Affine Lie algebras of sl_2

Twisted vertex operator representation of

Let be the [[Affine Lie algebras of sl_2|-twisted affine Lie algebra of ]], and be the corresponding [[Natural Heisenberg algebras|-natural Heisenberg module]] on . Defining

we construct the twisted vertex operator

where the second expression used the Normal ordered product. Skipping over a lot of detail¹, the representation of on extends to precisely two irreducible representations

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Footnotes

1. 1988. Vertex operator algebras and the Monster, §3, pp. 61ff. ↩