Triangular Decomposition of the Composition Algebra of the Kronecker Algebra

AbstractLetAbe a finite-dimensional hereditary algebra over a finite field, and letH(A) andC(A) be, respectively, the Ringel–Hall algebra and the composition algebra ofA. Definerdto be the element ∑[M]∈H(A), where [M] runs over the isomorphism classes of the regularA-modules with dimension vectord. We prove thatrdand the exceptionalA-modules all lie inC(A). LetKbe the Kronecker algebra,P(resp.I) the subalgebra ofC(K) generated by the preprojective (resp. preinjective)K-modules, andTthe subalgebra generated byr(n,n)forn≥0. Then we prove thatC(K)=P·T·Iand thenTis just the subalgebra ofC(K) generated by all regular elements