The Hall algebra of cyclic serial algebra (Q1340025)

The study of Hall algebras is strongly related to the study of quantum groups. In this paper, the author obtains some results for Hall algebras of a cyclic serial algebra which are similar to those of Hall algebras of representation directed algebras. Namely, let \(R\) be a finite cyclic algebra with \(n\) simples and such that all the indecomposable projective modules have length \(m\). The author introduces two orders \(\prec_1\) and \(\prec_2\) in \((\mathbb{Z}^+)^{nm}\) and through a one-to-one correspondence these orders can be regarded as orders on the isoclasses of finite \(R\)-modules. He proves then that a Hall algebra of a cyclic serial algebra can be identified with its Loewy subalgebra and its rational extension has a basis similar to the basis of Poincaré-Birkhoff-Witt type.