On Lie algebras associated with representation-directed algebras

Let BB be a representation-finite CC-algebra. The ZZ-Lie algebra L(B)L(B) associated with BB has been defined by Riedtmann in [Ch. Riedtmann, Lie algebras generated by indecomposables, J. Algebra 170 (1994) 526–546]. If BB is representation-directed, there is another ZZ-Lie algebra associated with BB defined by Ringel in [C.M. Ringel, Hall Algebras, vol. 26, Banach Center Publications, Warsaw, 1990, pp. 433–447] and denoted by K(B)K(B).We prove that the Lie algebras L(B)L(B) and K(B)K(B) are isomorphic for any representation-directed CC-algebra BB.