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Local systems of twisted vertex operators, vertex superalgebras and twisted modules
 Contemporary Math
"... Abstract. We introduce the notion of “local system of ZTtwisted vertex operators ” on a Z2graded vector space M, generalizing the notion of local system of vertex operators [Li]. First, we prove that any local system of ZTtwisted vertex operators on M has a vertex superalgebra structure with an a ..."
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Abstract. We introduce the notion of “local system of ZTtwisted vertex operators ” on a Z2graded vector space M, generalizing the notion of local system of vertex operators [Li]. First, we prove that any local system of ZTtwisted vertex operators on M has a vertex superalgebra structure with an automorphism σ of order T with M as a σtwisted module. Then we prove that for a vertex (operator) superalgebra V with an automorphism σ of order T, giving a σtwisted Vmodule M is equivalent to giving a vertex (operator) superalgebra homomorphism from V to some local system of ZTtwisted vertex operators on M. As applications, we study the twisted modules for vertex operator (super)algebras associated to some wellknown infinitedimensional Lie (super)algebras and we prove the complete reducibility of ZTtwisted modules for vertex operator algebras associated to standard modules for an affine Lie algebra. 1.
G.J.: Commutative quantum operator algebras
 J. Pure Appl. Algebra
"... ABSTRACT. A key notion bridging the gap between quantum operator algebras [22] and vertex operator algebras [4][8] is the definition of the commutativity of a pair of quantum operators (see section 2 below). This is not commutativity in any ordinary sense, but it is clearly the correct generalizatio ..."
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Cited by 40 (10 self)
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ABSTRACT. A key notion bridging the gap between quantum operator algebras [22] and vertex operator algebras [4][8] is the definition of the commutativity of a pair of quantum operators (see section 2 below). This is not commutativity in any ordinary sense, but it is clearly the correct generalization to the quantum context. The main purpose of the current paper is to begin laying the foundations for a complete mathematical theory of commutative quantum operator algebras. We give proofs of most of the relevant results announced in [22], and we carry out some calculations with sufficient detail to enable the interested reader to become proficient with the algebra of commuting quantum operators. We dedicate this paper to the memory of Feza Gürsey. 1
Classification of irreducible modules for the vertex operator algebra M(1
 J. Algebra
, 1999
"... We classify the irreducible modules for the fixed point vertex operator subalgebra of the vertex operator algebra associated to the Heisenberg algebra with the central charge 1 under the −1 automorphism. 1 ..."
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Cited by 39 (9 self)
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We classify the irreducible modules for the fixed point vertex operator subalgebra of the vertex operator algebra associated to the Heisenberg algebra with the central charge 1 under the −1 automorphism. 1
Compact automorphism groups of vertex operator algebras
 International Math. Research Notices
, 1996
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Vertex algebras and vertex Poisson algebras
 Commun. Contemp. Math
, 2004
"... This paper studies certain relations among vertex algebras, vertex Lie algebras and vertex poisson algebras. In this paper, the notions of vertex Lie algebra (conformal algebra) and vertex poisson algebra are revisited and certain general construction theorems of vertex poisson algebras are given. A ..."
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Cited by 32 (1 self)
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This paper studies certain relations among vertex algebras, vertex Lie algebras and vertex poisson algebras. In this paper, the notions of vertex Lie algebra (conformal algebra) and vertex poisson algebra are revisited and certain general construction theorems of vertex poisson algebras are given. A notion of filtered vertex algebra is formulated in terms of a notion of good filtration and it is proved that the associated graded vector space of a filtered vertex algebra is naturally a vertex poisson algebra. For any vertex algebra V, a general construction and a classification of good filtrations are given. To each Ngraded vertex algebra V = ∐ n∈N V (n) with V (0) = C1, a canonical (good) filtration is associated. Furthermore, a notion of ∗deformation of a vertex (poisson) algebra is formulated and a ∗deformation of vertex poisson algebras associated with vertex Lie algebras is constructed. 1
Vertex operator algebras and the representation theory of toroidal algebras
 in: Recent Developments in InfiniteDimensional Lie Algebras and Conformal Field Theory (Charlottesville, VA, 2000), Contemporary
, 2002
"... An explicit vertex operator algebra construction is given of a class of irreducible modules for toroidal Lie algebras. AMS (MOS) Subject Classifications:17B69, 17B68, 17B66, 17B10. ..."
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Cited by 24 (5 self)
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An explicit vertex operator algebra construction is given of a class of irreducible modules for toroidal Lie algebras. AMS (MOS) Subject Classifications:17B69, 17B68, 17B66, 17B10.
Intertwining operator algebras and vertex tensor categories for affine Lie algebras
, 1999
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Generalized rationality and a “Jacobi identity” for intertwining operator algebras
 Selecta Mathematica, New Series 6 (2000), 225–267. 35 [Hua7] [Hua8] [Hua9] [Hua10] [Hua11] [HL1] [HL2] [HL3] [HL4] [HL5] [HL6] Y.Z. Huang, Review of “Vertex Algebras and Algebraic Curves” by
"... We prove a generalized rationality property and a new identity that we call the “Jacobi identity ” for intertwining operator algebras. Most of the main properties of genuszero conformal field theories, including the main properties of vertex operator algebras, modules, intertwining operators, Verli ..."
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Cited by 23 (11 self)
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We prove a generalized rationality property and a new identity that we call the “Jacobi identity ” for intertwining operator algebras. Most of the main properties of genuszero conformal field theories, including the main properties of vertex operator algebras, modules, intertwining operators, Verlinde algebras, and fusing and braiding matrices, are incorporated into this identity. Together with associativity and commutativity for intertwining operators proved by the author in [H4] and [H6], the results of the present paper solve completely the problem of finding a natural purely algebraic structure on the direct sum of all inequivalent irreducible modules for a suitable vertex operator algebra. Two equivalent definitions of intertwining operator algebra in terms of this Jacobi identity are given. 0