Nakayama automorphisms of double Ore extensions of Koszul regular algebras

Let A be a Koszul Artin–Schelter regular algebra and σ an algebra homomorphism from A to M 2 × 2 ( A ) . We compute the Nakayama automorphisms of a trimmed double Ore extension A P [ y 1 , y 2 ; σ ] [introduced in Zhang and Zhang (J Pure Appl Algebra 212:2668–2690, 2008 )]. Using a similar method, we also obtain the Nakayama automorphism of a skew polynomial extension A [ t ; θ ] , where θ is a graded algebra automorphism of A . These lead to a characterization of the Calabi–Yau property of A P [ y 1 , y 2 ; σ ] , the skew Laurent extension A [ t ± 1 ; θ ] and A [ y 1 ± 1 , y 2 ± 1 ; σ ] with σ a diagonal type.