Differential Graded Down-up Algebras And Their Isomophism Problem And Some Homological Properties

A differential graded(DG for short)down-up algebra is a cochain DG algebra whose underlying graded algebra is a down-up graded algebra. Let(A,(?)A) be a DG down-up algebra such that its underlying graded algebra A# is generated by x,y and subject to the relations x2y-αxyx-βyx2=xy2-αyxy-βy2x=0, where α∈K and β ∈Kx =K \ {0}. We give a description of all possible differential of A.In particular,we prove that (?)A=0 unless 1+α-β=0 and β3=1. Besides the differential structures of DG down-up algebras,we also compute DG automorphism groups of DG down-up algebras and study the question of when two DG down-up algebras with non-trivial differential are isomorphic. In the last section,we compute the cohomology algebra of DG down-up algebra A(c1,c2,c3,d1,d2,d3). We prove that each DG down-up algebra A(c1,c2,c3,d1,d2,d3) is a Gorenstein DG algebra.Besides,we prove that each A(c1,c2,c3,d1,d2,d3) is homologically smooth.