We survey a recent progress on algebraic quantum field theory in connection with subfactor theory. We mainly concentrate on one-dimensional conformal quantum field theory.
This paper proposes a mapping method simplifying the Reed-Muller expansion(“RM expansion”)of a ternary function under fixed polarities and the transformation of the RM expansion coefficients with different fixed pol...This paper proposes a mapping method simplifying the Reed-Muller expansion(“RM expansion”)of a ternary function under fixed polarities and the transformation of the RM expansion coefficients with different fixed polarities.展开更多
Ⅰ. INTRODUCTIONIn the study of two-valued and multivalued logic, the algebra systems used most extensively are the lattice algebra and modular algrbra. People pay attention to modular algebra because of the following...Ⅰ. INTRODUCTIONIn the study of two-valued and multivalued logic, the algebra systems used most extensively are the lattice algebra and modular algrbra. People pay attention to modular algebra because of the following points: (i) the objects and results of the two basic oper-展开更多
The purpose of this research is to give a dual description of conformal blocks of <i>d</i>=2 rational CFT (conformal field theory) in terms of Hecke eigenforms and eigensheaves. In particular, partition fu...The purpose of this research is to give a dual description of conformal blocks of <i>d</i>=2 rational CFT (conformal field theory) in terms of Hecke eigenforms and eigensheaves. In particular, partition functions, conformal characters and lattice theta functions may be reconstructed from the action of Hecke operators. This method can be applied to: 1) rings of integers of Galois number fields equipped with the trace (or anti-trace) form;2) root lattices of affine Kac-Moody algebras and WZW-models;3) minimal models of Belavin-Polyakov-Zamolodchikov and related <i>d</i>=2 spin-chain/lattice models;4) vertex algebras of Leech and Niemeier lattices and others. We also use the original Witten’s idea to construct the 3-dimensional quantum gravity as the AdS/CFT-dual of <i>c</i>=24 Monster vertex algebra of Frenkel-Lepowsky- Meurman. Concerning the geometric Langlands duality, we use results of Beilinson-Drinfeld, Frenkel-Ben-Zvi, Gukov-Kapustin-Witten and many others (<i>cf.</i> references). The main new result in this paper is the construction of number-theoretical lattice vertex superalgebras in Section 5 and applications to conformal field theories and quantum gravity.展开更多
Let F be the underlying base field of characteristic p 〉 3 and denote by M the even part of the finite-dimensional simple modular Lie superalgebra M. In this paper, the generator sets of the Lie algebra M which will ...Let F be the underlying base field of characteristic p 〉 3 and denote by M the even part of the finite-dimensional simple modular Lie superalgebra M. In this paper, the generator sets of the Lie algebra M which will be heavily used to consider the derivation algebra Der（M） are given. Furthermore, the derivation algebra of M is determined by reducing derivations and a torus of M, i.e., Der（m）=ad（m） spanF{∏l ad （ξr＋1ξl）} spanF{adxi,ad（xiξ^v）∏ad（ξr＋1ξl）}. As a result, the derivation algebra of the even part of M does not equal the even part of the derivation superalgebra of M.展开更多
In this article the ■-graded transitive modular Lie superalgebra ⊕_(i≥-1)L_i,whose repre- sentation of L_o in L_(-1)is isomorphic to the natural representation of osp(L_(-1)),is determined.
The embedding theorem is established for Z-graded transitive modular Lie superalgebras g■■■(g-1)satisfying the conditions: (i)g0■■(g-1) and g0-module g-1 is isomorphic to the natural■(g-1)-module; (ii)dim g1=2/3...The embedding theorem is established for Z-graded transitive modular Lie superalgebras g■■■(g-1)satisfying the conditions: (i)g0■■(g-1) and g0-module g-1 is isomorphic to the natural■(g-1)-module; (ii)dim g1=2/3n(2n^2+1),where n=1/2dim g-1. In particular,it is proved that the finite-dimensional simple modular Lie superalgebras satisfying the conditions above are isomorphic to the odd Hamiltonian superalgebras.The restricted Lie superalgebras are also considered.展开更多
Y. Z. Zhang and Q. C. Zhang [J. Algebra, 2009, 321： 3601-3619] constructed a new family of finite-dimensional modular Lie superalgebra Ω. Let Ω denote the even part of the Lie superalgebra Ω. We first give the gen...Y. Z. Zhang and Q. C. Zhang [J. Algebra, 2009, 321： 3601-3619] constructed a new family of finite-dimensional modular Lie superalgebra Ω. Let Ω denote the even part of the Lie superalgebra Ω. We first give the generator sets of the Lie algebra Ω. Then, we reduce the derivation of Ω to a certain form. With the reduced derivation and a torus of Ω, we determine the derivation algebra of Ω.展开更多
文摘We survey a recent progress on algebraic quantum field theory in connection with subfactor theory. We mainly concentrate on one-dimensional conformal quantum field theory.
