Hostname: page-component-6bf8c574d5-7jkgd Total loading time: 0 Render date: 2025-02-17T12:21:24.089Z Has data issue: false hasContentIssue false
  • English
  • Français

SHUFFLE ALGEBRAS FOR QUIVERS AND R-MATRICES

Part of: Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  22 March 2022

Andrei Neguț Open the ORCID record for Andrei Neguț [Opens in a new window]
Andrei Neguț*
Affiliation:
MIT, Department of Mathematics, 77 Mass Ave, Cambridge, MA 02139, USA Simion Stoilow Institute of Mathematics, Calea Grivitei nr. 21, 010702 Bucharest, Romania
*
[email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We define slope subalgebras in the shuffle algebra associated to a (doubled) quiver, thus yielding a factorization of the universal R-matrix of the double of the shuffle algebra in question. We conjecture that this factorization matches the one defined by [1, 18, 32, 33, 34] using Nakajima quiver varieties.

Keywords

Hall algebras of quiversshuffle algebrasR-matrices

MSC classification

Primary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 22 , Issue 6 , November 2023 , pp. 2583 - 2618
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aganagic, M. and Okounkov, A., Elliptic stable envelopes, J. Amer. Math. Soc. 34(1) (2021), 79–133.10.1090/jams/954CrossRefGoogle Scholar
Bozec, T. and Schiffmann, O., Counting absolutely cuspidals for quivers, Math. Z. 292 (2019), 133–149.10.1007/s00209-018-2155-5CrossRefGoogle Scholar
Burban, I. and Schiffmann, O., On the Hall algebra of an elliptic curve, I, Duke Math. J. 161(7) (2012), 1171–1231.10.1215/00127094-1593263CrossRefGoogle Scholar
Davison, B., ‘BPS Lie algebras and the less perverse filtration on the preprojective CoHA’, Preprint, 2020, https://arxiv.org/abs/2007.03289.Google Scholar
Davison, B. and Meinhardt, S., Cohomological Donaldson-Thomas theory of a quiver with potential and quantum enveloping algebras, Invent. Math. 221 (2020), 777–871.CrossRefGoogle Scholar
Drinfeld, V. G., Quantum groups, in Proceedings of the International Congress of Mathematicians, Vol. 1 (Berkeley CA, 1986), pp. 798–820 (American Mathematical Society, Providence, RI, 1987).Google Scholar
Enriquez, B., On correlation functions of Drinfeld currents and shuffle algebras, Transform. Groups 5(2) (2000), 111–120.CrossRefGoogle Scholar
Feigin, B., Hashizume, K., Hoshino, A., Shiraishi, J. and Yanagida, S., A commutative algebra on degenerate $\mathbb{C}{\mathbb{P}}^1$ and Macdonald polynomials, J. Math. Phys. 50(9) (2009), 42, 095215.CrossRefGoogle Scholar
Feigin, B. and Odesskii, A., Quantized moduli spaces of the bundles on the elliptic curve and their applications , in Integrable structures of Exactly Solvable Two-Dimensional Models of Quantum Field Theory (Kiev, 2000), NATO Science Series II: Mathematics, Physics and Chemistry, 35, pp. 123–137 (Kluwer Academic, Dordrecht, the Netherlands, 2001).Google Scholar
Ginzburg, V. and Vasserot, E., Langlands reciprocity for affine quantum groups of type ${A}_n$ , Int. Math. Res. Not. IMRN 1993(3) (1993), 67–85.CrossRefGoogle Scholar
Hausel, T., Letellier, E. and Rodriguez-Villegas, F., Positivity for Kac polynomials and DT-invariants of quivers, Ann. of Math. (2) 177(3) (2013), 1147–1168.CrossRefGoogle Scholar
Kac, V., Infinite root systems, representations of graphs and invariant theory, Invent. Math. 56(1) (1980), 57–92.10.1007/BF01403155CrossRefGoogle Scholar
Khoroshkin, S. and Tolstoy, V., The universal $R$ -matrix for quantum untwisted affine Lie algebras, Funct. Anal. Appl. 26(1) (1992), 69–71.Google Scholar
Kirillov, A. and Reshetikhin, N., $q$ -Weyl group and a multiplicative formula for universal $R$ -matrices, Comm. Math. Phys. 184 (1990), 421–424.CrossRefGoogle Scholar
Kontsevich, M. and Soibelman, Y., Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants, Commun. Number Theory Phys. 5(2) (2011), 231–352.CrossRefGoogle Scholar
Levendorsky, S. and Soibelman, Y., Some applications of quantum Weyl groups, J. Geom. Phys. 7 (1990), 241–254.CrossRefGoogle Scholar
