• James Lepowsky
  • Haisheng Li
Part of the Progress in Mathematics book series (PM, volume 227)


In this chapter we discuss the notion of module for a vertex (operator) algebra V. A V-module is defined, as expected, to be a vector space W equipped with a linear map
$$ Y_W :V \to (End W)[[x,x^{ - 1} ]] $$
such that all the defining properties of a vertex algebra that make sense hold. (Actually, this is the typical principle for defining the notion of module for categories of algebras in general—Lie algebras, associative algebras, etc. A module is a vector space equipped with a linear action of the algebra such that all the algebra axioms that make sense hold.) Specifically, these defining properties are the truncation condition, the vacuum property and, the Jacobi identity. (The creation property, for instance, would not make sense, so it will not be an axiom.) Accordingly, almost all of the assertions in Chapter 3 that make sense also hold and, a large amount of material in Chapter 3 carries over in the obvious ways, often without change. In this chapter, we carry out many of these analogues, and we also discuss some additional concepts.


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Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • James Lepowsky
    • 1
  • Haisheng Li
    • 2
  1. 1.Department of MathematicsRutgers UniversityPiscatawayUSA
  2. 2.Department of MathematicsRutgers UniversityCamdenUSA

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