Dans cet article, nous démontrons que l’algèbre vertex affine universelle associée à une algèbre de Lie simple est simple si et seulement si la variété associée à son unique quotient simple est égale à . Nous en déduisons un résultat analogue pour la réduction quantique de Drinfeld-Sokolov appliquée à l’algèbre vertex affine universelle.
In this note, we prove that the universal affine vertex algebra associated with a simple Lie algebra is simple if and only if the associated variety of its unique simple quotient is equal to . We also derive an analogous result for the quantized Drinfeld-Sokolov reduction applied to the universal affine vertex algebra.
Accepté le :
Publié le :
DOI : 10.5802/jep.144
Keywords: Associated variety, affine Kac-Moody algebra, affine vertex algebra, singular vector, affine $W$-algebra
Mot clés : Variété associée, algèbre de Kac-Moody, algèbre vertex affine, vecteur singulier, $W$-algèbre affine
@article{JEP_2021__8__169_0, author = {Arakawa, Tomoyuki and Jiang, Cuipo and Moreau, Anne}, title = {Simplicity of vacuum modules and associated~varieties}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {169--191}, publisher = {Ecole polytechnique}, volume = {8}, year = {2021}, doi = {10.5802/jep.144}, mrnumber = {4201804}, language = {en}, url = {http://www.numdam.org/articles/10.5802/jep.144/} }
TY - JOUR AU - Arakawa, Tomoyuki AU - Jiang, Cuipo AU - Moreau, Anne TI - Simplicity of vacuum modules and associated varieties JO - Journal de l’École polytechnique — Mathématiques PY - 2021 SP - 169 EP - 191 VL - 8 PB - Ecole polytechnique UR - http://www.numdam.org/articles/10.5802/jep.144/ DO - 10.5802/jep.144 LA - en ID - JEP_2021__8__169_0 ER -
%0 Journal Article %A Arakawa, Tomoyuki %A Jiang, Cuipo %A Moreau, Anne %T Simplicity of vacuum modules and associated varieties %J Journal de l’École polytechnique — Mathématiques %D 2021 %P 169-191 %V 8 %I Ecole polytechnique %U http://www.numdam.org/articles/10.5802/jep.144/ %R 10.5802/jep.144 %G en %F JEP_2021__8__169_0
Arakawa, Tomoyuki; Jiang, Cuipo; Moreau, Anne. Simplicity of vacuum modules and associated varieties. Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 169-191. doi : 10.5802/jep.144. http://www.numdam.org/articles/10.5802/jep.144/
[ABD04] Rationality, regularity, and -cofiniteness, Trans. Amer. Math. Soc., Volume 356 (2004) no. 8, pp. 3391-3402 | DOI | MR | Zbl
[AK18] Quasi-lisse vertex algebras and modular linear differential equations, Lie groups, geometry, and representation theory (Progress in Math.), Volume 326, Birkhäuser/Springer, Cham, 2018, pp. 41-57 | DOI | MR | Zbl
[AM17] Sheets and associated varieties of affine vertex algebras, Adv. Math., Volume 320 (2017), pp. 157-209 Corrigendum: Ibid 372 (2020), article Id. 107302 | DOI | MR | Zbl
[AM18a] Joseph ideals and lisse minimal -algebras, J. Inst. Math. Jussieu, Volume 17 (2018) no. 2, pp. 397-417 | DOI | MR | Zbl
[AM18b] On the irreducibility of associated varieties of -algebras, J. Algebra, Volume 500 (2018), pp. 542-568 | DOI | MR | Zbl
[Ara05] Representation theory of superconformal algebras and the Kac-Roan-Wakimoto conjecture, Duke Math. J., Volume 130 (2005) no. 3, pp. 435-478 | DOI | MR | Zbl
[Ara07] Representation theory of -algebras, Invent. Math., Volume 169 (2007) no. 2, pp. 219-320 | DOI | Zbl
[Ara11] Representation theory of -algebras, II, Exploring new structures and natural constructions in mathematical physics (Adv. Stud. Pure Math.), Volume 61, Math. Soc. Japan, Tokyo, 2011, pp. 51-90 | DOI | MR | Zbl
[Ara12a] A remark on the -cofiniteness condition on vertex algebras, Math. Z., Volume 270 (2012) no. 1-2, pp. 559-575 | DOI | MR | Zbl
[Ara12b] -algebras at the critical level, Algebraic groups and quantum groups (Contemp. Math.), Volume 565, American Mathematical Society, Providence, RI, 2012, pp. 1-13 | DOI | MR | Zbl
[Ara15a] Associated varieties of modules over Kac-Moody algebras and -cofiniteness of -algebras, Internat. Math. Res. Notices (2015) no. 22, pp. 11605-11666 | DOI | MR | Zbl
