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Two constructions of grading-restricted vertex (super)algebras
We give two constructions of grading-restricted vertex (super)algebras. We
first give a new construction of a class of grading-restricted vertex
(super)algebras originally obtained by Meurman and Primc using a different
method. This construction is based on a new definition of vertex operators and
a new method. Our second construction is a generalization of the author's
construction of the moonshine module vertex operator algebra and a related
vertex operator superalgebra. This construction needs properties of
intertwining operators formulated and proved by the author.Comment: 26 pages. Misprints are corrected. To appear in Journal of Pure and
Applied Algebr
Virasoro vertex operator algebras, the (nonmeromorphic) operator product expansion and the tensor product theory
In the references [HL1]--[HL5] and [H1], a theory of tensor products of
modules for a vertex operator algebra is being developed. To use this theory,
one first has to verify that the vertex operator algebra satisfies certain
conditions. We show in the present paper that for any vertex operator algebra
containing a vertex operator subalgebra isomorphic to a tensor product algebra
of minimal Virasoro vertex operator algebras (vertex operator algebras
associated to minimal models), the tensor product theory can be applied. In
particular, intertwining operators for such a vertex operator algebra satisfy
the (nonmeromorphic) commutativity (locality) and the (nonmeromorphic)
associativity (operator product expansion). Combined with a result announced in
[HL4], the results of the present paper also show that the category of modules
for such a vertex operator algebra has a natural structure of a braided tensor
category. In particular, for any pair of relatively prime positive
integers larger than , the category of minimal modules of central charge
for the Virasoro algebra has a natural structure of a
braided tensor category.Comment: LaTeX file. 37 page
A functional-analytic theory of vertex (operator) algebras, I
This paper is the first in a series of papers developing a
functional-analytic theory of vertex (operator) algebras and their
representations. For an arbitrary Z-graded finitely-generated vertex algebra
(V, Y, 1) satisfying the standard grading-restriction axioms, a locally convex
topological completion H of V is constructed. By the geometric interpretation
of vertex (operator) algebras, there is a canonical linear map from the tensor
product of V and V to the algebraic completion of V realizing linearly the
conformal equivalence class of a genus-zero Riemann surface with analytically
parametrized boundary obtained by deleting two ordered disjoint disks from the
unit disk and by giving the obvious parametrizations to the boundary
components. We extend such a linear map to a linear map from the completed
tensor product of H and H to H, and prove the continuity of the extension. For
any finitely-generated C-graded V-module (W, Y_W) satisfying the standard
grading-restriction axioms, the same method also gives a topological completion
H^W of W and gives the continuous extensions from the completed tensor product
of H and H^W to H^W of the linear maps from the tensor product of V and W to
the algenbraic completion of W realizing linearly the above conformal
equivalence classes of the genus-zero Riemann surfaces with analytically
parametrized boundaries.Comment: LaTeX file. 31 pages, 1 figur
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