Triangular Decomposition of Tame Non-Simply Laced Composition Algebras,

Let A be a finite dimensional hereditary algebra over a finite field, and let (A) be the composition algebra of A. Let (resp. ) be the subalgebra of (A) generated by the preprojective (resp. preinjective) A-modules, and let be the subalgebra generated by rd, where rd = ∑[M] (A), [M] runs over the isomorphism classes of the regular A-modules with dimension vector d, and (A) is the Ringel–Hall algebra of A. Then we prove that (A) = • • , if A is one of the types n, n, n, n, n, 41, 42, 21, 22, or Ã11.