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

TL;DR

This paper introduces a twisted bi-symplectic structure on Koszul twisted Calabi-Yau algebras, extending noncommutative symplectic geometry and linking it to derived representation schemes.

Contribution

It establishes a noncommutative, twisted analogue of shifted symplectic structures for Koszul twisted Calabi-Yau algebras, enhancing the understanding of their geometric and duality properties.

Findings

Defines a twisted bi-symplectic structure for these algebras

Shows the structure induces a twisted symplectic form on derived schemes

Provides a DG enhancement of Van den Bergh's noncommutative Poincaré duality

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.