CALCULATION FOR RINGEL—HALL NUMBERS OF CYCLIC SERIAL ALGEBRAS

Abstract

C. M. Ringel in [1] introduced the Hall algebra, now also called the Ringel—Hall algebra, of a finitary ring. This kind of algebras turns out to be strongly related to Lie algebras and quantum groups (see, for example, 2-8). Let R be a finitary ring. For any finite (left) R-modules L, M, and N, denote by F LN M the number of submodules U of M such that U ≃ N and M/U ≃ L. Then the Ringel—Hall algebra of R is an associative ring, which is a free Z-module with a basis {u [M]}[M] indexed by the isomorphism classes of all finite R-modules with the multiplication defined by For this reason we call the structural constants F LN M as the Ringel--Hall numbers of R. In general, it is not easy to calculate Ringel—Hall numbers, even for some special finitary rings (see, for example, [5]). For a cyclic serial algebra, Guo [9] gave a calculating formula of F LN M when L or N is semisimple. In this note, we give another algorithm for F LN M with L or N indecomposable. Our proofs are more simple and the algorithm is more direct. Then one can use a general method to calculate any Ringel—Hall number. In addition, we obtain a very simple formula of F LN M when L and N both are indecomposable.