Representations Of Valued Graph With Cycles And Double Ringel-Hall Algebras

The main conclusion of Kac Theorem is the correspondence between the set of the dimension vectors of indecomposable representations of a graph and the set of positive roots of the matrix associated to the graph. This thesis consider this subject for the case of valued quivers with cycles. And then we prove it by using the method of Ringel-Hall algebras.Firstly, for any symmetrizable Borcherds-Cartan matrix A with integer entries and even diagonal entries, we can define a Borcherds datum C of A, and there exists a valued quiver Γ and a F_q-species S of Γ such that the symmetric Euler form of S is the Borcherds datum C of A. Given any quiver (Q, σ) (called as a α-quiver) with automorphism (admissible or non-admissible), we can construct a valued quiver Γ and a F_q-species S. But any valued quiver Γ can be obtained from a α-quiver (Q,σ) without loops. By introducing Frobenius morphism F on the path algebra of (Q, σ), we can connect the F_q-representations of (Γ, S) with F-stable representations of (Q,σ) over F_q. So by using the Burnside orbit counting formula, we can get the formula of the number of representations (indecomposable representations) of any valued quiver with fixed dimension vector. And then we get the Kac Theorem for the case of valued quiver with cycles by using the method of Frobeninus morphisms and quiver automorphisms.By Green formula, comultiplication on the (extended twisted) Ringel-Hall algebra generated by the basis indexed by the isomorphic classes of A-modules where A is a finite dimensional hereditary algebra can be defined. And then it is a Hopf algebra by defining antipode on it. The Double Ringel-Hall algebra is canonically isomorphic to the quantized enveloping algebra of a Borcherds generalized Kac-Moody algebra. In this thesis, we begin with valued quivers (Γ,S) which may have cycles. First, we study the Ringel-Hall algebra H((?)) and Double Ringel-Hall algebra D((?)) defined by nilpotent representations of S. By studying the Double Generic Composition algebra C~*((?)) , we get that C~*((?)) is canoni-cally isomorphic to the quantized enveloping algebra of a generalized Kac-Moody algebra. By decomposing D((?)), on one hand, we get that D((?)) is canonically isomorphic to the quantum enveloping algebra of a generalized Kac-Moody algebra. On the other hand, the complete reducibility of integrable highest weight D((?))-modules are considered and we get the Weyl-Kac character formula for the irreducible highest weight D((?))-modules with dominant highest weights. Then we prove the Kac Theorem for the case of valued quiver of any type while using the method of Ringel-Hall algebras.