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Commutativity of quantization and reduction for quiver representations

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Abstract

Given a finite quiver, its double may be viewed as its non-commutative “cotangent” space, and hence is a non-commutative symplectic space. Crawley-Boevey, Etingof and Ginzburg constructed the non-commutative reduction of this space while Schedler constructed its quantization. We show that the non-commutative quantization and reduction commute with each other. Via the quantum and classical trace maps, such a commutativity induces the commutativity of the quantization and reduction on the space of quiver representations.

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References

  1. Alekseev, A., Kosmann-Schwarzbach, Y., Meinrenken, E.: Quasi-Poisson manifolds. Can. J. Math. 54(1), 3–29 (2002)

    Article  MathSciNet  Google Scholar 

  2. Alekseev, A., Malkin, A., Meinrenken, E.: Lie group valued moment maps. J. Differ. Geom. 48(3), 445–495 (1998)

    Article  MathSciNet  Google Scholar 

  3. Bocklandt, R., Le Bruyn, L.: Necklace Lie algebras and noncommutative symplectic geometry. Math. Z. 240(1), 141–167 (2002)

    Article  MathSciNet  Google Scholar 

  4. Chen, X., Eshmatov, F.: Calabi–Yau algebras and the shifted noncommutative symplectic structure. Adv. Math. 367, 107126, 40 (2020)

    Article  MathSciNet  Google Scholar 

  5. Crawley-Boevey, W.: Geometry of the moment map for representations of quivers. Compos. Math. 126(3), 257–293 (2001)

    Article  MathSciNet  Google Scholar 

  6. Crawley-Boevey, W.: Poisson structures on moduli spaces of representations. J. Algebra 325, 205–215 (2011)

    Article  MathSciNet  Google Scholar 

  7. Crawley-Boevey, W., Etingof, P., Ginzburg, V.: Noncommutative geometry and quiver algebras. Adv. Math. 209(1), 274–336 (2007)

    Article  MathSciNet  Google Scholar 

  8. Fedosov, B.V.: Non-abelian reduction in deformation quantization. Lett. Math. Phys. 43(2), 137–154 (1998)

    Article  MathSciNet  Google Scholar 

  9. Gan, W., Ginzburg, V.: Almost-commuting variety, \(\cal{D}\)-modules, and Cherednik algebras. With an appendix by Ginzburg, Int. Math. Res. Pap., pp. 1–54 (2006)

  10. Ginzburg, V.: Non-commutative symplectic geometry, quiver varieties, and operads. Math. Res. Lett. 8(3), 377–400 (2001)

    Article  MathSciNet  Google Scholar 

  11. Guillemin, V., Sternberg, S.: Geometric quantization and multiplicities of group representations. Invent. Math. 67(3), 515–538 (1982)

    Article  MathSciNet  Google Scholar 

  12. Holland, M.P.: Quantization of the Marsden–Weinstein reduction for extended Dynkin quivers. Ann. Sci. École Norm. Sup. (4) 32(6), 813–834 (1999)

    Article  MathSciNet  Google Scholar 

  13. Kontsevich, M.: Formal (non)commutative symplectic geometry. In: The Gelfand Mathematical Seminars, 1990–1992, pp. 173–187. Birkhäuser Boston, Boston (1993)

  14. Kontsevich, M.: Homological algebra of mirror symmetry. In: Proceedings of the International Congress of Mathematicians, vol. 1, 2 (Zürich, 1994), pp. 120–139. Birkhäuser, Basel (1995)

  15. Kontsevich, M., Rosenberg, A.L.: Noncommutative smooth spaces. In: The Gelfand Mathematical Seminars, 1996–1999. In: Gelfand Math. Sem., pp. 85–108. Birkhäuser, Boston (2000)

  16. Losev, I.: Isomorphisms of quantizations via quantization of resolutions. Adv. Math. 231(3–4), 1216–1270 (2012)

    Article  MathSciNet  Google Scholar 

  17. Lu, J.H.: Moment maps at the quantum level. Commun. Math. Phys. 157(2), 389–404 (1993)

    Article  MathSciNet  Google Scholar 

  18. Meinrenken, E.: Twisted K-homology and group-valued moment maps. Int. Math. Res. Not. 20, 4563–4618 (2012)

    Article  MathSciNet  Google Scholar 

  19. Schedler, T.: A Hopf algebra quantizing a necklace Lie algebra canonically associated to a quiver. Int. Math. Res. Not. 20, 725–760 (2005)

    Article  MathSciNet  Google Scholar 

  20. Schedler, T.: Deformations of algebras in noncommutative geometry. In: Noncommutative Algebraic Geometry. Math. Sci. Res. Inst. Publ., vol. 64, pp. 71–165. Cambridge Univ. Press, New York (2016)

  21. Van den Bergh, M.: Double Poisson algebras. Trans. Am. Math. Soc. 360(11), 5711–5769 (2008)

    Article  MathSciNet  Google Scholar 

  22. Van den Bergh, M.: Non-commutative quasi-Hamiltonian spaces. In: Poisson Geometry in Mathematics and Physics. Contemp. Math., vol. 450, pp. 273–299. Amer. Math. Soc., Providence (2008)

  23. Xu, P.: Fedosov \(\ast \)-products and quantum momentum maps. Commun. Math. Phys. 197(1), 167–197 (1998)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author would like to thank Professors Xiaojun Chen and Farkhod Eshmatov as well as Jieheng Zeng for many helpful conversations; he also thanks Professor Yongbin Ruan for inviting him to visit IASM, Zhejiang University during the preparation of the paper. This work is partially supported by NSFC (Nos. 11890660 and 11890663).

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

    Hu Zhao

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  1. Hu Zhao
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Correspondence to Hu Zhao.

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Zhao, H. Commutativity of quantization and reduction for quiver representations. Math. Z. 301, 3525–3554 (2022). https://doi.org/10.1007/s00209-022-03028-1

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