Simplicity of vacuum modules and associated varieties

Tomoyuki Arakawa; Cuipo Jiang; Anne Moreau

doi:10.5802/jep.144

Simplicity of vacuum modules and associated varieties

Tomoyuki Arakawa ¹; Cuipo Jiang ²; Anne Moreau ³

¹ Research Institute for Mathematical Sciences, Kyoto University Kyoto, 606-8502, Japan
² School of Mathematical Sciences, Shanghai Jiao Tong University Shanghai, 200240, China
³ Faculté des Sciences d’Orsay, Université Paris-Saclay 91405 Orsay, France

Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 169-191.

Abstract
Résumé

In this note, we prove that the universal affine vertex algebra associated with a simple Lie algebra $𝔤$ is simple if and only if the associated variety of its unique simple quotient is equal to $𝔤^{*}$ . We also derive an analogous result for the quantized Drinfeld-Sokolov reduction applied to the universal affine vertex algebra.

Dans cet article, nous démontrons que l’algèbre vertex affine universelle associée à une algèbre de Lie simple $𝔤$ est simple si et seulement si la variété associée à son unique quotient simple est égale à $𝔤^{*}$ . Nous en déduisons un résultat analogue pour la réduction quantique de Drinfeld-Sokolov appliquée à l’algèbre vertex affine universelle.

Received: 2020-04-03
Accepted: 2021-01-08
Published online: 2021-01-19

MR: 4201804

DOI: 10.5802/jep.144
Classification: 17B69
Keywords: Associated variety, affine Kac-Moody algebra, affine vertex algebra, singular vector, affine

W

-algebra
Author's affiliations:

Tomoyuki Arakawa ¹; Cuipo Jiang ²; Anne Moreau ³

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Tomoyuki Arakawa; Cuipo Jiang; Anne Moreau. Simplicity of vacuum modules and associated varieties. Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 169-191. doi : 10.5802/jep.144. https://jep.centre-mersenne.org/articles/10.5802/jep.144/

References
Cited by

[ABD04] T. Abe, G. Buhl & C. Dong - “Rationality, regularity, and $C_{2}$ -cofiniteness”, Trans. Amer. Math. Soc. 356 (2004) no. 8, p. 3391-3402 | Article | MR: 2052955 | Zbl: 1070.17011

[AK18] T. Arakawa & K. Kawasetsu - “Quasi-lisse vertex algebras and modular linear differential equations”, in Lie groups, geometry, and representation theory, Progress in Math., vol. 326, Birkhäuser/Springer, Cham, 2018, p. 41-57 | Article | MR: 3890204 | Zbl: 07065394

[AM17] T. Arakawa & A. Moreau - “Sheets and associated varieties of affine vertex algebras”, Adv. Math. 320 (2017), p. 157-209, Corrigendum: Ibid 372 (2020), article Id. 107302 | Article | MR: 3709103 | Zbl: 1425.17032

[AM18a] T. Arakawa & A. Moreau - “Joseph ideals and lisse minimal $W$ -algebras”, J. Inst. Math. Jussieu 17 (2018) no. 2, p. 397-417 | Article | MR: 3773273 | Zbl: 1416.17014

[AM18b] T. Arakawa & A. Moreau - “On the irreducibility of associated varieties of $W$ -algebras”, J. Algebra 500 (2018), p. 542-568 | Article | MR: 3765468 | Zbl: 1383.81102

[Ara05] T. Arakawa - “Representation theory of superconformal algebras and the Kac-Roan-Wakimoto conjecture”, Duke Math. J. 130 (2005) no. 3, p. 435-478 | Article | MR: 2184567 | Zbl: 1112.17026

[Ara07] T. Arakawa - “Representation theory of $𝒲$ -algebras”, Invent. Math. 169 (2007) no. 2, p. 219-320 | Article | Zbl: 1172.17019

[Ara11] T. Arakawa - “Representation theory of $W$ -algebras, II”, in Exploring new structures and natural constructions in mathematical physics, Adv. Stud. Pure Math., vol. 61, Math. Soc. Japan, Tokyo, 2011, p. 51-90 | Article | MR: 2867144 | Zbl: 1262.17014

