Ringel-Hall Algebras Of Infinite Quivers And Corresponding Quantum Groups

This PhD dissertation is on Ringel-Hall algebras of infinite quivers without oriented cycles or loops and related topics. The theory of Ringel-Hall algebras of infinite quivers without oriented cycles is first founded in this thesis, some important property of this type Ringel-Hall algebra is exposed, and a realization of the quantum group of the infinite quiver without oriented cycles is firstly made by using Ringel-Hall algebra methods corresponding to the infinite quiver without oriented cycles. The dissertation consists of three main parts.Given a infinite quiver Q without oriented cycles, we have the corresponding Cartan datum, then we have the corresponding quantum group U_q(Q).1. We find a series of finite subquivers of Q, i.e. Q1, Q2, …, such that Q\ C Q₂ C… and Q = U(+∞)/(i＝1)Qi. Let k be a finite field. Then we construct the twisted Ringel-Hall algebra H_*(kQ) with basis of the set of iso-classes of finite dimension kQ— modules by using extension of modules. We prove that the twisted Ringel-Hall algebra H_*(kQi) can be viewed as a subalgebra of the twisted Ringel-Hall algebra H_*(kQj) for any i, j ∈ (?)>0, i < j and that the twisted Ringel-Hall algebra H_*(kQi) can be viewed as a subalgebra of the twisted Ringel-Hall algebra H_*(kQ); Let the inclusion map from H_*(kQi) to H_*(kQ) be (?)ij, then {(?)ij|i,j ∈ (?)>0,i < j} is a directed system, hence there exists the directed limit lim_i→+∞H_*(kQi), we prove that H_*(kQ) ≌ 2. Since the set (?) of all finite dimensional kQ— simple modules and the symmetric Ringel form (—, —) defined on the category of finite dimensional kQ— modules constitute a Cartan datum ((?), (—,—)), we derive the...