In this paper, we introduce and study differential graded (DG for short) polynomial algebras. In brief, a DG polynomial algebra A is a connected cochain DG algebra such that its underlying graded algebra A # is a polynomial algebra [x 1 , x 2 , · · · , xn] with |x i | = 1, for any i ∈ {1, 2, · · · , n}.We describe all possible differential structures on DG polynomial algebras; compute their DG automorphism groups; study their isomorphism problems; and show that they are all homologically smooth and Gorestein DG algebras. Furthermore, it is proved that the DG polynomial algebra A is a Calabi-Yau DG algebra when its differential ∂ A = 0 and the trivial DG polynomial algebra (A, 0) is Calabi-Yau if and only if n is an odd integer.2010 Mathematics Subject Classification. Primary 16E45, 16E65, 16W20,16W50.

No abstract

In this paper, we introduce the notion of DG Sklyanin algebras, which are connected cochain DG algebras whose underlying graded algebras are Sklyanin algebras. Let [Formula: see text] be a [Formula: see text]-dimensional DG Sklyanin algebra with [Formula: see text], where [Formula: see text] and [Formula: see text] We systematically study its differential structures and various homological properties. Especially, we figure out the conditions for [Formula: see text] to be Calabi–Yau, Koszul, Gorenstein and homologically smooth, respectively.