Smash Products of Calabi-Yau Algebras by Hopf Algebras

Patrick Le Meur (1, 2)

1 LMBP - Laboratoire de Mathématiques Blaise Pascal
2 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche

Let H be a Hopf algebra and A be an H-module algebra. The smash product A♯H is studied from the viewpoint of Calabi-Yau duality. When H has Van den Bergh duality it is proved that its antipode is invertible. When both A and H have Van den Bergh duality (or, are skew-Calabi-Yau) it is proved that A♯H has Van den Bergh duality (or, is skew-Calabi-Yau, respectively). In the case of skew-Calabi-Yau algebras, a Nakayama automorphism of A♯H is described and some sufficient conditions for A♯H to be Calabi-Yau are derived. These results are based on the study of the inverse dualising object of A♯H in the more general situation where A is a dg algebra.

Domaines: Théorie des représentations [math.RT]

Patrick Le Meur. Smash Products of Calabi-Yau Algebras by Hopf Algebras. Journal of Noncommutative Geometry, 2019, 13 (3), pp.887-961. ⟨10.4171/JNCG/341⟩. ⟨hal-01314228v2⟩