Dong, Chongying; Griess, Robert L. jun.; Ryba, Alex Rank one lattice type vertex operator algebras and their automorphism groups. II: \(E\)-series. (English) Zbl 1028.17019 J. Algebra 217, No. 2, 701-710 (1999). Summary: Let \(L\) be the \(A_1\) root lattice and let \(G\) be a finite subgroup of \(\text{Aut}(V)\), where \(V=V_L\) is the associated lattice VOA (in this case, \(\text{Aut}(V)\cong \text{PSL}(2,\mathbb{C}))\). The fixed point sub-VOA, \(V^G\), was studied previously by the authors [C. Dong and R. L. Griess jun., J. Algebra 208, 262-275 (1998; Zbl 0918.17023)], who found a set of generators and determined the automorphism group when \(G\) is cyclic (from the “\(A\)-series”) or dihedral (from the “\(D\)-series”). In the present article, they obtain analogous results for the remaining possibilities for \(G\), that is \(G\) belongs to the “\(E\)-series”: \(G\cong \text{Alt}_4\), \(\text{Alt}_5\), or \(\text{Sym}_4\). For such \(L\) and \(G\), the above \(V_L^G\) may be rational VOAs. Cited in 7 Documents MSC: 17B69 Vertex operators; vertex operator algebras and related structures 17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras Keywords:vertex operator algebras; root lattice; automorphism group Citations:Zbl 0918.17023 × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Borcherds, R. E., Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Nat. Acad. Sci. U.S.A., 83, 3068-3071 (1986) · Zbl 0613.17012 [2] Curtis, C.; Reiner, L., Representation Theory of Finite Groups and Associative Algebras (1962), Wiley-Interscience: Wiley-Interscience New York · Zbl 0131.25601 [3] Dong, C., Vertex algebras associated with even lattices, J. Algebra, 160, 245-265 (1993) · Zbl 0807.17022 [4] Dong, C.; Griess, R. L., Rank one lattice type vertex operator algebras and their automorphism groups, J. Algebra, 208, 262-275 (1998) · Zbl 0918.17023 [5] Dong, C.; Li, H.; Mason, G., Compact automorphism groups of vertex operator algebras, Internat. Math. Res. Notices, 18, 913-921 (1996) · Zbl 0873.17028 [6] Dong, C.; Li, H.; Mason, G., Regularity of rational vertex operator algebras, Adv. Math., 132, 148-166 (1997) · Zbl 0902.17014 [7] Dong, C.; Mason, G., On quantum Galois theory, Duke Math. J., 86, 305-321 (1997) · Zbl 0890.17031 [8] Dong, C.; Mason, G., Quantum Galois theory for compact Lie groups, J. Algebra, 241, 92-102 (1991) · Zbl 0929.17031 [9] Dong, C.; Nagamoto, K., Representations of vertex operator algebra \(V^+_L\) for rank one lattice \(L\), Comm. Math. Phys., 202, 169-195 (1999) · Zbl 0929.17033 [10] C. Dong, and, K. Nagamoto, Automorphism groups and twisted modules for lattice vertex algebras, Contemp. Math, to appear.; C. Dong, and, K. Nagamoto, Automorphism groups and twisted modules for lattice vertex algebras, Contemp. Math, to appear. [11] Frenkel, I. B.; Lepowsky, J.; Meurmann, A., Vertex Operator Algebras and the Monster. Vertex Operator Algebras and the Monster, Pure and Applied Mathematics, 134 (1988), Academic Press: Academic Press San Diego · Zbl 0674.17001 [12] Ginsparg, P., Curiosities at \(c=1\), Nuclear Phys., 295, 153-170 (1988) [13] Humphreys, J., Introduction to Lie Algebras and Representation Theory. Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, 9 (1972), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0254.17004 [14] Kiritsis, E., Proof of the completeness of the classification of rational conformal field theories with \(c=1\), Phys. Lett. B, 217, 427-430 (1989) [15] Parker, R. A., The computer calculation of modular characters (the Meat-Axe), (Atkinson, M. D., Computational Group Theory (1984), Academic Press: Academic Press London) · Zbl 0555.20001 [16] Springer, T. A., Invariant Theory. Invariant Theory, Lecture Notes in Mathematics, 585 (1997), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0346.20020 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.