangchang and Du Xia~nneng ect. , studied these algebras systematically.In 3,by the important tool of twisted extension algebra, we are going to give a constructive characterization of this class of the lean quasi-hereditary algebras.