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JOSEPH IDEALS AND LISSE MINIMAL $W$-ALGEBRAS

Part of: Lie algebras and Lie superalgebras Groups and algebras in quantum theory

Published online by Cambridge University Press:  07 March 2016

Tomoyuki Arakawa  and
Anne Moreau
Tomoyuki Arakawa
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan ([email protected])
Anne Moreau
Affiliation:
Laboratoire de Mathématiques et Applications, Téléport 2 - BP 30179, Boulevard Marie et Pierre Curie, 86962 Futuroscope Chasseneuil Cedex, France ([email protected])
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Abstract

We consider a lifting of Joseph ideals for the minimal nilpotent orbit closure to the setting of affine Kac–Moody algebras and find new examples of affine vertex algebras whose associated varieties are minimal nilpotent orbit closures. As an application we obtain a new family of lisse ($C_{2}$-cofinite) $W$-algebras that are not coming from admissible representations of affine Kac–Moody algebras.

Keywords

Joseph idealassociated varietyDeligne exceptional seriesaffine Kac–Moody algebraaffine W-algebra

MSC classification

Primary: 17B67: Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 17B69: Vertex operators; vertex operator algebras and related structures 81R10: Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $W$-algebras and other current algebras and their representations
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 17 , Issue 2 , April 2018 , pp. 397 - 417
Copyright
© Cambridge University Press 2016 

