Triangular Decomposition of the Composition Algebra of the Kronecker Algebra Original Research Article

LetAbe a finite-dimensional hereditary algebra over a finite field, and letimage(A) andimage(A) be, respectively, the Ringel–Hall algebra and the composition algebra ofA. Definerdto be the element ∑[M]set membership, variantimage(A), where [M] runs over the isomorphism classes of the regularA-modules with dimension vectord. We prove thatrdand the exceptionalA-modules all lie inimage(A). LetKbe the Kronecker algebra,image(resp.image) the subalgebra ofimage(K) generated by the preprojective (resp. preinjective)K-modules, andimagethe subalgebra generated byr(n, n)forn≥0. Then we prove thatimage(K)=image·image·imageand thenimageis just the subalgebra ofimage(K) generated by all regular elements.