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On VOA associated with special Jordan algebras. (English) Zbl 0934.17016

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)

MSC:

17B69 Vertex operators; vertex operator algebras and related structures
17C10 Structure theory for Jordan algebras
Full Text: DOI

References:

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