1 Introduction 1
1.1 Background 1
1.2 Main results 2
1.3 Organization 8
2 Preliminaries 10
2.1 Finitary algebras and their Ringel-Hall algebras 10
2.2 Grothendieck groups and twisted Ringel-Hall algebras 12
2.3 Double Ringel-Hall algebras of hereditary algebras 13
2.4 Representations of tame hereditary algebras 18
3 Ringel-Hall algebras and the fundamental relations 22
3.1 Fundamental relations 22
3.2 Higher order fundamental relations 29
3.3 Composition monoid algebra QMc 32
3.4 Quotients of Ringel-Hall algebras 38
4 Subalgebras of Ringel-Hall algebras of tame quivers 45
4.1 Hopf subalgebras of D(A) 45
4.2 Double Ringel-Hall algebras of cyclic quivers 47
4.3 Acyclic quivers of type ？ 55
5 Finite dimensional weight D(△2)-modules 61
5.1 Finite dimensional irreducible representations of Uv(？l2） 61
5.2 Finite dimensional weight D(△2)-modules arising from Uv(？l2)-modules 65
5.3 Finite dimensional weight D(△2)-modules arising from Uv(？l2)-modules 75
References 81