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Chongying Dong - One of the best experts on this subject based on the ideXlab platform.
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determinants for integral forms in lattice type vertex Operator Algebras
Journal of Algebra, 2020Co-Authors: Chongying Dong, Robert L. GriessAbstract:Abstract We prove a determinant formula for the standard integral form of a lattice vertex Operator Algebra.
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retracted article modular framed vertex Operator Algebras
European Journal of Mathematics, 2018Co-Authors: Chongying Dong, Ching Hung Lam, Li RenAbstract:Framed vertex Operator Algebras over any Algebraically closed field whose characteristic is different from 2 and 7 are studied. In particular, the rationality of framed vertex Operator Algebras is established. For a code vertex Operator Algebra, the irreducible modules are constructed and classified. Moreover, a Z[\frac{1}{2}]-form for any framed vertex Operator Algebra over C is constructed. As a result, one can obtain a modular framed vertex Operator Algebra from any framed vertex Operator Algebra over C.
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Modular framed vertex Operator Algebras
arXiv: Quantum Algebra, 2017Co-Authors: Chongying Dong, Ching Hung Lam, Li RenAbstract:Framed vertex Operator Algebras over any Algebraically closed field whose characteristic is different from 2 and 7 are studied. In particular, the rationality of framed vertex Operator Algebras is established. For a code vertex Operator Algebra, the irreducible modules are constructed and classified. Moreover, a Z[\frac{1}{2}]-form for any framed vertex Operator Algebra over C is constructed. As a result, one can obtain a modular framed vertex Operator Algebra from any framed vertex Operator Algebra over C.
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the 3 permutation orbifold of a lattice vertex Operator Algebra
Journal of Pure and Applied Algebra, 2017Co-Authors: Chongying DongAbstract:Abstract Irreducible modules of the 3-permutation orbifold of a rank one lattice vertex Operator Algebra are listed explicitly. Fusion rules are determined by using the quantum dimensions. The S-matrix is also given.
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representations of vertex Operator Algebras over an arbitrary field
Journal of Algebra, 2014Co-Authors: Chongying DongAbstract:An associative Algebra A(V) for a vertex Operator Algebra over an arbitrary Algebraically closed field F is constructed such that there is a one to one correspondence between irreducible A(V)-modules and irreducible admissible V-modules. Moreover, A(V) is semisimple if V is rational. An A(V)-bimodule A(M) is also constructed for any admissible V-module M and its connection with the fusion rules is investigated. These results are then used to compute the fusion rules for the rational vertex Operator Algebra L(12,0)F associated to the irreducible highest weight module for the Virasoro Algebra with central charge 12.
Cuipo Jiang - One of the best experts on this subject based on the ideXlab platform.
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representations of z2 orbifold of the parafermion vertex Operator Algebra k sl2 k
Journal of Algebra, 2019Co-Authors: Cuipo Jiang, Qing WangAbstract:Abstract In this paper, the irreducible modules for the Z 2 -orbifold vertex Operator subAlgebra of the parafermion vertex Operator Algebra associated to the irreducible highest weight modules for the affine Kac-Moody Algebra A 1 ( 1 ) of level k are classified and constructed.
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the commutant of lslˆ2 n 0 in the vertex Operator Algebra lslˆ2 1 0 n
Advances in Mathematics, 2016Co-Authors: Cuipo Jiang, Zongzhu LinAbstract:Abstract We study the commutant L sl ˆ 2 ( n , 0 ) c of L sl ˆ 2 ( n , 0 ) in the vertex Operator Algebra L sl ˆ 2 ( 1 , 0 ) ⊗ n , for n ≥ 2 . The main results include a complete classification of all irreducible L sl ˆ 2 ( n , 0 ) c -modules and that L sl ˆ 2 ( n , 0 ) c is a rational vertex Operator Algebra. As a consequence, every irreducible L sl ˆ 2 ( n , 0 ) c -module arises from the coset construction as conjectured in [37] .
