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Extensions of the Jacobi Identity for Vertex Operators, and Standard $$A^{(1)}_1$$-Modules
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Memoirs of the American Mathematical Society
1993; 85 pp; softcover
Volume: 106
ISBN-10: 0-8218-2571-2
ISBN-13: 978-0-8218-2571-6
List Price: US$34 Individual Members: US$20.40
Institutional Members: US\$27.20
Order Code: MEMO/106/507

This book extends the Jacobi identity, the main axiom for a vertex operator algebra, to multi-operator identities. Based on constructions of Dong and Lepowsky, relative $${\mathbf Z}_2$$-twisted vertex operators are then introduced, and a Jacobi identity for these operators is established. Husu uses these ideas to interpret and recover the twisted Z -operators and corresponding generating function identities developed by Lepowsky and Wilson for the construction of the standard $$A^{(1)}_1$$-modules. The point of view of the Jacobi identity also shows the equivalence between these twisted Z-operator algebras and the (twisted) parafermion algebras constructed by Zamolodchikov and Fadeev. The Lepowsky-Wilson generating function identities correspond to the identities involved in the construction of a basis for the space of C-disorder fields of such parafermion algebras.

• Standard representations of the twisted affine Lie algebra $$A^{(1)}_1$$