Twisted bi-symplectic structure on Koszul twisted Calabi-Yau algebras

Abstract: For a Koszul Artin-Schelter regular algebra (also called twisted Calabi-Yau algebra), we show that it has a "twisted" bi-symplectic structure, which may be viewed as a noncommutative and twisted analogue of the shifted symplectic structure introduced by Pantev, To\"en, Vaqui\'e and Vezzosi. This structure gives a quasi-isomorphism between the tangent complex and the twisted cotangent complex of the algebra, and may be viewed as a DG enhancement of Van den Bergh's noncommutative Poincar\'e duality; it also induces a twisted symplectic structure on its derived representation schemes.