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

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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 analog of the shifted symplectic structure introduced by Pantev, Toën, Vaquié 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 Poincaré duality; it also induces a twisted symplectic structure on its derived representation schemes.

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Notes

  1. In [6], the derived representation scheme of A is simply the DG commutative algebra \(\varvec{L}(A)_V\), and is also denoted by \(\mathrm {DRep}_V(A)\). Here we reserve this terminology to mean its affine scheme, which is consistent with Yeung in [39].

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Acknowledgements

This work is partially supported by NSFC (No. 11671281, 11890660 and 11890663). We are grateful to Youming Chen, Song Yang and Xiangdong Yang for helpful conversations, and to the anonymous referee for suggestions which improve the presentation of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Sichuan University, Chengdu, 610064, Sichuan, People’s Republic of China

    Xiaojun Chen

  2. Department of Mathematics and Statistics, University of Toledo, Toledo, OH, 43606, USA

    Alimjon Eshmatov

  3. AKFA University, Tashkent, 100095, Uzbekistan

    Farkhod Eshmatov

  4. V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, Tashkent, 100174, Uzbekistan

    Farkhod Eshmatov

  5. School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai, 519082, Guangdong, People’s Republic of China

    Leilei Liu

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  1. Xiaojun Chen
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Correspondence to Alimjon Eshmatov.

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Chen, X., Eshmatov, A., Eshmatov, F. et al. Twisted bi-symplectic structure on Koszul twisted Calabi-Yau algebras. Sel. Math. New Ser. 28, 62 (2022). https://doi.org/10.1007/s00029-022-00774-2

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