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extension of vertex Operator Algebra v_ widehat h _ 4 ell 0
Algebra Colloquium, 2014Co-Authors: Cuipo Jiang, Song WangAbstract:We classify the irreducible restricted modules for the affine Nappi-Witten Lie Algebra with some natural conditions. It turns out that the representation theory of is quite different from the theory of representations of Heisenberg Algebras. We also study the extension of the vertex Operator Algebra by the even lattice L. We give the structure of the extension and its irreducible modules via irreducible representations of viewed as a vertex Algebra.
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a characterization of the rational vertex Operator Algebra vzα ii
Advances in Mathematics, 2013Co-Authors: Chongying Dong, Cuipo JiangAbstract:Abstract A characterization of vertex Operator Algebra V L + for any rank one positive definite even lattice L is given in terms of dimensions of homogeneous subspaces of small weights. This result reduces the classification of rational vertex Operator Algebras of central charge 1 to the characterization of three vertex Operator Algebras in the E-series of central charge one.
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Representations of the vertex Operator Algebra VL2A4
Journal of Algebra, 2013Co-Authors: Chongying Dong, Cuipo JiangAbstract:Abstract The rationality and C 2 -cofiniteness of the orbifold vertex Operator Algebra V L 2 A 4 are established and all the irreducible modules are constructed and classified. This is part of classification of rational vertex Operator Algebras with c = 1 .
Zongzhu Lin - One of the best experts on this subject based on the ideXlab platform.
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the commutant of lslˆ2 n 0 in the vertex Operator Algebra lslˆ2 1 0 n
Advances in Mathematics, 2016Co-Authors: Cuipo Jiang, Zongzhu LinAbstract:Abstract We study the commutant L sl ˆ 2 ( n , 0 ) c of L sl ˆ 2 ( n , 0 ) in the vertex Operator Algebra L sl ˆ 2 ( 1 , 0 ) ⊗ n , for n ≥ 2 . The main results include a complete classification of all irreducible L sl ˆ 2 ( n , 0 ) c -modules and that L sl ˆ 2 ( n , 0 ) c is a rational vertex Operator Algebra. As a consequence, every irreducible L sl ˆ 2 ( n , 0 ) c -module arises from the coset construction as conjectured in [37] .
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the commutant of l_ widehat frak sl _ 2 n 0 in the vertex Operator Algebra l_ widehat frak sl _ 2 1 0 otimes n
arXiv: Quantum Algebra, 2013Co-Authors: Zongzhu LinAbstract:We study the commutant $L_{\widehat{\frak{sl}}_{2}}(n,0)^c$ of $L_{\widehat{\frak{sl}}_{2}}(n,0)$ in the vertex Operator Algebra $L_{\widehat{\frak{sl}}_{2}}(1,0)^{\otimes n}$, for $n\geq 2$. The main results include a complete classification of all irreducible $L_{\widehat{\frak{sl}}_{2}}(n,0)^c$-modules and a proof that $L_{\widehat{\frak{sl}}_{2}}(n,0)^c$ is a rational vertex Operator Algebra. As a consequence, every irreducible $L_{\widehat{\frak{sl}}_{2}}(n,0)^c$-module arises from the coset construction as conjectured in \cite{LS}.
Ching Hung Lam - One of the best experts on this subject based on the ideXlab platform.
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sigma involutions associated with parafermion vertex Operator Algebra k mathfrak sl _2 k
arXiv: Quantum Algebra, 2020Co-Authors: Ching Hung Lam, Hiromichi YamadaAbstract:An irreducible module for the parafermion vertex Operator Algebra $K(\mathfrak{sl}_2,k)$ is said to be of $\sigma$-type if an automorphism of the fusion Algebra of $K(\mathfrak{sl}_2,k)$ of order $k$ is trivial on it. For any integer $k \ge 3$, we show that there exists an automorphism of order $2$ of the subAlgebra of the fusion Algebra of $K(\mathfrak{sl}_2,k)^{\langle \theta \rangle}$ spanned by the irreducible direct summands of $\sigma$-type irreducible $K(\mathfrak{sl}_2,k)$-modules, where $\theta$ is an involution of $K(\mathfrak{sl}_2,k)$. We discuss some examples of such an automorphism as well.
