## McKay correspondence, cohomological Hall algebras and categorification

HTML articles powered by AMS MathViewer

- by Duiliu-Emanuel Diaconescu, Mauro Porta and Francesco Sala;
- Represent. Theory
**27**(2023), 933-972 - DOI: https://doi.org/10.1090/ert/649
- Published electronically: October 5, 2023
- PDF | Request permission

## Abstract:

Let $\pi \colon Y\to X$ denote the canonical resolution of the two dimensional Kleinian singularity $X$ of type ADE. In the present paper, we establish isomorphisms between the cohomological and K-theoretical Hall algebras of $\omega$-semistable properly supported sheaves on $Y$ with fixed slope $\mu$ and $\zeta$-semistable finite-dimensional representations of the preprojective algebra of affine type ADE of slope zero respectively, under some conditions on $\zeta$ depending on the polarization $\omega$ and $\mu$. These isomorphisms are induced by the derived McKay correspondence. In addition, they are interpreted as decategorified versions of a monoidal equivalence between the corresponding categorified Hall algebras. In the type A case, we provide a finer description of the cohomological, K-theoretical and categorified Hall algebra of $\omega$-semistable properly supported sheaves on $Y$ with fixed slope $\mu$: for example, in the cohomological case, the algebra can be given in terms of Yangians of finite type ADE Dynkin diagrams.## References