[Ara12a] T. Arakawa - “A remark on the $C_{2}$ -cofiniteness condition on vertex algebras”, Math. Z. 270 (2012) no. 1-2, p. 559-575 | Article | MR: 2875849 | Zbl: 1351.17031

[Ara12b] T. Arakawa - “ $W$ -algebras at the critical level”, in Algebraic groups and quantum groups, Contemp. Math., vol. 565, American Mathematical Society, Providence, RI, 2012, p. 1-13 | Article | MR: 2932418 | Zbl: 1328.17021

[Ara15a] T. Arakawa - “Associated varieties of modules over Kac-Moody algebras and $C_{2}$ -cofiniteness of $W$ -algebras”, Internat. Math. Res. Notices (2015) no. 22, p. 11605-11666 | Article | MR: 3456698 | Zbl: 1335.17011

[Ara15b] T. Arakawa - “Rationality of $W$ -algebras: principal nilpotent cases”, Ann. of Math. (2) 182 (2015) no. 2, p. 565-604 | Article | MR: 3418525 | Zbl: 1394.17058

[AvE19] T. Arakawa & J. van Ekeren - “Rationality and fusion rules of exceptional W-algebras”, 2019 | 1905.11473

[BD] A. Beilinson & V. Drinfeld - “Quantization of Hitchin’s integrable system and Hecke eigensheaves”, preprint, available at http://math.uchicago.edu/~drinfeld/langlands/QuantizationHitchin.pdf

[BFM] A. Beilinson, B. Feigin & B. Mazur - “Introduction to algebraic field theory on curves”, preprint

[BLL + 15] C. Beem, M. Lemos, P. Liendo, W. Peelaers, L. Rastelli & B. C. van Rees - “Infinite chiral symmetry in four dimensions”, Comm. Math. Phys. 336 (2015) no. 3, p. 1359-1433 | Article | MR: 3324147 | Zbl: 1320.81076

[BR18] C. Beem & L. Rastelli - “Vertex operator algebras, Higgs branches, and modular differential equations”, J. High Energy Phys. (2018) no. 8, article ID 114, 70 pages | Article | MR: 3861208 | Zbl: 1396.81191

[CM16] J.-Y. Charbonnel & A. Moreau - “The symmetric invariants of centralizers and Slodowy grading”, Math. Z. 282 (2016) no. 1-2, p. 273-339 | Article | MR: 3448383 | Zbl: 1371.17011

[DM06] C. Dong & G. Mason - “Integrability of $C_{2}$ -cofinite vertex operator algebras”, Internat. Math. Res. Notices (2006), article ID 80468, 15 pages | Article | MR: 2219226 | Zbl: 1106.17033

[DSK06] A. De Sole & V. G. Kac - “Finite vs affine $W$ -algebras”, Japan. J. Math. 1 (2006) no. 1, p. 137-261 | Article | MR: 2261064 | Zbl: 1161.17015

[EF01] D. Eisenbud & E. Frenkel - “Appendix to [Mus01]”, 2001

[FF90] B. Feigin & E. Frenkel - “Quantization of the Drinfelʼd-Sokolov reduction”, Phys. Lett. B 246 (1990) no. 1-2, p. 75-81 | Article | Zbl: 1242.17023

[FF92] B. Feigin & E. Frenkel - “Affine Kac-Moody algebras at the critical level and Gelʼfand-Dikiĭ algebras”, in Infinite analysis, Part A, B (Kyoto, 1991), Adv. Ser. Math. Phys., vol. 16, World Sci. Publ., River Edge, NJ, 1992, p. 197-215 | Article | Zbl: 0925.17022

[FG04] E. Frenkel & D. Gaitsgory - “ $D$ -modules on the affine Grassmannian and representations of affine Kac-Moody algebras”, Duke Math. J. 125 (2004) no. 2, p. 279-327 | Article | MR: 2096675 | Zbl: 1107.17013

[Fre05] E. Frenkel - “Wakimoto modules, opers and the center at the critical level”, Adv. Math. 195 (2005) no. 2, p. 297-404 | Article | MR: 2146349 | Zbl: 1129.17014