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retracted article modular framed vertex Operator Algebras
European Journal of Mathematics, 2018Co-Authors: Chongying Dong, Ching Hung Lam, Li RenAbstract:Framed vertex Operator Algebras over any Algebraically closed field whose characteristic is different from 2 and 7 are studied. In particular, the rationality of framed vertex Operator Algebras is established. For a code vertex Operator Algebra, the irreducible modules are constructed and classified. Moreover, a Z[\frac{1}{2}]-form for any framed vertex Operator Algebra over C is constructed. As a result, one can obtain a modular framed vertex Operator Algebra from any framed vertex Operator Algebra over C.
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Modular framed vertex Operator Algebras
arXiv: Quantum Algebra, 2017Co-Authors: Chongying Dong, Ching Hung Lam, Li RenAbstract:Framed vertex Operator Algebras over any Algebraically closed field whose characteristic is different from 2 and 7 are studied. In particular, the rationality of framed vertex Operator Algebras is established. For a code vertex Operator Algebra, the irreducible modules are constructed and classified. Moreover, a Z[\frac{1}{2}]-form for any framed vertex Operator Algebra over C is constructed. As a result, one can obtain a modular framed vertex Operator Algebra from any framed vertex Operator Algebra over C.
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on orbifold constructions associated with the leech lattice vertex Operator Algebra
arXiv: Quantum Algebra, 2017Co-Authors: Ching Hung Lam, Hiroki ShimakuraAbstract:In this article, we study orbifold constructions associated with the Leech lattice vertex Operator Algebra. As an application, we prove that the structure of a strongly regular holomorphic vertex Operator Algebra of central charge $24$ is uniquely determined by its weight one Lie Algebra if the Lie Algebra has the type $A_{3,4}^3A_{1,2}$, $A_{4,5}^2$, $D_{4,12}A_{2,6}$, $A_{6,7}$, $A_{7,4}A_{1,1}^3$, $D_{5,8}A_{1,2}$ or $D_{6,5}A_{1,1}^2$ by using the reverse orbifold construction. Our result also provides alternative constructions of these vertex Operator Algebras (except for the case $A_{6,7}$) from the Leech lattice vertex Operator Algebra.
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the structure of parafermion vertex Operator Algebras
arXiv: Quantum Algebra, 2009Co-Authors: Qing Wang, Chongying Dong, Ching Hung Lam, Hiromichi YamadaAbstract:It is proved that the parafermion vertex Operator Algebra associated to the irreducible highest weight module for the affine Kac-Moody Algebra A_1^{(1)} of level k coincides with a certain W-Algebra. In particular, a set of generators for the parafermion vertex Operator Algebra is determined.
Geoffrey Mason - One of the best experts on this subject based on the ideXlab platform.
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Regularity of Rational Vertex Operator Algebras
Advances in Mathematics, 1997Co-Authors: Chongying Dong, Geoffrey MasonAbstract:Rational vertex Operator Algebras, which play a fundamental role in rational conformal field theory (see [BPZ] and [MS]), single out an important class of vertex Operator Algebras. Most vertex Operator Algebras which have been studied so far are rational vertex Operator Algebras. Familiar examples include the moonshine module V ♮ ([B], [FLM], [D2]), the vertex Operator Algebras VL associated with positive definite even lattices L ([B], [FLM], [D1]), the vertex Operator Algebras L(l, 0) associated with integrable representations of affine Lie Algebras [FZ] and the vertex Operator Algebras L(cp,q, 0) associated with irreducible highest weight representations for the discrete series of the Virasoro Algebra ([DMZ] and [W]). A rational vertex Operator Algebra as studied in this paper is a vertex Operator Algebra such that any admissible module is a direct sum of simple ordinary modules (see Section 2). It is natural to ask if such complete reducibility holds for an arbitrary weak module (defined in Section 2). A rational vertex Operator Algebra with this property is called a regular vertex Operator Algebra. One motivation for studying such vertex Operator Algebras arises in trying to understand the appearance of negative fusion rules (which are computed by the Verlinde formula) for vertex Operator Algebras L(l, 0) for certain rational l (cf. [KS] and [MW]). In this paper we give several sufficient conditions under which a rational vertex Operator Algebra is regular. We prove that the rational vertex Operator Algebras V , L(l, 0) for positive integers l, L(cp,q, 0) and VL for positive definite even lattices L are regular. Our result for L(l, 0) implies that any restricted integrable module of level l for the corresponding affine Lie Algebra is a direct sum of irreducible highest weight integrable modules. This result is expected to be useful in comparing the construction of tensor product of modules for L(l, 0) in [F] based on Kazhdan-Lusztig’s approach [KL] with the construction of tensor product of modules [HL] in this special case. We should remark that VL in general is a vertex Algebra in the sense of [DL] if L is not positive definite. In this case we establish the complete reducibility of any weak module. Since the definition of vertex Operator Algebra is by now well-known, we do not define vertex Operator Algebra in this paper. We refer the reader to [FLM] and [FHL] for their elementary properties. The reader can find the details of the constructions of V ♮ and VL in [FLM], and L(l, 0) and L(cp,q, 0) in [DMZ], [DL], [FLM], [FZ], [L1] and [W].