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References

Adamović, D., Some rational vertex algebras, Glas. Mat. Ser. III 29(49)(1) (1994), 25–40.Google Scholar
Adamović, D., Vertex operator algebras and irreducibility of certain modules for affine Lie algebras, Math. Res. Lett. 4(6) (1997), 809–821.Google Scholar
Adamović, D. and Milas, A., Vertex operator algebras associated to modular invariant representations for A 1 (1) , Math. Res. Lett. 2(5) (1995), 563–575.Google Scholar
Arakawa, T., Representation theory of superconformal algebras and the Kac–Roan–Wakimoto conjecture, Duke Math. J. 130(3) (2005), 435–478.Google Scholar
Arakawa, T., A remark on the C 2 cofiniteness condition on vertex algebras, Math. Z. 270(1–2) (2012), 559–575.Google Scholar
Arakawa, T., Rationality of Bershadsky–Polyakov vertex algebras, Comm. Math. Phys. 323(2) (2013), 627–633.Google Scholar
Arakawa, T., Associated varieties of modules over Kac–Moody algebras and C 2 -cofiniteness of W-algebras, Int. Math. Res. Not. IMRN 2015 (2015), 11605–11666.Google Scholar
Arakawa, T., Rationality of W-algebras: principal nilpotent cases, Ann. of Math. (2) 182(2) (2015), 565–694.Google Scholar
Arakawa, T., Rationality of admissible affine vertex algebras in the category 𝓞, Duke Math. J. 165(1) (2016), 67–93.Google Scholar
Arakawa, T., Lam, C. H. and Yamada, H., Zhu’s algebra, C 2 -algebra and C 2 -cofiniteness of parafermion vertex operator algebras, Adv. Math. 264 (2014), 261–295.Google Scholar
Axtell, J. D. and Lee, K.-H., Vertex operator algebras associated to type G affine Lie algebras, J. Algebra 337 (2011), 195–223.CrossRefGoogle Scholar
Bakalov, B. and Kirillov, A. Jr, Lectures on Tensor Categories and Modular Functors, University Lecture Series, Volume 21 (American Mathematical Society, Providence, RI, 2001).Google Scholar
Beem, C., Lemos, M., Liendo, P., Peelaers, W., Rastelli, L. and van Rees, B. C., Infinite chiral symmetry in four dimensions, Comm. Math. Phys. 336(3) (2015), 1359–1433.Google Scholar
Beilinson, A. and Drinfeld, V., Quantization of Hitchin’s integrable system and Hecke eigensheaves, Preprint.Google Scholar
Creutzig, T. and Ridout, D., Modular data and Verlinde formulae for fractional level WZW models I, Nuclear Phys. B 865(1) (2012), 83–114.Google Scholar
Creutzig, T. and Ridout, D., Modular data and Verlinde formulae for fractional level WZW models II, Nuclear Phys. B 875(2) (2013), 423–458.Google Scholar
Deligne, P., La série exceptionnelle de groupes de Lie, C. R. Acad. Sci. Paris, Ser I 322(4) (1996), 321–326.Google Scholar
Deligne, P. and Gross, B., On the exceptional series, and its descendants, C. R. Acad. Sci. Paris, Ser I 335(11) (2002), 877–881.Google Scholar
De Sole, A. and Kac, V. G., Finite vs affine W-algebras, Jpn. J. Math. 1(1) (2006), 137–261.Google Scholar
Dong, C. and Mason, G., Integrability of $C_{2}$ -cofinite vertex operator algebras, Int. Math. Res. Not. (2006), Art ID 80468, 15 pp.CrossRefGoogle Scholar
Feigin, B. and Malikov, F., Modular functor and representation theory of sl̂2 at a rational level, in Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), Contemporary Mathematics, Volume 202, pp. 357–405 (American Mathematical Society, Providence, RI, 1997).Google Scholar
Frenkel, E. and Ben-Zvi, D., Vertex Algebras and Algebraic Curves, 2nd edn, Mathematical Surveys and Monographs, Volume 88 (American Mathematical Society, Providence, RI, 2004).Google Scholar
Frenkel, I. and Zhu, Y., Vertex operator algebras associated to representations of affine and Virasoro algebras, Duke Math. J. 66(1) (1992), 123–168.CrossRefGoogle Scholar
Gan, W. T. and Savin, G., Uniqueness of Joseph ideal, Math. Res. Lett. 11(5–6) (2004), 589–597.CrossRefGoogle Scholar
Garfinkle, D., A new construction of the Joseph ideal. PhD thesis, MIT, 1982.Google Scholar
Huang, Y.-Z., Rigidity and modularity of vertex tensor categories, Commun. Contemp. Math. 10(suppl. 1) (2008), 871–911.Google Scholar
Joseph, A., The minimal orbit in a simple Lie algebra and its associated maximal ideal, Ann. Sci. Éc. Norm. Supér. (4) 9(1) (1976), 1–29.Google Scholar
Joseph, A., Orbital varietes of the minimal orbit, Ann. Sci. Éc. Norm. Supér. (4) 31(1) (1998), 17–45.CrossRefGoogle Scholar
Kac, V., Roan, S.-S. and Wakimoto, M., Quantum reduction for affine superalgebras, Comm. Math. Phys. 241(2–3) (2003), 307–342.Google Scholar
Kac, V. and Wakimoto, M., Modular invariant representations of infinite-dimensional Lie algebras and superalgebras, Proc. Natl Acad. Sci. USA 85(14) (1988), 4956–4960.Google Scholar
Kac, V. and Wakimoto, M., Classification of modular invariant representations of affine algebras, in Infinite-dimensional Lie Algebras and Groups (Luminy-Marseille, 1988), Adv. Ser. Math. Phys., Volume 7, pp. 138–177 (World Scientific Publishers, Teaneck, NJ, 1989).Google Scholar
Kac, V. and Wakimoto, M., Quantum reduction and representation theory of superconformal algebras, Adv. Math. 185(2) (2004), 400–458.CrossRefGoogle Scholar
Kac, V. and Wakimoto, M., On rationality of W-algebras, Transform. Groups 13(3–4) (2008), 671–713.Google Scholar
Kawasetsu, K., ${\mathcal{W}}$ -algebras with non-admissible levels and the Deligne exceptional series, Preprint, arXiv:1505.06985.Google Scholar
Miyamoto, M., Modular invariance of vertex operator algebras satisfying C 2 -cofiniteness, Duke Math. J. 122(1) (2004), 51–91.Google Scholar
Perše, O., Vertex operator algebras associated to type B affine Lie algebras on admissible half-integer levels, J. Algebra 307(1) (2007), 215–248.Google Scholar
Perše, O., Vertex operator algebra analogue of embedding of B 4 into F 4 , J. Pure Appl. Algebra 211(3) (2007), 702–720.CrossRefGoogle Scholar
Perše, O., A note on representations of some affine vertex algebras of type D , Glas. Mat. Ser. III 48(68)(1) (2013), 81–90.Google Scholar
Premet, A., Special transverse slices and their enveloping algebras, Adv. Math. 170(1) (2002), 1–55. With an appendix by Serge Skryabin.Google Scholar
Premet, A., Enveloping algebras of Slodowy slices and the Joseph ideal, J. Eur. Math. Soc. 9(3) (2007), 487–543.CrossRefGoogle Scholar
Wang, W., Dimension of a minimal nilpotent orbit, Proc. Amer. Math. Soc. 127(3) (1999), 935–936.CrossRefGoogle Scholar
Zhu, Y., Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9(1) (1996), 237–302.Google Scholar