Triangular Decomposition of the Composition Algebra of the Kronecker Algebra

Let A be a nite dimensional hereditary algebra over a nite eld, H(A) and C(A) be respectively the Ringel-Hall algebra and the composition algebra of A. Deene r d to be the element P M] 2 H(A), where M] runs over the isomorphism classes of the regular A?modules with dimension vector d. We prove that r d and the exceptional A?modules all lie in C(A). Let K be the Kronecker algebra, P (resp. I) the subalgebra of C(K) generated by the preprojective (resp. preinjective) K?modules, and T the subalgebra generated by r (n;n) for n 0, then we prove that C(K) = P T I and then T is just the subalgebra of C(K) generated by all regular elements. Throughout this paper A is assumed to be a nite dimensional hereditary k-algebra, where k is a nite eld with q elements. All modules are nite dimensional over k (hence nite as sets). For the representation theory of nite dimensional hereditary algebras we refer to R1]. By the recent works of C.M.Ringel R3-9] and J.A.Green G, Theorem 2] we know that Ringel's twisted generic composition algebra C (A) is isomorphic to G. Lusztig's algebra f (and hence isomorphic to the positive part U + of the Drinfeld-Jimbo quantization U = U ? U 0 U + associated to a Kac-Moody Lie algebra of type (I;) ((L], chap.33), where (I;) is the Cartan datum associated to A). This remarkable result provides the Ringel-Hall algebra approach to quantum group and makes it possible that results in the representations of quiver have applications or interpretations in quantum group. To be simple, we consider the untwisted case (all considerations remain true under the twisted multiplication introduced in R7]). Let H(A) and C(A) denote the Ringel-Hall algebra and Ringel's composition algebra of A respectively. By deenition H(A) is the Q-vector space with basis the set of isomorphism classes X] of nite modules, with the multiplication