Lam, Ching Hung On VOA associated with special Jordan algebras. (English) Zbl 0934.17016 Commun. Algebra 27, No. 4, 1665-1681 (1999). If \(V=\oplus V_n\) is a vertex operator algebra (VOA), then the weight two space \(V_2\) has an algebra structure with the operation \(a_1 b\); in the case when \(V_n=0\) for \(n<0\), \(V_0=\mathbb C \mathbf 1\) and \(V_1=0\) this operation is commutative (and in general nonassociative). The most interesting and motivating example is the Moonshine module, but there are other examples as well [cf. C. H. Lam, Commun. Algebra 24, 4339-4360 (1996; Zbl 0892.17020)]. In the paper under review the author starts with a simple Jordan algebra \(A\) of type A, B or C and constructs a vertex operator algebra \(V\) such that \(V_2\cong A\). Roughly speaking, \(V\) is constructed as a vertex operator subalgebra generated by quadratic elements \(h_i(-1)h_j(-1)\mathbf 1\) in the VOA associated with a Heisenberg Lie algebra. In addition, the automorphism groups of these vertex operator algebras are computed by using the results in [N. Jacobson, The structure and representations of Jordan algebras, AMS Colloq. Publ. 39 (1968; Zbl 0218.17010)]. Reviewer: M.Primc (Zagreb) Cited in 1 ReviewCited in 8 Documents MSC: 17B69 Vertex operators; vertex operator algebras and related structures 17C10 Structure theory for Jordan algebras Keywords:vertex operator algebra; Moonshine module; simple Jordan algebra; automorphism groups Citations:Zbl 0892.17020; Zbl 0218.17010 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] DOI: 10.2307/1969128 · Zbl 0029.01003 · doi:10.2307/1969128 [2] Borcherds R., Pro. Natl. Acad. Sci 83 pp 3068– (1986) [3] DOI: 10.1215/S0012-7094-97-08609-9 · Zbl 0890.17031 · doi:10.1215/S0012-7094-97-08609-9 [4] Frenkel I.B., Mem. Amer. Math. Soc 104 (1993) [5] Frenkel I.B., Vertex operator algebras and the Monster (1988) · Zbl 0674.17001 [6] DOI: 10.1007/BF01389186 · Zbl 0498.20013 · doi:10.1007/BF01389186 [7] Jacobson F.D., Trans. Amer.Math. Soc 65 pp 141– (1949) [8] Jacobson N., Amer. Math. Soc 39 (1968) [9] DOI: 10.1080/00927879608825819 · Zbl 0892.17020 · doi:10.1080/00927879608825819 [10] Lam C.H., Ph.D. dissertation (1996) [11] DOI: 10.1007/BF02102011 · Zbl 0823.17039 · doi:10.1007/BF02102011 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.