Strongly graded vertex algebras generated by vertex Lie algebras

2019 ◽  
Vol 21 (08) ◽  
pp. 1850069
Author(s):  
Yufeng Pei ◽  
Jinwei Yang
Keyword(s):  
Lie Algebras ◽  
Category Theory ◽  
Tensor Category ◽  
Vertex Algebra ◽  
Vertex Algebras ◽  
Intertwining Operators ◽  
Infinite Dimensional ◽  
Conformal Field ◽  
Convergence And Extension ◽  
Logarithmic Intertwining 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.

scholarly journals Differential equations and logarithmic intertwining operators for strongly graded vertex algebras

2017 ◽  
Vol 19 (02) ◽  
pp. 1650009 ◽  
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.

Vertex algebras associated to the affine Lie algebras of abelian polynomial current algebras

2016 ◽  
Vol 27 (05) ◽  
pp. 1650046 ◽  
Author(s):  
Jinwei Yang
Keyword(s):  
Differential Equations ◽  
Lie Algebras ◽  
Tensor Category ◽  
Extension Property ◽  
Vertex Algebras ◽  
Affine Lie Algebras ◽  
Systems Of Differential Equations ◽  
Convergence And Extension ◽  
Logarithmic Intertwining Operators ◽  
Current Algebras

We construct a family of vertex algebras associated to the affine Lie algebra of polynomial current algebras of finite-dimensional abelian Lie algebras, along with their modules and logarithmic modules. These vertex algebras and their (logarithmic) modules are strongly [Formula: see text]-graded and quasi-conformal. We then show that matrix elements of products and iterates of logarithmic intertwining operators among these logarithmic modules satisfy certain systems of differential equations. Using these systems of differential equations, we verify the convergence and extension property needed in the logarithmic tensor category theory developed by Huang, Lepowsky and Zhang.

scholarly journals A SMASH PRODUCT CONSTRUCTION OF NONLOCAL VERTEX ALGEBRAS

2007 ◽  
Vol 09 (05) ◽  
pp. 605-637 ◽  
Author(s):  
HAISHENG LI
Keyword(s):  
Lie Algebras ◽  
Suitable Condition ◽  
Vertex Algebra ◽  
Vertex Algebras ◽  
Smash Product ◽  
Module Algebra ◽  
Primitive Elements ◽  
Product Construction

A notion of vertex bialgebra and a notion of nonlocal vertex module-algebra for a vertex bialgebra are studied and then a smash product construction of nonlocal vertex algebras is presented. For every nonlocal vertex algebra V satisfying a suitable condition, a canonical bialgebra B(V) is constructed such that primitive elements of B(V) are essentially pseudo-derivations and group-like elements are essentially pseudo-endomorphisms. As an application, vertex algebras associated with the Heisenberg Lie algebras as well as those associated with the nondegenerate even lattices are reconstructed through smash products, and furthermore, a different approach to the construction of modules for the lattice vertex algebras is given.

scholarly journals Structure of the Extended Schrödinger–Virasoro Lie Algebra $\widetilde{\mathfrak{sv}}$

2009 ◽  
Vol 16 (04) ◽  
pp. 549-566 ◽  
Author(s):  
Shoulan Gao ◽  
Cuipo Jiang ◽  
Yufeng Pei
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Statistical Physics ◽  
Two Dimensional ◽  
Universal Central Extension ◽  
Central Extensions ◽  
Infinite Dimensional ◽  
Conformal Field

We study the derivations, the central extensions and the automorphism group of the extended Schrödinger–Virasoro Lie algebra [Formula: see text], introduced by Unterberger in the context of two-dimensional conformal field theory and statistical physics. Moreover, we show that [Formula: see text] is an infinite-dimensional complete Lie algebra, and the universal central extension of [Formula: see text] in the category of Leibniz algebras is the same as that in the category of Lie algebras.

scholarly journals VERTEX ALGEBRAS AND VERTEX POISSON ALGEBRAS

2004 ◽  
Vol 06 (01) ◽  
pp. 61-110 ◽  
Author(s):  
HAISHENG LI
Keyword(s):  
Vector Space ◽  
Lie Algebras ◽  
Poisson Algebra ◽  
Vertex Algebra ◽  
General Construction ◽  
Vertex Algebras ◽  
Poisson Algebras ◽  
Formal Deformation ◽  
Good Filtration

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 ℕ-graded vertex algebra V=∐n∈ℕV(n) with [Formula: see text], a canonical (good) filtration is associated and certain results about generating subspaces of certain types of V are also obtained. Furthermore, a notion of formal deformation of a vertex (Poisson) algebra is formulated and a formal deformation of vertex Poisson algebras associated with vertex Lie algebras is constructed.

INFINITE DIMENSIONAL LIE ALGEBRAS IN 4D CONFORMAL FIELD THEORY

2008 ◽  
Vol 05 (08) ◽  
pp. 1361-1371
Author(s):  
IVAN TODOROV
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Recent Work ◽  
Lie Algebras ◽  
Scalar Fields ◽  
Operator Product ◽  
Positive Energy ◽  
Infinite Dimensional ◽  
Conformal Field ◽  
Compact Gauge Group

It is known that there are no scalar Lie fields in more than two space-time dimensions [4]. Bilocal fields, however, which naturally arise in conformal operator product expansions, do generate infinite Lie algebras. Recent work, [2, 3], is reviewed, in which we classify such algebras and their unitary positive energy representations in a theory of a system of scalar fields of dimension two. The results are linked to the Doplicher–Haag–Roberts theory of superselection sectors governed by a (global) compact gauge group.

scholarly journals On complete reducibility for infinite-dimensional Lie algebras

2011 ◽  
Vol 226 (2) ◽  
pp. 1911-1972 ◽  
Author(s):  
Maria Gorelik ◽  
Victor Kac
Keyword(s):  
Lie Algebras ◽  
Infinite Dimensional ◽  
Complete Reducibility
Sign in / Sign up

Export Citation Format

Share Document