Generating System Of Some Subalgebras Of Hall Algebras

It is well known that the generic Ringel-Hall algebra is one of the most successful model of the quantum group. Most important feature of Ringel-Hall algebra approach is that it makes the use of methods in representation theory of algebra, especially the use of methods in homological algebra, and the Auslander-Reiten theory on quantum groups become possible. Rather, we let k is a finite field and A is a finite dimensional hereditary k-algebra. It is well-known that there is a one to one correspondence between the valued graph A and the generalized Cartan matrixΔ. Ringel and Green's work (see) show that under this correspondence, there is an isomorphism between the composition subalgebra of Ringel-Hall algebra of A and the positive part of the quantum group of typeΔ. This important result means that the Ringel-Hall algebra give a very powerful method to study quantum groups and the representation theory of A get an interpretation and application in quantum groups. Therefore, the information on the category ofΛ-modules can provide some new information about the structure of corresponding quantum group. In Ringel gave a generating system for the Ringel-Hall subalgebras corresponding to APR-tilting modules in the category of representation directed algebras. In this paper, we give a generating system for the Ringel-Hall subalgebras corresponding to any tilting modules in the module category of representation directed algebras by using the results on homologically finite subcategories of the category ofΛ-modules. However, the two generating systems are not coinside in the APR-tilting case, in general.