Skew polynomial algebras with coefficients in Koszul Artin–Schelter regular algebras

Abstract

Let A be a Koszul Artin–Schelter regular algebra with Nakayama automorphism ξ. We show that the Yoneda Ext-algebra of the skew polynomial algebra A[z;ξ] is a trivial extension of a Frobenius algebra. Then we prove that A[z;ξ] is Calabi–Yau; and hence each Koszul Artin–Schelter regular algebra is a subalgebra of a Koszul Calabi–Yau algebra. A superpotential wˆ is also constructed so that the Calabi–Yau algebra A[z;ξ] is isomorphic to the derivation quotient of wˆ. The Calabi–Yau property of a skew polynomial algebra with coefficients in a PBW-deformation of a Koszul Artin–Schelter regular algebra is also discussed.