The notion of Calabi-Yau algebras was introduced by Ginzburg in 2007, and has been widely studied by mathematicians in recent years. In this talk, we study several geometric structures on Calabi-Yau algebras, and show that there is a noncommutative version of “shifted" symplectic structure on Calabi-Yau algebras, which induces a shifted symplectic structure on their moduli spaces of derived representations. If time permits we also discuss the derived noncommutative Poisson structure on Calabi-Yau algebras as well as its quantization. This talk is based on the works joint with Eshmatov.
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