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simple currents and extensions of vertex Operator Algebras
Communications in Mathematical Physics, 1996Co-Authors: Chongying Dong, Geoffrey MasonAbstract:We consider how a vertex Operator Algebra can be extended to an abelian interwining Algebra by a family of weak twisted modules which aresimple currents associated with semisimple weight one primary vectors. In the case that the extension is again a vertex Operator Algebra, the rationality of the extended Algebra is discussed. These results are applied to affine Kac-Moody Algebras in order to construct all the simple currents explicitly (except forE 8) and to get various extensions of the vertex Operator Algebras associated with integrable representations.
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certain associative Algebras similar to u sl_ 2 and zhu s Algebra a v_ l
arXiv: Quantum Algebra, 1996Co-Authors: Chongying Dong, Geoffrey MasonAbstract:It is proved that Zhu's Algebra for vertex Operator Algebra associated to a positive-definite even lattice of rank one is a finite-dimensional semiprimitive quotient Algebra of certain associative Algebra introduced by Smith. Zhu's Algebra for vertex Operator Algebra associated to any positive-definite even lattice is also calculated and is related to a generalization of Smith's Algebra.
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Simple currents and extensions of vertex Operator Algebras
Communications in Mathematical Physics, 1996Co-Authors: Chongying Dong, Geoffrey MasonAbstract:We consider how a vertex Operator Algebra can be extended to an abelian intertwining Algebra by a family of weak twisted modules which are {\em simple currents} associated with semisimple weight one primary vectors. In the case that the extension is again a vertex Operator Algebra, the rationality of the extended Algebra is discussed. These results are applied to affine Kac-Moody Algebras in order to construct all the simple currents explicitly (except for $E_8$) and to get various extensions of the vertex Operator Algebras associated with integrable representations.
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Regularity of rational vertex Operator Algebras
arXiv: Quantum Algebra, 1995Co-Authors: Chongying Dong, Geoffrey MasonAbstract:A regular vertex Operator Algebra is a vertex Operator Algebra such that any weak module (without grading) is a direct sum of ordinary irreducible modules. In this paper we give several sufficient conditions under which a rational vertex Operator Algebra is regular. We prove that the moonshine module vertex Operator Algebra $V^{\natural},$ the vertex Operator Algebras $L(l,0)$ associated with the integrable representations of affine Algebras of level $l,$ the vertex Operator Algebras $L(c_{p,q},0)$ associated with irreducible highest weight representations for the discrete series of the Virasoro Algebra and the vertex Operator Algebras $V_L$ associated with positive definite even lattices $L$ are regular. Our result for $L(l,0)$ implies that any restricted integrable module of level $l$ for the corresponding affine Lie Algebra is a direct sum of irreducible highest weight integrable modules. The space $V_L$ in general is a vertex Algebra if $L$ is not positive definite. In this case we establish the complete reducibility of any weak module.