Logarithmic Intertwining Operators and Vertex Operators

We construct three families of vertex algebras along with their modules from appropriate vertex Lie algebras, using the constructions in [Vertex Lie algebra, vertex Poisson algebras and vertex algebras, in Recent Developments in Infinite-Dimensional Lie Algebras and Conformal Field Theory[Formula: see text] Proceedings of an International Conference at University of Virginia[Formula: see text] May 2000, in Contemporary Mathematics, Vol. 297 (American Mathematical Society, 2002), pp. 69–96] by Dong, Li and Mason. These vertex algebras are strongly graded vertex algebras introduced in [Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, I: Introduction and strongly graded algebras and their generalized modules, in Conformal Field Theories and Tensor Categories[Formula: see text] Proceedings of a Workshop Held at Beijing International Center for Mathematics Research, eds. C. Bai, J. Fuchs, Y.-Z. Huang, L. Kong, I. Runkel and C. Schweigert, Mathematical Lectures from Beijing University, Vol. 2 (Springer, New York, 2014), pp. 169–248] by Huang, Lepowsky and Zhang in their logarithmic tensor category theory and can also be realized as vertex algebras associated to certain well-known infinite dimensional Lie algebras. We classify irreducible [Formula: see text]-gradable weak modules for these vertex algebras by determining their Zhu’s algebras. We find examples of strongly graded generalized modules for these vertex algebras that satisfy the [Formula: see text]-cofiniteness condition introduced in [Differential equations and logarithmic intertwining operators for strongly graded vertex algebra, Comm. Contemp. Math. 19(2) (2017) 1650009] by the second author. In particular, by a result of the second author [Differential equations and logarithmic intertwining operators for strongly graded vertex algebra, Comm. Contemp. Math. 19(2) (2017) 1650009, 26 pp.], the convergence and extension property for products and iterates of logarithmic intertwining operators in [Y.-Z. Huang, J. Lepowsky and L. Zhang, Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, VII: Convergence and extension properties and applications to expansion for intertwining maps, preprint (2011); arXiv:1110.1929 ] among such strongly graded generalized modules is verified.

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8-Vertex Correlation Functions and Twist Covariance of q-KZ Equation

Reviews in Mathematical Physics ◽

10.1142/s0129055x98000331 ◽

1998 ◽

Vol 10 (08) ◽

pp. 1027-1059

Author(s):

C. Frønsdal ◽

A. Galindo

Keyword(s):

Hopf Algebras ◽

Casimir Operator ◽

Intertwining Operators ◽

Vertex Operators ◽

Vertex Model ◽

Standard Quantum ◽

R Matrix ◽

Point Correlation ◽

Point Correlation Function ◽

Kz Equation

We study the vertex operators Φ(z) associated with standard quantum groups. The element Z=RR t is a "Casimir operator" for quantized Kac–Moody algebras and the quantum Knizhnik–Zamolodchikov (q-KZ) equation is interpreted as the statement :ZΦ(z):=Φ(z). We study the covariance of the q-KZ equation under twisting, first within the category of Hopf algebras, and then in the wider context of quasi Hopf algebras. We obtain the intertwining operators associated with the elliptic R-matrix and calculate the two-point correlation function for the eight-vertex model.

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Logarithmic intertwining operators and W(2,2p−1) algebras

Journal of Mathematical Physics ◽

10.1063/1.2747725 ◽

2007 ◽

Vol 48 (7) ◽

pp. 073503 ◽

Cited By ~ 34

Author(s):

Dražen Adamović ◽

Antun Milas

Keyword(s):

Intertwining Operators ◽

Logarithmic Intertwining Operators

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Differential equations and logarithmic intertwining operators for strongly graded vertex algebras

Communications in Contemporary Mathematics ◽

10.1142/s0219199716500097 ◽

2017 ◽

Vol 19 (02) ◽

pp. 1650009 ◽

Cited By ~ 2

Author(s):

Jinwei Yang

Keyword(s):

Differential Equations ◽

Category Theory ◽

Tensor Category ◽

Extension Property ◽

Vertex Algebra ◽

Intertwining Operators ◽

Matrix Elements ◽

Systems Of Differential Equations ◽

Convergence And Extension ◽

Logarithmic Intertwining Operators

We derive certain systems of differential equations for matrix elements of products and iterates of logarithmic intertwining operators among strongly graded generalized modules for a strongly graded vertex algebra under a certain finiteness condition and a condition related to the horizontal gradings. Using these systems of differential equations, we verify the convergence and extension property needed in the logarithmic tensor category theory for such strongly graded generalized modules developed by Huang, Lepowsky and Zhang.

