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A generalized vertex operator algebra for Heisenberg intertwiners. (English) Zbl 1287.17050
In the paper under review, the authors consider the extension of rank \(l\) Heisenberg vertex operator algebra \(M(1)\) to the space \({\mathcal M}= \bigoplus _{\alpha \in {\mathbb C} ^l } M(1 , \alpha)\) given by the direct sum of all irreducible \(M(1)\)-modules. For any \(\alpha \in {\mathbb C} ^l\), they construct an intertwining operator of type \( {M(1 , \alpha) \choose M(1 , \alpha) \;M(1)} \), called the creative intertwining operator. The extension of these intertwining operators on \({\mathcal M}\), give \({\mathcal M}\) a structure of \({\mathbb C}\)-parameterized generalized vertex operator algebra. This construction is similar to the standard construction of lattice vertex operator algebras from [I. Frenkel, J. Lepowsky and A. Meurman, Vertex operator algebras and the Monster. Boston etc.: Academic Press (1988; Zbl 0674.17001); V. Kac, Vertex algebras for beginners. 2nd ed. Providence, RI: AMS (1998; Zbl 0924.17023)], and gives a natural generalization from rational to complex parameters of the notion of generalized vertex operator algebra from [C. Dong and J. Lepowsky, Generalized vertex algebras and relative vertex operators. Basel: Birkhäuser (1993; Zbl 0803.17009)].
The authors also discuss skew-symmetry, the existence of invariant symmetric bilinear form on \({\mathcal M}\), and apply these results to lattice vertex operator (super)algebras.

17B69 Vertex operators; vertex operator algebras and related structures
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