Notes on piecewise-Koszul algebras

The relationships between piecewise-Koszul algebras and other .Koszul-type. algebras are discussed. The Yoneda-Ext algebra and the dual algebra of a piecewise-Koszul algebra are studied, and a sufficient condition for the dual algebra A ! to be piecewise-Koszul is given. Finally, by studying the trivial extension algebras of the path algebras of Dynkin quivers in bipartite orientation, we give explicit constructions for piecewise-Koszul algebras with arbitrary .period. and piecewise-Koszul algebras with arbitrary .jump-degree..