The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform
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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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.
Karapet Mkrtchyan - One of the best experts on this subject based on the ideXlab platform.
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notes on higher spin Algebras minimal representations and structure constants
Journal of High Energy Physics, 2014Co-Authors: Euihun Joung, Karapet MkrtchyanAbstract:The higher-spin (HS) Algebras relevant to Vasiliev’s equations in various dimensions can be interpreted as the symmetries of the minimal representation of the isometry Algebra. After discussing this connection briefly, we generalize this concept to any classical Lie Algebra and consider the corresponding HS Algebras. For $ \mathfrak{s}{{\mathfrak{p}}_{2N }} $ and $ \mathfrak{s}{{\mathfrak{o}}_N} $ , the minimal representations are unique so we get unique HS Algebras. For $ \mathfrak{s}{{\mathfrak{l}}_N} $ , the minimal representation has one-parameter family, so does the corresponding HS Algebra. The $ \mathfrak{s}{{\mathfrak{o}}_N} $ HS Algebra is what underlies the Vasiliev theory while the $ \mathfrak{s}{{\mathfrak{l}}_2} $ one coincides with the 3D HS Algebra hs[λ]. Finally, we derive the explicit expression of the structure constant of these Algebras — more precisely, their bilinear and trilinear forms. Several consistency checks are carried out for our results.
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notes on higher spin Algebras minimal representations and structure constants
arXiv: High Energy Physics - Theory, 2014Co-Authors: Euihun Joung, Karapet MkrtchyanAbstract:The higher-spin (HS) Algebras so far known can be interpreted as the symmetries of the minimal representation of the isometry Algebra. After discussing this connection briefly, we generalize this concept to any classical Lie Algebras and consider the corresponding HS Algebras. For sp(2N) and so(N), the minimal representations are unique so we get unique HS Algebras. For sl(N), the minimal representation has one-parameter family, so does the corresponding HS Algebra. The so(N) HS Algebra is what underlies the Vasiliev theory while the sl(2) one coincides with the 3D HS Algebra hs[lambda]. Finally, we derive the explicit expression of the structure constant of these Algebras --- more precisely, their bilinear and trilinear forms. Several consistency checks are carried out for our results.
Chongying Dong - One of the best experts on this subject based on the ideXlab platform.
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quantum dimensions and quantum galois theory
Transactions of the American Mathematical Society, 2013Co-Authors: Chongying Dong, Xiangyu JiaoAbstract:The quantum dimensions of modules for vertex operator Algebras are defined and their properties are discussed systematically. The quantum dimensions of the Heisenberg vertex operator Algebra modules, the Virasoro vertex operator Algebra modules and the lattice vertex operator Algebra modules are computed. A criterion for simple current modules of a rational vertex operator Algebra is given. The possible values of the quantum dimensions are obtained for rational vertex operator Algebras. A full Galois theory for rational vertex operator Algebras is established using the quantum dimensions.
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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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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.
Euihun Joung - One of the best experts on this subject based on the ideXlab platform.
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notes on higher spin Algebras minimal representations and structure constants
Journal of High Energy Physics, 2014Co-Authors: Euihun Joung, Karapet MkrtchyanAbstract:The higher-spin (HS) Algebras relevant to Vasiliev’s equations in various dimensions can be interpreted as the symmetries of the minimal representation of the isometry Algebra. After discussing this connection briefly, we generalize this concept to any classical Lie Algebra and consider the corresponding HS Algebras. For $ \mathfrak{s}{{\mathfrak{p}}_{2N }} $ and $ \mathfrak{s}{{\mathfrak{o}}_N} $ , the minimal representations are unique so we get unique HS Algebras. For $ \mathfrak{s}{{\mathfrak{l}}_N} $ , the minimal representation has one-parameter family, so does the corresponding HS Algebra. The $ \mathfrak{s}{{\mathfrak{o}}_N} $ HS Algebra is what underlies the Vasiliev theory while the $ \mathfrak{s}{{\mathfrak{l}}_2} $ one coincides with the 3D HS Algebra hs[λ]. Finally, we derive the explicit expression of the structure constant of these Algebras — more precisely, their bilinear and trilinear forms. Several consistency checks are carried out for our results.
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notes on higher spin Algebras minimal representations and structure constants
arXiv: High Energy Physics - Theory, 2014Co-Authors: Euihun Joung, Karapet MkrtchyanAbstract:The higher-spin (HS) Algebras so far known can be interpreted as the symmetries of the minimal representation of the isometry Algebra. After discussing this connection briefly, we generalize this concept to any classical Lie Algebras and consider the corresponding HS Algebras. For sp(2N) and so(N), the minimal representations are unique so we get unique HS Algebras. For sl(N), the minimal representation has one-parameter family, so does the corresponding HS Algebra. The so(N) HS Algebra is what underlies the Vasiliev theory while the sl(2) one coincides with the 3D HS Algebra hs[lambda]. Finally, we derive the explicit expression of the structure constant of these Algebras --- more precisely, their bilinear and trilinear forms. Several consistency checks are carried out for our results.
Alexander Zimmermann - One of the best experts on this subject based on the ideXlab platform.
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the hochschild cohomology ring of a frobenius Algebra with semisimple nakayama automorphism is a batalin vilkovisky Algebra
Journal of Algebra, 2016Co-Authors: Thierry Lambre, Guodong Zhou, Alexander ZimmermannAbstract:Abstract In analogy with a recent result of N. Kowalzig and U. Krahmer for twisted Calabi–Yau Algebras, we show that the Hochschild cohomology ring of a Frobenius Algebra with semisimple Nakayama automorphism is a Batalin–Vilkovisky Algebra, thus generalizing a result of T. Tradler for finite dimensional symmetric Algebras. We give a criterion to determine when a Frobenius Algebra given by quiver with relations has semisimple Nakayama automorphism and apply it to some known classes of tame Frobenius Algebras. We also provide ample examples including quantum complete intersections, finite dimensional Hopf Algebras defined over an Algebraically closed field of characteristic zero and the Koszul duals of Koszul Artin–Schelter regular Algebras of dimension three.
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the hochschild cohomology ring of a frobenius Algebra with semisimple nakayama automorphism is a batalin vilkovisky Algebra
arXiv: K-Theory and Homology, 2014Co-Authors: Thierry Lambre, Guodong Zhou, Alexander ZimmermannAbstract:Analogous to a recent result of N. Kowalzig and U. Kr\"{a}hmer for twisted Calabi-Yau Algebras, we show that the Hochschild cohomology ring of a Frobenius Algebra with semisimple Nakayama automorphism is a Batalin-Vilkovisky Algebra, thus generalizing a result of T.Tradler for finite dimensional symmetric Algebras. We give a criterion to determine when a Frobenius Algebra given by quiver with relations has semisimple Nakayama automorphism and apply it to some known classes of tame Frobenius Algebras. We also provide ample examples including quantum complete intersections, finite dimensional Hopf Algebras defined over an Algebraically closed field of characteristic zero and Koszul duals of Koszul Artin-Schelter regular Algebras of dimension three.
