Abstract
These lecture notes are based on Yang’s talk at the MATRIX program Geometric R-Matrices: from Geometry to Probability, at the University of Melbourne, Dec. 18–22, 2017, and Zhao’s talk at Perimeter Institute for Theoretical Physics in January 2018. We give an introductory survey of the results in Yang and Zhao (Quiver varieties and elliptic quantum groups, 2017. arxiv1708.01418). We discuss a sheafified elliptic quantum group associated to any symmetric Kac-Moody Lie algebra. The sheafification is obtained by applying the equivariant elliptic cohomological theory to the moduli space of representations of a preprojective algebra. By construction, the elliptic quantum group naturally acts on the equivariant elliptic cohomology of Nakajima quiver varieties. As an application, we obtain a relation between the sheafified elliptic quantum group and the global affine Grassmannian over an elliptic curve.
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References
Aganagic, M., Okounkov, A.: Elliptic stable envelopes (2016, preprint). arXiv:1604.00423
Ando, M.: Power operations in elliptic cohomology and representations of loop groups. Trans. Am. Math. Soc. 352(12), 5619–5666 (2000). MR1637129
Ando, M.: The Sigma-orientation for circle-equivariant elliptic cohomology. Geom. Topol. 7, 91–153 (2003). MR1988282
Braverman, A., Finkelberg, M., Nakajima, H.: Coulomb branches of \(3d, \mathcal {N}=4\) quiver gauge theories and slices in the affine Grassmannian (with appendices by Alexander Braverman, Michael Finkelberg, Joel Kamnitzer, Ryosuke Kodera, Hiraku Nakajima, Ben Webster, and Alex Weekes). arXiv:1604.03625
Chen, H.-Y.: Torus equivariant elliptic cohomology and sigma orientation. Ph.D. Thesis, University of Illinois at Urbana-Champaign, 109pp. (2010). MR2873496
Drinfeld, V.: Quasi-Hopf algebras. Algebra i Analiz 1(6), 114–148 (1989). MR1047964
Felder, G.: Elliptic quantum groups. In: XIth International Congress of Mathematical Physics (Paris, 1994), pp. 211–218. Int. Press, Cambridge (1995). MR1370676
Felder, G., Rimanyi, R., Varchenko, A.: Elliptic dynamical quantum groups and equivariant elliptic cohomology (2017, preprint). arXiv:1702.08060
Gautam, S., Toledano Laredo, V.: Elliptic quantum groups and their finite-dimensional representations (2017, preprint). arXiv:1707.06469
Gepner, D.: Equivariant elliptic cohomology and homotopy topoi. Ph.D. thesis, University of Illinois (2006)
Ginzburg, V., Kapranov, M., Vasserot, E.: Elliptic algebras and equivariant elliptic cohomology. (1995, Preprint). arXiv:9505012
Goerss, P., Hopkins, M.: Moduli spaces of commutative ring spectra. In: Structured Ring Spectra, London Mathematical Society Lecture Note Series, vol. 315, pp. 151–200. Cambridge University Press, Cambridge (2004)
Konno, H.: Elliptic weight functions and elliptic q-KZ equation. J. Integr. Syst. 2(1), xyx011 (2017)
Kontsevich, M., Soibelman, Y.: Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants. Commun. Number Theory Phys. 5(2), 231–352 (2011). MR2851153
Levine, M., Morel, F.: Algebraic Cobordism Theory. Springer, Berlin (2007). MR2286826
Lurie, J.: A survey of elliptic cohomology. In: Algebraic Topology. Abel Symposia, vol. 4, pp. 219–277. Springer, Berlin (2009). MR2597740
Maulik, D., Okounkov, A.: Quantum groups and quantum cohomology (preprint). arXiv:1211.1287v1
McGerty, K., Nevins, T.: Kirwan surjectivity for quiver varieties. Invent. Math. 212, 161–187 (2018)
Mirkovic, I.: The loop Grassmannians in the framework of local spaces over a curve . In: Recent Advances in Representation Theory, Quantum Groups, Algebraic Geometry, and Related Topics. Contemporary Mathematics, vol. 623, pp. 215–226. American Mathematical Society, Providence (2014). MR3288629
Mirković, I.: Some extensions of the notion of loop Grassmannians. Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 21(532), 53–74 (2017). MR3697895
Nakajima, H.: Quiver varieties and finite dimensional representations of quantum affine algebras. J. Am. Math. Soc. 14(1), 145–238 (2001). MR1808477
Okounkov, A., Smirnov, A.: Quantum difference equation for Nakajima varieties. arXiv:1602.09007
Schiffmann, O., Vasserot, E.: The elliptic Hall algebra and the K-theory of the Hilbert scheme of \(\mathbb {A}^2\). Duke Math. J. 162(2), 279–366 (2013). MR3018956
Varagnolo, M.: Quiver varieties and yangians. Lett. Math. Phys. 53(4), 273–283 (2000). MR1818101
Yang, Y., Zhao, G.: On two cohomological Hall algebras. Proc. R. Soc. Edinb. Sect. A (to appear). arXiv:1604.01477
Yang, Y., Zhao, G.: Quiver varieties and elliptic quantum groups (2017, preprint). arxiv1708.01418
Yang, Y., Zhao, G.: Cohomological Hall algebras and affine quantum groups. Sel. Math. 24(2), 1093–1119 (2018). arXiv:1604.01865
Yang, Y., Zhao, G.: The cohomological Hall algebra of a preprojective algebra. Proc. Lond. Math. Soc. 116, 1029–1074. arXiv:1407.7994
Zhao, G., Zhong, C.: Elliptic affine Hecke algebra and its representations (2015, preprint). arXiv:1507.01245
Acknowledgements
Y.Y. would like to thank the organizers of the MATRIX program Geometric R-Matrices: from Geometry to Probability for their kind invitation, and many participants of the program for useful discussions, including Vassily Gorbounov, Andrei Okounkov, Allen Knutson, Hitoshi Konno, Paul Zinn-Justin. Proposition 1 and Sect. 3.3 are new, for which we thank Hitoshi Konno for interesting discussions and communications. These notes were written when both authors were visiting the Perimeter Institute for Theoretical Physics (PI). We are grateful to PI for the hospitality.
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Yang, Y., Zhao, G. (2019). How to Sheafify an Elliptic Quantum Group. In: de Gier, J., Praeger, C., Tao, T. (eds) 2017 MATRIX Annals. MATRIX Book Series, vol 2. Springer, Cham. https://doi.org/10.1007/978-3-030-04161-8_54
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