On Lie algebras associated with representation-finite algebras

Let Λ be a representation-finite CC-algebra which has Hall polynomials, with the universal cover Λ˜ which is a locally bounded directed CC-algebra. In this paper we prove that the ZZ-Lie algebra L(Λ)L(Λ) associated with Λ which is defined by Riedtmann in [17] and the ZZ-Lie algebra K(Λ)K(Λ) associated with Λ which is defined by Ringel in [19] are isomorphic. As an application we show that if Λ is a representation-finite (generalized) cluster-tilted algebra or representation-finite trivial extension algebra, then K(Λ)≅L(Λ)K(Λ)≅L(Λ).