Hochschild Homology And Cyclic Homology Of Koszul Algebras

Hochschild homology of algebras can be regarded as a noncommutative gener-alization of differential form modules.And the cyclic homology,closely related to Hochschild homology,is not only a noncommutative version of de Rham homology,but also a Lie algebra simulation of K-theory.They are both important contents of noncommutative geometry.Hochschild homology and cohomology are invariants un-der Morita equivalence,Tilting equivalence and Derived equivalence of algebras.And Hochschild homology and cohomology have important applications in the representa-tion theory of finite-dimensional algebras.This paper studies Hochschild homology groups and cyclic homology groups of two classes of Koszul algebras.Based on the minimal projective bimodule resolutions of these two algebras con-structed by Furuya,we first give an explicit description of the so-called " cycle struc-ture".Then we give pure combinational descriptions of Hochschild homology com-plexes of these two algebras by using the "cycle structure".Thus we calculate the dimension and k-basis of each Hochschild homology space of these two algebras by using the methods of linear algebra,which is helpful to reveal the close connection between Hochschild homology groups and cycles of the Gabriel quivers of algebras.Moreover,we obtain the dimensions of cyclic homology groups of these two algebras when the base field k is of zero characteristic.