- Claudio Bartocci, Ugo Bruzzo, and Daniel Hernández Ruipérez,
*Fourier-Mukai and Nahm transforms in geometry and mathematical physics*, Progress in Mathematics, vol. 276, Birkhäuser Boston, Inc., Boston, MA, 2009. MR**2511017**, DOI 10.1007/b11801 - Jonathan Beck,
*Braid group action and quantum affine algebras*, Comm. Math. Phys.**165**(1994), no. 3, 555–568. MR**1301623**, DOI 10.1007/BF02099423 - Tom Bridgeland, Alastair King, and Miles Reid,
*The McKay correspondence as an equivalence of derived categories*, J. Amer. Math. Soc.**14**(2001), no. 3, 535–554. MR**1824990**, DOI 10.1090/S0894-0347-01-00368-X - T. Bozec, D. Calaque, and S. Scherotzke,
*Relative critical loci and quiver moduli*, arXiv:2006.01069, 2020. - Tristan Bozec, Olivier Schiffmann, and Éric Vasserot,
*On the number of points of nilpotent quiver varieties over finite fields*, Ann. Sci. Éc. Norm. Supér. (4)**53**(2020), no. 6, 1501–1544 (English, with English and French summaries). MR**4203037**, DOI 10.24033/asens.2452 - Brian Conrad,
*Grothendieck duality and base change*, Lecture Notes in Mathematics, vol. 1750, Springer-Verlag, Berlin, 2000. MR**1804902**, DOI 10.1007/b75857 - Tim Cramer,
*Double Hall algebras and derived equivalences*, Adv. Math.**224**(2010), no. 3, 1097–1120. MR**2628805**, DOI 10.1016/j.aim.2009.12.021 - William Crawley-Boevey and Martin P. Holland,
*Noncommutative deformations of Kleinian singularities*, Duke Math. J.**92**(1998), no. 3, 605–635. MR**1620538**, DOI 10.1215/S0012-7094-98-09218-3 - B. Davison,
*The integrality conjecture and the cohomology of preprojective stacks*, arXiv:1602.02110, 2016. - Ben Davison and Sven Meinhardt,
*Cohomological Donaldson-Thomas theory of a quiver with potential and quantum enveloping algebras*, Invent. Math.**221**(2020), no. 3, 777–871. MR**4132957**, DOI 10.1007/s00222-020-00961-y - D.-E. Diaconescu, M. Porta, and F. Sala,
*Cohomological Hall algebras and their representations via torsion pairs*, arXiv:2207.08926, 2022. - J.-M. Drezet and J. Le Potier,
*Fibrés stables et fibrés exceptionnels sur $\textbf {P}_2$*, Ann. Sci. École Norm. Sup. (4)**18**(1985), no. 2, 193–243 (French, with English summary). MR**816365**, DOI 10.24033/asens.1489 - Tobias Dyckerhoff and Mikhail Kapranov,
*Higher Segal spaces*, Lecture Notes in Mathematics, vol. 2244, Springer, Cham, 2019. MR**3970975**, DOI 10.1007/978-3-030-27124-4 - Alexander I. Efimov,
*Homotopy finiteness of some DG categories from algebraic geometry*, J. Eur. Math. Soc. (JEMS)**22**(2020), no. 9, 2879–2942. MR**4127943**, DOI 10.4171/jems/979 - Victor Ginzburg, Mikhail Kapranov, and Éric Vasserot,
*Langlands reciprocity for algebraic surfaces*, Math. Res. Lett.**2**(1995), no. 2, 147–160. MR**1324698**, DOI 10.4310/MRL.1995.v2.n2.a4 - James A. Green,
*Hall algebras, hereditary algebras and quantum groups*, Invent. Math.**120**(1995), no. 2, 361–377. MR**1329046**, DOI 10.1007/BF01241133 - G. Gonzalez-Sprinberg and J.-L. Verdier,
*Construction géométrique de la correspondance de McKay*, Ann. Sci. École Norm. Sup. (4)**16**(1983), no. 3, 409–449 (1984) (French). MR**740077**, DOI 10.24033/asens.1454 - Robin Hartshorne,
*Algebraic geometry*, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR**463157**, DOI 10.1007/978-1-4757-3849-0 - Lutz Hille and David Ploog,
*Tilting chains of negative curves on rational surfaces*, Nagoya Math. J.**235**(2019), 26–41. MR**3986709**, DOI 10.1017/nmj.2017.40 - Daniel Huybrechts and Manfred Lehn,
*The geometry of moduli spaces of sheaves*, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2010. MR**2665168**, DOI 10.1017/CBO9780511711985 - Akira Ishii, Kazushi Ueda, and Hokuto Uehara,
*Stability conditions on $A_n$-singularities*, J. Differential Geom.**84**(2010), no. 1, 87–126. MR**2629510** - M. Kalck and J. Karmazyn,
*Noncommutative Knörrer type equivalences via noncommutative resolutions of singularities*, arXiv:1707.02836, 2017. - M. M. Kapranov,
*Eisenstein series and quantum affine algebras*, J. Math. Sci. (New York)**84**(1997), no. 5, 1311–1360. Algebraic geometry, 7. MR**1465518**, DOI 10.1007/BF02399194 - M. Kapranov and E. Vasserot,
*Kleinian singularities, derived categories and Hall algebras*, Math. Ann.**316**(2000), no. 3, 565–576. MR**1752785**, DOI 10.1007/s002080050344 - M. Kapranov and E. Vasserot,
*The cohomological Hall algebra of a surface and factorization cohomology*, arXiv:1901.07641, 2019. - B. Keller,
*Erratum to “Deformed Calabi-Yau completions”*, arXiv:1809.01126, 2018. - Maxim Kontsevich and Yan Soibelman,
*Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants*, Commun. Number Theory Phys.**5**(2011), no. 2, 231–352. MR**2851153**, DOI 10.4310/CNTP.2011.v5.n2.a1 - S. Mozgovoy,
*Motivic Donaldson-Thomas invariants and McKay correspondence*, arXiv:1107.6044, 2011. - Kentaro Nagao,
*Derived categories of small toric Calabi-Yau 3-folds and curve counting invariants*, Q. J. Math.**63**(2012), no. 4, 965–1007. MR**2999994**, DOI 10.1093/qmath/har025 - Kentaro Nagao and Hiraku Nakajima,
*Counting invariant of perverse coherent sheaves and its wall-crossing*, Int. Math. Res. Not. IMRN**17**(2011), 3885–3938. MR**2836398**, DOI 10.1093/imrn/rnq195 - Mauro Porta and Francesco Sala,
*Two-dimensional categorified Hall algebras*, J. Eur. Math. Soc. (JEMS)**25**(2023), no. 3, 1113–1205. MR**4577961**, DOI 10.4171/jems/1303 - Jie Ren and Yan Soibelman,
*Cohomological Hall algebras, semicanonical bases and Donaldson-Thomas invariants for 2-dimensional Calabi-Yau categories (with an appendix by Ben Davison)*, Algebra, geometry, and physics in the 21st century, Progr. Math., vol. 324, Birkhäuser/Springer, Cham, 2017, pp. 261–293. MR**3727563**, DOI 10.1007/978-3-319-59939-7_{7} - Claus Michael Ringel,
*Hall algebras and quantum groups*, Invent. Math.**101**(1990), no. 3, 583–591. MR**1062796**, DOI 10.1007/BF01231516 - Francesco Sala and Olivier Schiffmann,
*Cohomological Hall algebra of Higgs sheaves on a curve*, Algebr. Geom.**7**(2020), no. 3, 346–376. MR**4087863**, DOI 10.14231/ag-2020-010 - Olivier G. Schiffmann,
*Kac polynomials and Lie algebras associated to quivers and curves*, Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lectures, World Sci. Publ., Hackensack, NJ, 2018, pp. 1393–1424. MR**3966814** - O. Schiffmann and É. Vasserot,
*On cohomological Hall algebras of quivers: Yangians*, arXiv:1705.07491, 2017. - Olivier Schiffmann and Eric Vasserot,
*On cohomological Hall algebras of quivers: generators*, J. Reine Angew. Math.**760**(2020), 59–132. MR**4069884**, DOI 10.1515/crelle-2018-0004 - Yuhi Sekiya and Kota Yamaura,
*Tilting theoretical approach to moduli spaces over preprojective algebras*, Algebr. Represent. Theory**16**(2013), no. 6, 1733–1786. MR**3127356**, DOI 10.1007/s10468-012-9380-0 - The Stacks Project authors, 2020,
*The Stacks Project*, https://stacks.math.columbia.edu. - Bertrand Toën and Michel Vaquié,
*Moduli of objects in dg-categories*, Ann. Sci. École Norm. Sup. (4)**40**(2007), no. 3, 387–444 (English, with English and French summaries). MR**2493386**, DOI 10.1016/j.ansens.2007.05.001 - R. Vakil,
*The rising sea: foundations of algebraic geometry*, 2017. - Michel Van den Bergh,
*Three-dimensional flops and noncommutative rings*, Duke Math. J.**122**(2004), no. 3, 423–455. MR**2057015**, DOI 10.1215/S0012-7094-04-12231-6 - M. Varagnolo and E. Vasserot,
*K-theoretic Hall algebras, quantum groups and super quantum groups*, Selecta Math. (N.S.)**28**(2022), no. 1, Paper No. 7, 56. MR**4340640**, DOI 10.1007/s00029-021-00723-5 - Michael Wemyss,
*The $\textrm {GL}(2,\Bbb C)$ McKay correspondence*, Math. Ann.**350**(2011), no. 3, 631–659. MR**2805639**, DOI 10.1007/s00208-010-0572-9 - K\B{o}ta Yoshioka,
*Perverse coherent sheaves and Fourier-Mukai transforms on surfaces, I*, Kyoto J. Math.**53**(2013), no. 2, 261–344. MR**3079307**, DOI 10.1215/21562261-2081234 - Yaping Yang and Gufang Zhao,
*On two cohomological Hall algebras*, Proc. Roy. Soc. Edinburgh Sect. A**150**(2020), no. 3, 1581–1607. MR**4091073**, DOI 10.1017/prm.2018.162