文摘This paper proposes a mapping method simplifying the Reed-Muller expansion(“RM expansion”)of a ternary function under fixed polarities and the transformation of the RM expansion coefficients with different fixed polarities.
基金Project supported by the National Natural Science Foundation of China.
文摘Ⅰ. INTRODUCTIONIn the study of two-valued and multivalued logic, the algebra systems used most extensively are the lattice algebra and modular algrbra. People pay attention to modular algebra because of the following points: (i) the objects and results of the two basic oper-
文摘The purpose of this research is to give a dual description of conformal blocks of <i>d</i>=2 rational CFT (conformal field theory) in terms of Hecke eigenforms and eigensheaves. In particular, partition functions, conformal characters and lattice theta functions may be reconstructed from the action of Hecke operators. This method can be applied to: 1) rings of integers of Galois number fields equipped with the trace (or anti-trace) form;2) root lattices of affine Kac-Moody algebras and WZW-models;3) minimal models of Belavin-Polyakov-Zamolodchikov and related <i>d</i>=2 spin-chain/lattice models;4) vertex algebras of Leech and Niemeier lattices and others. We also use the original Witten’s idea to construct the 3-dimensional quantum gravity as the AdS/CFT-dual of <i>c</i>=24 Monster vertex algebra of Frenkel-Lepowsky- Meurman. Concerning the geometric Langlands duality, we use results of Beilinson-Drinfeld, Frenkel-Ben-Zvi, Gukov-Kapustin-Witten and many others (<i>cf.</i> references). The main new result in this paper is the construction of number-theoretical lattice vertex superalgebras in Section 5 and applications to conformal field theories and quantum gravity.
基金supported by the National Natural Science Foundation of China(Nos.11171055,11471090)the Fundamental Research Funds for the Central University(No.14ZZ2221)+3 种基金the Jilin Provincial Natural Science Foundation of China(No.201115006)the Heilongjiang Provincial Natural Science Foundation of China(No.A201210)the Technology Program of Education Department of Heilongjiang Province(No.12531763)the Program for Young Teachers Scientific Research in Qiqihar University(No.2012K-M32)
文摘Let F be the underlying base field of characteristic p 〉 3 and denote by M the even part of the finite-dimensional simple modular Lie superalgebra M. In this paper, the generator sets of the Lie algebra M which will be heavily used to consider the derivation algebra Der（M） are given. Furthermore, the derivation algebra of M is determined by reducing derivations and a torus of M, i.e., Der（m）=ad（m） spanF{∏l ad （ξr＋1ξl）} spanF{adxi,ad（xiξ^v）∏ad（ξr＋1ξl）}. As a result, the derivation algebra of the even part of M does not equal the even part of the derivation superalgebra of M.
基金Project supported by the NNSF (10271076)EMNSF (99036) of China
文摘In this article the ■-graded transitive modular Lie superalgebra ⊕_(i≥-1)L_i,whose repre- sentation of L_o in L_(-1)is isomorphic to the natural representation of osp(L_(-1)),is determined.
基金Acknowledgements Part of the work was done during the author＇s visit to SCMS （Shanghai Center for Mathematical Sciences）, and the author would like to thank for the hospitality. The author also thank the referees for their careful reading, helpful suggestions and comments. This work was supported by the National Natural Science Foundation of China （Grant No. 11301180）.
文摘We compute explicitly the modular derivations for Poisson-Ore extensions and tensor products of Poisson algebras.
基金This work is partially supported by the National Natural Science Foundation of China(Grant No.10671160)China Postdoctoral Science Foundation(Grant No.20060400107)
文摘The embedding theorem is established for Z-graded transitive modular Lie superalgebras g■■■(g-1)satisfying the conditions: (i)g0■■(g-1) and g0-module g-1 is isomorphic to the natural■(g-1)-module; (ii)dim g1=2/3n(2n^2+1),where n=1/2dim g-1. In particular,it is proved that the finite-dimensional simple modular Lie superalgebras satisfying the conditions above are isomorphic to the odd Hamiltonian superalgebras.The restricted Lie superalgebras are also considered.
文摘Y. Z. Zhang and Q. C. Zhang [J. Algebra, 2009, 321： 3601-3619] constructed a new family of finite-dimensional modular Lie superalgebra Ω. Let Ω denote the even part of the Lie superalgebra Ω. We first give the generator sets of the Lie algebra Ω. Then, we reduce the derivation of Ω to a certain form. With the reduced derivation and a torus of Ω, we determine the derivation algebra of Ω.