Levendorsky, S., Soibelman, Y. and Stukopin, V., The Quantum Weyl group and the universal quantum R-Matrix for affine Lie algebra ${A}_1^{(1)}$ , Lett. Math. Phys. 27 (1993), 253–264.CrossRefGoogle Scholar
Maulik, D. and Okounkov, A., Quantum Groups and Quantum Cohomology , Astérisque 408 (2019).Google Scholar
McGerty, K. and Nevins, T., Kirwan Surjectivity for quiver varieties, Invent. Math. 212 (2018), 161–187.CrossRefGoogle Scholar
Nakajima, H., Quiver varieties and Kac-Moody algebras, Duke Math. J. 91(3) (1998), 515–560.CrossRefGoogle Scholar
Nakajima, H., Quiver varieties and finite dimensional representations of quantum affine algebras, J. Amer. Math. Soc. 14(1) (2000), 145–238.CrossRefGoogle Scholar
Nakajima, H., Quiver varieties and tensor products, Invent. Math. 146 (2001), 399–449.CrossRefGoogle Scholar
Neguț, A., The shuffle algebra revisited, Int. Math. Res. Not. IMRN 2014(22) (2014), 6242–6275.10.1093/imrn/rnt156CrossRefGoogle Scholar
Neguț, A., Moduli of flags of sheaves and their K-theory, Algebr. Geom. 2(1) (2015), 19–43.10.14231/AG-2015-002CrossRefGoogle Scholar
Neguț, A., Shuffle algebras associated to surfaces, Selecta Math. (N.S.) 25 (2019), 36.CrossRefGoogle Scholar
Neguț, A., Quantum toroidal and shuffle algebras, Adv. Math. 372 (2020), 107288.CrossRefGoogle Scholar
Neguț, A., Quantum Algebras and Cyclic Quiver Varieties , PhD thesis, 2015, https://arxiv.org/abs/1504.06525.Google Scholar
Neguț, A., ‘Affine Laumon spaces and a conjecture of Kuznetsov’, Preprint, 2018, https://arxiv.org/abs/1811.01011.Google Scholar
Neguț, A., ‘A tale of two shuffle algebras’, Preprint, 2019, https://arxiv.org/abs/1908.08395.Google Scholar
Neguț, A., ‘The $R$ -matrix of the quantum toroidal algebra’, Preprint, 2020, https://arxiv.org/abs/2005.14182.Google Scholar
Neguț, A., ‘Shuffle algebras for quivers and wheel conditions’, Preprint, 2021, https://arxiv.org/abs/2108.08779.Google Scholar
Okounkov, A., Inductive construction of stable envelopes, Lett. Math. Phys. 111(6) (2021), 141.10.1007/s11005-021-01472-4CrossRefGoogle Scholar
Okounkov, A., ‘Nonabelian stable envelopes, vertex functions with descendents, and integral solutions of $q$ –difference equations’, Preprint, 2020, https://arxiv.org/abs/2010.13217.Google Scholar
Okounkov, A. and Smirnov, A., ‘Quantum difference equation for Nakajima varieties’, Preprint, 2016, https://arxiv.org/abs/1602.09007.Google Scholar
Pădurariu, T., K-Theoretic Hall Algebras for Quivers with Potential, PhD thesis, 2019, https://arxiv.org/abs/1911.05526.Google Scholar
Pădurariu, T., ‘Categorical and $K$ -theoretic Hall algebras for quivers with potential’, Preprint, 2021, https://arxiv.org/abs/2107.13642.Google Scholar
Rosso, M., An analogue of P.B.W. theorem and the universal R-matrix for ${U}_h\mathrm{sl}\left(N+1\right)$ , Comm. Math. Phys. 124 (1989), 307–318.CrossRefGoogle Scholar
Schiffmann, O., ‘Kac polynomials and Lie algebras associated to quivers and curves’, Preprint, 2018, https://arxiv.org/abs/1802.09760.Google Scholar
Schiffmann, O. and Vasserot, E., Hall algebras of curves, commuting varieties and Langlands duality, Math. Ann. 353 (2012), 1399–1451.CrossRefGoogle Scholar
Schiffmann, O. and Vasserot, E., ‘On cohomological Hall algebras of quivers: Yangians’, Preprint, 2017, https://arxiv.org/abs/1705.07491.Google Scholar
Varagnolo, M. and Vasserot, E., On the $K$ -theory of the cyclic quiver variety, Int. Math. Res. Not. IMRN 1999(18) (1999), 1005–1028.CrossRefGoogle Scholar
Varagnolo, M. and Vasserot, E., ‘K-theoretic Hall algebras, quantum groups and super quantum groups’, Selecta Math. (N.S.) 28(1) (2022), 56, Paper No. 7.CrossRefGoogle Scholar
Yang, Y. and Zhao, G., The cohomological Hall algebra of a preprojective algebra, Proc. Lond. Math. Soc. (3) 116(5) (2018), 1029–1074.CrossRefGoogle Scholar
Zhao, Y., ‘The Feigin-Odesskii wheel conditions and sheaves on surfaces’, Preprint, 2019, https://arxiv.org/abs/1909.07870.Google Scholar