## Bibliographic Information

**Duiliu-Emanuel Diaconescu**- Affiliation: New High Energy Theory Center - Serrin Building, Rutgers, The State University of New Jersey, 126 Frelinghuysen Rd., Piscataway, New Jersey 08854-8019
- MR Author ID: 629570
- Email: duiliu@physics.rutgers.edu
**Mauro Porta**- Affiliation: Institut de recherche mathématique avancée (IRMA), Université de Strasbourg, France
- MR Author ID: 1177756
- ORCID: 0000-0002-1239-3409
- Email: porta@math.unistra.fr
**Francesco Sala**- Affiliation: Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa (PI), Italy; Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan
- MR Author ID: 980911
- ORCID: 0000-0002-8779-9999
- Email: francesco.sala@unipi.it
- Received by editor(s): June 17, 2020
- Received by editor(s) in revised form: April 25, 2022, January 25, 2023, and April 18, 2023
- Published electronically: October 5, 2023
- Additional Notes: The work of the first author was partially supported by NSF grant DMS-1802410, while the work of the third author was partially supported by JSPS KAKENHI Grant Numbers JP21K03197 and JP18K13402.
- © Copyright 2023 American Mathematical Society
- Journal: Represent. Theory
**27**(2023), 933-972 - MSC (2020): Primary 14A20; Secondary 17B37, 55P99
- DOI: https://doi.org/10.1090/ert/649
- MathSciNet review: 4650513