[Ara15b] Rationality of -algebras: principal nilpotent cases, Ann. of Math. (2), Volume 182 (2015) no. 2, pp. 565-604 | DOI | MR | Zbl
[AvE19] Rationality and fusion rules of exceptional W-algebras, 2019 | arXiv
[BD] Quantization of Hitchin’s integrable system and Hecke eigensheaves (preprint, available at http://math.uchicago.edu/~drinfeld/langlands/QuantizationHitchin.pdf)
[BFM] Introduction to algebraic field theory on curves (preprint)
[BLL + 15] Infinite chiral symmetry in four dimensions, Comm. Math. Phys., Volume 336 (2015) no. 3, pp. 1359-1433 | DOI | MR | Zbl
[BR18] Vertex operator algebras, Higgs branches, and modular differential equations, J. High Energy Phys. (2018) no. 8, 114, 70 pages | DOI | MR | Zbl
[CM16] The symmetric invariants of centralizers and Slodowy grading, Math. Z., Volume 282 (2016) no. 1-2, pp. 273-339 | DOI | MR | Zbl
[DM06] Integrability of -cofinite vertex operator algebras, Internat. Math. Res. Notices (2006), 80468, 15 pages | DOI | MR | Zbl
[DSK06] Finite vs affine -algebras, Japan. J. Math., Volume 1 (2006) no. 1, pp. 137-261 | DOI | MR | Zbl
[EF01] Appendix to [Mus01], 2001
[FF90] Quantization of the Drinfelʼd-Sokolov reduction, Phys. Lett. B, Volume 246 (1990) no. 1-2, pp. 75-81 | DOI | Zbl
[FF92] Affine Kac-Moody algebras at the critical level and Gelʼfand-Dikiĭ algebras, Infinite analysis, Part A, B (Kyoto, 1991) (Adv. Ser. Math. Phys.), Volume 16, World Sci. Publ., River Edge, NJ, 1992, pp. 197-215 | DOI | Zbl
[FG04] -modules on the affine Grassmannian and representations of affine Kac-Moody algebras, Duke Math. J., Volume 125 (2004) no. 2, pp. 279-327 | DOI | MR | Zbl
[Fre05] Wakimoto modules, opers and the center at the critical level, Adv. Math., Volume 195 (2005) no. 2, pp. 297-404 | DOI | MR | Zbl
[FZ92] Vertex operator algebras associated to representations of affine and Virasoro algebras, Duke Math. J., Volume 66 (1992) no. 1, pp. 123-168 | DOI | MR | Zbl
[GG02] Quantization of Slodowy slices, Internat. Math. Res. Notices (2002) no. 5, pp. 243-255 | DOI | MR | Zbl
[GK07] On simplicity of vacuum modules, Adv. Math., Volume 211 (2007) no. 2, pp. 621-677 | DOI | MR | Zbl
[Har77] Algebraic geometry, Graduate Texts in Math., 52, Springer-Verlag, New York-Heidelberg, 1977 | Zbl
[Hum72] Introduction to Lie algebras and representation theory, Graduate Texts in Math., 9, Springer-Verlag, New York-Berlin, 1972 | MR | Zbl
[Kac90] Infinite-dimensional Lie algebras, Cambridge University Press, Cambridge, 1990 | DOI | Zbl
[KRW03] Quantum reduction for affine superalgebras, Comm. Math. Phys., Volume 241 (2003) no. 2-3, pp. 307-342 | DOI | MR | Zbl
[KW89] Classification of modular invariant representations of affine algebras, Infinite-dimensional Lie algebras and groups (Luminy-Marseille, 1988) (Adv. Ser. Math. Phys.), Volume 7, World Sci. Publ., Teaneck, NJ, 1989, pp. 138-177 | MR
[KW08] On rationality of -algebras, Transform. Groups, Volume 13 (2008) no. 3-4, pp. 671-713 | DOI | MR | Zbl
[Li05] Abelianizing vertex algebras, Comm. Math. Phys., Volume 259 (2005) no. 2, pp. 391-411 | DOI | MR | Zbl
[LL04] Introduction to vertex operator algebras and their representations, Progress in Math., 227, Birkhäuser Boston, Inc., Boston, MA, 2004 | DOI | MR | Zbl
[Miy04] Modular invariance of vertex operator algebras satisfying -cofiniteness, Duke Math. J., Volume 122 (2004) no. 1, pp. 51-91 | DOI | MR | Zbl
[Mus01] Jet schemes of locally complete intersection canonical singularities, Invent. Math., Volume 145 (2001) no. 3, pp. 397-424 | DOI | MR | Zbl
[Pre02] Special transverse slices and their enveloping algebras, Adv. Math., Volume 170 (2002) no. 1, pp. 1-55 | DOI | MR | Zbl
[RT92] Indice et polynômes invariants pour certaines algèbres de Lie, J. reine angew. Math., Volume 425 (1992), pp. 123-140 | Zbl
[Slo80] Simple singularities and simple algebraic groups, Lect. Notes in Math., 815, Springer, Berlin, 1980 | MR | Zbl
[XY19] 4d SCFTs and lisse W-algebras, 2019 | arXiv
[Zhu96] Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc., Volume 9 (1996) no. 1, pp. 237-302 | DOI | MR | Zbl
Cité par Sources :