A family of regular vertex operator algebras with two generators

Dražen Adamović

Open Mathematics, Tome 5 (2007), p. 1-18 / Harvested from The Polish Digital Mathematics Library

Résumé

For every m ∈ ℂ ∖ 0, −2 and every nonnegative integer k we define the vertex operator (super)algebra D m,k having two generators and rank 3mm+2 . If m is a positive integer then D m,k can be realized as a subalgebra of a lattice vertex algebra. In this case, we prove that D m,k is a regular vertex operator (super) algebra and find the number of inequivalent irreducible modules.

Détail

Publié le : 2007-01-01

Zbl 1197.17016

EUDML-ID : urn:eudml:doc:268977

Bibliographie

[1] D. Adamović: “Rationality of Neveu-Schwarz vertex operator superalgebras”, Int. Math. Res. Not., Vol. 17, (1997), pp. 865–874 http://dx.doi.org/10.1155/S107379289700055X | Zbl 0885.17018

[2] D. Adamović: “Representations of the N = 2 superconformal vertex algebra”, Int. Math. Res. Not., Vol. 2, (1999), pp. 61–79 http://dx.doi.org/10.1155/S1073792899000033 | Zbl 0920.17013

[3] D. Adamović: “Vertex algebra approach to fusion rules for N = 2 superconformal minimal models”, J. Algebra, Vol. 239, (2001), pp. 549–572 http://dx.doi.org/10.1006/jabr.2000.8728 | Zbl 1022.17019

[4] D. Adamović: Regularity of certain vertex operator superalgebras, Contemp. Math., Vol. 343, Amer. Math. Soc., Providence, 2004, pp. 1–16. | Zbl 1058.17017

[5] T. Abe, G. Buhl and C. Dong: “Rationality, regularity and C 2-cofiniteness”, Trans. Amer. Math. Soc., Vol. 356, (2004), pp. 3391–3402. http://dx.doi.org/10.1090/S0002-9947-03-03413-5 | Zbl 1070.17011

[6] D. Adamović and A. Milas: “Vertex operator algebras associated to the modular invariant representations for A 1(1)”, Math. Res. Lett., Vol. 2, (1995), pp. 563–575 | Zbl 0848.17033

[7] C. Dong: “Vertex algebras associated with even lattices”, J. Algebra, Vol. 160, (1993), pp. 245–265. http://dx.doi.org/10.1006/jabr.1993.1217

[8] C. Dong and J. Lepowsky: Generalized vertex algebras and relative vertex operators, Birkhäuser, Boston, 1993. | Zbl 0803.17009

[9] C. Dong, H. Li and G. Mason: “Regularity of rational vertex operator algebras”, Adv. Math., Vol. 132, (1997), pp. 148–166 http://dx.doi.org/10.1006/aima.1997.1681

[10] C. Dong, G. Mason and Y. Zhu: “Discrete series of the Virasoro algebra and the Moonshine module”, Proc. Sympos. Math. Amer. Math. Soc., Vol. 56(2), (1994), pp. 295–316 | Zbl 0813.17019

[11] W. Eholzer and M.R. Gaberdiel: “Unitarity of rational N = 2 superconformal theories”, Comm. Math. Phys., Vol. 186, (1997), pp. 61–85. | Zbl 0897.17003

[12] I.B. Frenkel, Y.-Z. Huang and J. Lepowsky: “On axiomatic approaches to vertex operator algebras and modules”, Memoris Am. Math. Soc., Vol. 104, 1993. | Zbl 0789.17022

[13] I. B. Frenkel, J. Lepowsky and A. Meurman: Vertex Operator Algebras and the Monster, Pure Appl. Math., Vol. 134, Academic Press, New York, 1988. | Zbl 0674.17001

[14] I.B. Frenkel and Y. Zhu: “Vertex operator algebras associated to representations of affine and Virasoro algebras”, Duke Math. J., Vol. 66, (1992), pp. 123–168. http://dx.doi.org/10.1215/S0012-7094-92-06604-X | Zbl 0848.17032

[15] B.L. Feigin, A.M. Semikhatov and I.Yu. Tipunin: “Equivalence between chain categories of representations of affine sl(2) and N = 2 superconformal algebras”, J. Math. Phys., Vol. 39, (1998), pp. 3865–3905 http://dx.doi.org/10.1063/1.532473 | Zbl 0935.17011

[16] B.L. Feigin, A.M. Semikhatov and I.Yu. Tipunin: “A semi-infinite construction of unitary N=2 modules”, Theor. Math. Phys., Vol. 126(1), (2001), pp. 1–47. http://dx.doi.org/10.1023/A:1005286813871

[17] Y.-Z Huang and A. Milas: “Intertwining operator superalgebras and vertex tensor categories for superconformal algebras”, II. Trans. Amer. Math. Soc., Vol. 354, (2002), pp. 363–385. http://dx.doi.org/10.1090/S0002-9947-01-02869-0 | Zbl 1082.17014

[18] V.G. Kac: Vertex Algebras for Beginners, University Lecture Series, Vol. 10, 2nd ed., AMS, 1998. | Zbl 0924.17023

[19] Y. Kazama and H. Suzuki: “New N=2 superconformal field theories and superstring compactifications”, Nuclear Phys. B, Vol. 321, (1989), pp. 232–268. http://dx.doi.org/10.1016/0550-3213(89)90250-2

[20] H. Li: “Local systems of vertex operators, vertex superalgebras and modules”, J. Pure Appl. Algebra, Vol. 109, (1996), pp. 143–195. http://dx.doi.org/10.1016/0022-4049(95)00079-8

[21] H. Li: “Extension of Vertex Operator Algebras by a Self-Dual Simple Module”, J. Algebra, Vol. 187, (1997), pp. 236–267. http://dx.doi.org/10.1006/jabr.1997.6795

[22] H. Li: “Certain extensions of vertex operator algebras of affine type”, Comm. Math. Phys., Vol. 217, (2001), pp. 653–696. http://dx.doi.org/10.1007/s002200100386 | Zbl 0985.17019

[23] H. Li: “Some finiteness properties of regular vertex operator algebras”, J. Algebra, Vol. 212, (1999), pp. 495–514. http://dx.doi.org/10.1006/jabr.1998.7654

[24] M. Wakimoto: Lectures on infinite-dimensional Lie algebra, algebra, World Scientific Publishing Co., Inc., River Edge, NJ, 2001.

[25] W. Wang: “Rationality of Virasoro Vertex operator algebras”, Internat. Math. Res. Notices, Vol. 71(1), (1993), PP. 197–211. http://dx.doi.org/10.1155/S1073792893000212 | Zbl 0791.17029

[26] Xu Xiaoping: Introduction to vertex operator superalgebras and their modules, Mathematics and Its Applications, Vol. 456, Kluwer Academic Publishers, 1998.