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Logarithmic intertwining operators and genus-one correlation functions

Communications in Contemporary Mathematics ◽

10.1142/s0219199716500267 ◽

2016 ◽

Vol 18 (06) ◽

pp. 1650026 ◽

Cited By ~ 2

Author(s):

Francesco Fiordalisi

Keyword(s):

Differential Equations ◽

Correlation Functions ◽

Modular Group ◽

Vertex Operator Algebra ◽

Formal Series ◽

Positive Energy ◽

Intertwining Operators ◽

System Of Differential Equations ◽

Logarithmic Intertwining Operators ◽

Genus One

This is the first of two papers in which we study the modular invariance of pseudotraces of logarithmic intertwining operators. We construct and study genus-one correlation functions for logarithmic intertwining operators among generalized modules over a positive-energy and [Formula: see text]-cofinite vertex operator algebra [Formula: see text]. We consider grading-restricted generalized [Formula: see text]-modules which admit a right action of some associative algebra [Formula: see text], and intertwining operators among such modules which commute with the action of [Formula: see text] ([Formula: see text]-intertwining operators). We obtain duality properties, i.e. suitable associativity and commutativity properties, for [Formula: see text]-intertwining operators. Using pseudotraces introduced by Miyamoto and studied by Arike, we define formal [Formula: see text]-traces of products of [Formula: see text]-intertwining operators, and obtain certain identities for these formal series. This allows us to show that the formal [Formula: see text]-traces satisfy a system of differential equations with regular singular points, and therefore are absolutely convergent in a suitable region and can be extended to yield multivalued analytic functions, called genus-one correlation functions. Furthermore, we show that the space of solutions of these differential equations is invariant under the action of the modular group.

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Polynomial ring representations of endomorphisms of exterior powers

Collectanea mathematica ◽

10.1007/s13348-020-00310-5 ◽

2021 ◽

Author(s):

Ommolbanin Behzad ◽

André Contiero ◽

Letterio Gatto ◽

Renato Vidal Martins

Keyword(s):

Lie Algebra ◽

Polynomial Ring ◽

Vertex Operators ◽

Exterior Power ◽

Infinite Dimensional ◽

Symmetric Structure ◽

Rational Polynomials ◽

Exterior Powers ◽

Vertex Representation ◽

Countably Infinite

AbstractAn explicit description of the ring of the rational polynomials in r indeterminates as a representation of the Lie algebra of the endomorphisms of the k-th exterior power of a countably infinite-dimensional vector space is given. Our description is based on results by Laksov and Throup concerning the symmetric structure of the exterior power of a polynomial ring. Our results are based on approximate versions of the vertex operators occurring in the celebrated bosonic vertex representation, due to Date, Jimbo, Kashiwara and Miwa, of the Lie algebra of all matrices of infinite size, whose entries are all zero but finitely many.

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NONRELATIVISTIC HOLOGRAPHY — A GROUP-THEORETICAL PERSPECTIVE

International Journal of Modern Physics A ◽

10.1142/s0217751x14300014 ◽

2014 ◽

Vol 29 (03n04) ◽

pp. 1430001 ◽

Cited By ~ 10

Author(s):

V. K. DOBREV

Keyword(s):

Representation Theory ◽

Theoretical Perspective ◽

Explicit Construction ◽

Intertwining Operators ◽

Difference Operators ◽

Semisimple Lie Groups ◽

Boundary Operators ◽

Boundary Fields ◽

Theoretical Results ◽

Schrödinger Algebra

We give a review of some group-theoretical results related to nonrelativistic holography. Our main playgrounds are the Schrödinger equation and the Schrödinger algebra. We first recall the interpretation of nonrelativistic holography as equivalence between representations of the Schrödinger algebra describing bulk fields and boundary fields. One important result is the explicit construction of the boundary-to-bulk operators in the framework of representation theory, and that these operators and the bulk-to-boundary operators are intertwining operators. Further, we recall the fact that there is a hierarchy of equations on the boundary, invariant with respect to Schrödinger algebra. We also review the explicit construction of an analogous hierarchy of invariant equations in the bulk, and that the two hierarchies are equivalent via the bulk-to-boundary intertwining operators. The derivation of these hierarchies uses a mechanism introduced first for semisimple Lie groups and adapted to the nonsemisimple Schrödinger algebra. These require development of the representation theory of the Schrödinger algebra which is reviewed in some detail. We also recall the q-deformation of the Schrödinger algebra. Finally, the realization of the Schrödinger algebra via difference operators is reviewed.

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