Parafermion vertex operator algebras

This paper reviews some recent results on the parafermion vertex operator algebra associated to the integrable highest weight module L(k, 0) of positive integer level k for any affine Kac-Moody Lie algebra ĝ, where g is a finite dimensional simple Lie algebra. In particular, the generators and the C ₂-cofiniteness of the parafermion vertex operator algebras are discussed. A proof of the well-known fact that the parafermion vertex operator algebra can be realized as the commutant of a lattice vertex operator algebra in L(k, 0) is also given.     

A family of regular vertex operator algebras with two generators

For every m ∈ ℂ ∖ {0, −2} and every nonnegative integer k we define the vertex operator (super)algebra D _m,k having two generators and rank . If m is a positive integer then D _m,k can be realized as a subalgebra of a lattice vertex algebra. In this case, we prove that D _m,k is a regular vertex operator (super) algebra and find the number of inequivalent irreducible modules.      

Vertex operator algebras associated to modified regular representations of affine Lie algebras

Let G be a simply-connected complex Lie group with simple Lie algebra g and let be its affine Lie algebra. We use intertwining operators and Knizhnik-Zamolodchikov equations to construct a family of N-graded vertex operator algebras (VOAs) associated to g. These vertex operator algebras contain the algebra of regular functions on G as the conformal weight 0 subspaces and are -modules of dual levels in the sense that , where h^∨ is the dual Coxeter number of g. This family of VOAs was previously studied by Arkhipov-Gaitsgory and Gorbounov-Malikov-Schechtman from different points of view. We show that when k is irrational, the vertex envelope of the vertex algebroid associated to G and the level k is isomorphic to the vertex operator algebra we constructed above. The case of rational levels is also discussed.     

Bimodules associated to vertex operator superalgebras

Let V be a vertex operator superalgebra and m,n ∈ 21Z . We construct an An(V ) -Am(V )-bimodule An,m(V ) which characterizes the action of V from the level m subspace to level n subspace of an admissible V -module. We also construct the Verma type admissible V -module from an Am(V )-module by using bimodules     

Local conformal nets arising from framed vertex operator algebras

We apply an idea of framed vertex operator algebras to a construction of local conformal nets of (injective type III₁) factors on the circle corresponding to various lattice vertex operator algebras and their twisted orbifolds. In particular, we give a local conformal net corresponding to the moonshine vertex operator algebras of Frenkel-Lepowsky-Meurman. Its central charge is 24, it has a trivial representation theory in the sense that the vacuum sector is the only irreducible DHR sector, its vacuum character is the modular invariant J-function and its automorphism group (the gauge group) is the Monster group. We use our previous tools such as α-induction and complete rationality to study extensions of local conformal nets.     

Conformal designs based on vertex operator algebras

We introduce the notion of a conformal design based on a vertex operator algebra. This notation is a natural analog of the notion of block designs or spherical designs when the elements of the design are based on self-orthogonal binary codes or integral lattices, respectively. It is shown that the subspaces of an extremal self-dual vertex operator algebra of fixed degree form conformal 11-, 7-, or 3-designs, generalizing similar results of Assmus and Mattson and Venkov for extremal doubly-even codes and extremal even lattices. Other examples are coming from group actions on vertex operator algebras, the case studied first by Matsuo. The classification of conformal 6- and 8-designs is investigated. Again, our results are analogous to similar results for codes and lattices.     

Associating quantum vertex algebras to deformed Heisenberg Lie algebras

We associate quantum vertex algebras and their ϕ-coordinated quasi modules to certain deformed Heisenberg algebras.     

P-twisted affine Lie algebra and its realizations by twisted vertex operators

In this paper, we define a P-twisted affine Lie algebra, and construct its realizations by twisted vertex operators.     

Intertwining operator superalgebras and vertex tensor categories for superconformal algebras, II

We construct the intertwining operator superalgebras and vertex tensor categories for the superconformal unitary minimal models and other related models.

     

Regularity of vertex operator superalgebras

We prove that the regularity of a vertex operator superalgebra can be reduced to the semisimplicity of the category of its weak modules.Moreover,the rationality can be replaced by requiring that each 1/2Z+-graded module is a direct sum of irreducible weak modules.     

On twisted representations of vertex algebras

In this paper, we develop a formalism for working with representations of vertex and conformal algebras by generalized fields—formal power series involving non-integer powers of the variable. The main application of our technique is the construction of a large family of representations for the vertex superalgebra corresponding to an integer lattice Λ. For an automorphism coming from a finite-order automorphism we find the conditions for existence of twisted modules of . We show that the category of twisted representations of is semisimple with finitely many isomorphism classes of simple objects.     

G2型2-Toroidal李代数的顶点表示

在[4]和[5]中已经研究了sim p ly-laced型T oro idal李代数的顶点表示,[6]文据此给出了Bl型T oro idal李代数顶点表示的构造.受[6]文启发,本文给出了G2型T oro idal李代数的顶点表示的构造,这种构造方式与D(41)的D ynk in图的顶点粘合和一个2上循环有着紧密联系.     

Commutants of certain analytic operator algebras

We prove that algebraic commutants of maximal subdiagonal algebras and of analytic operator algebras determined by flows in a -finite von Neumann algebra are self-adjoint.

     

Local automorphisms of standard operator algebras

Let X be an infinite-dimensional separable real or complex Banach space and A a closed standard operator algebra on X. Then every local automorphism of A is an automorphism. The assumptions of infinite-dimensionality, separability, and closeness are all indispensable.