[FZ92] I. B. Frenkel & Y. Zhu - “Vertex operator algebras associated to representations of affine and Virasoro algebras”, Duke Math. J. 66 (1992) no. 1, p. 123-168 | Article | MR: 1159433 | Zbl: 0848.17032

[GG02] W. L. Gan & V. Ginzburg - “Quantization of Slodowy slices”, Internat. Math. Res. Notices (2002) no. 5, p. 243-255 | Article | MR: 1876934 | Zbl: 0989.17014

[GK07] M. Gorelik & V. G. Kac - “On simplicity of vacuum modules”, Adv. Math. 211 (2007) no. 2, p. 621-677 | Article | MR: 2323540 | Zbl: 1112.17023

[Har77] R. Hartshorne - Algebraic geometry, Graduate Texts in Math., vol. 52, Springer-Verlag, New York-Heidelberg, 1977 | Zbl: 0367.14001

[Hum72] J. E. Humphreys - Introduction to Lie algebras and representation theory, Graduate Texts in Math., vol. 9, Springer-Verlag, New York-Berlin, 1972 | MR: 323842 | Zbl: 0254.17004

[Kac90] V. G. Kac - Infinite-dimensional Lie algebras, Cambridge University Press, Cambridge, 1990 | Article | Zbl: 0716.17022

[KRW03] V. G. Kac, S.-S. Roan & M. Wakimoto - “Quantum reduction for affine superalgebras”, Comm. Math. Phys. 241 (2003) no. 2-3, p. 307-342 | Article | MR: 2013802 | Zbl: 1106.17026

[KW89] V. G. Kac & M. Wakimoto - “Classification of modular invariant representations of affine algebras”, in Infinite-dimensional Lie algebras and groups (Luminy-Marseille, 1988), Adv. Ser. Math. Phys., vol. 7, World Sci. Publ., Teaneck, NJ, 1989, p. 138-177 | MR: 1026952

[KW08] V. G. Kac & M. Wakimoto - “On rationality of $W$ -algebras”, Transform. Groups 13 (2008) no. 3-4, p. 671-713 | Article | MR: 2452611 | Zbl: 1221.17033

[Li05] H. Li - “Abelianizing vertex algebras”, Comm. Math. Phys. 259 (2005) no. 2, p. 391-411 | Article | MR: 2172688 | Zbl: 1129.17015

[LL04] J. Lepowsky & H. Li - Introduction to vertex operator algebras and their representations, Progress in Math., vol. 227, Birkhäuser Boston, Inc., Boston, MA, 2004 | Article | MR: 2023933 | Zbl: 1055.17001

[Miy04] M. Miyamoto - “Modular invariance of vertex operator algebras satisfying $C_{2}$ -cofiniteness”, Duke Math. J. 122 (2004) no. 1, p. 51-91 | Article | MR: 2046807 | Zbl: 1165.17311

[Mus01] M. Mustaţă - “Jet schemes of locally complete intersection canonical singularities”, Invent. Math. 145 (2001) no. 3, p. 397-424 | Article | MR: 1856396 | Zbl: 1091.14004

[Pre02] A. Premet - “Special transverse slices and their enveloping algebras”, Adv. Math. 170 (2002) no. 1, p. 1-55 | Article | MR: 1929302 | Zbl: 1005.17007

[RT92] M. Raïs & P. Tauvel - “Indice et polynômes invariants pour certaines algèbres de Lie”, J. reine angew. Math. 425 (1992), p. 123-140 | Zbl: 0743.17008

[Slo80] P. Slodowy - Simple singularities and simple algebraic groups, Lect. Notes in Math., vol. 815, Springer, Berlin, 1980 | MR: 584445 | Zbl: 0441.14002

[XY19] D. Xie & W. Yan - “4d $𝒩 = 2$ SCFTs and lisse W-algebras”, 2019 | 1910.02281

[Zhu96] Y. Zhu - “Modular invariance of characters of vertex operator algebras”, J. Amer. Math. Soc. 9 (1996) no. 1, p. 237-302 | Article | MR: 1317233 | Zbl: 0854.17034

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