Construction of vertex operator algebras from commutative associative algebras

Given a commutative associative algebra A with an associative form (’), we construct a vertex operator algebra V with the weight two space V₂;? A If in addition the form (’) is nondegenerate, we show that there is a simple vertex operator algebra with V₂;? A We also show that if A is semisimple, then the vertex operator algebra constructed is the tensor products of a certain number of Virasoro vertex operator algebras.     

扭的Heisenberg-Virasoro顶点算子代数的一个特征化（英文）

The twisted Heisenberg-Virasoro algebra is the universal central extension of the Lie algebra of differential operators on a circle of order at most one. In this paper, we first study the variety of semi-conformal vectors of the twisted Heisenberg-Virasoro vertex operator algebra, which is a finite set consisting of two nontrivial elements. Based on this property,we also show that the twisted Heisenberg-Virasoro vertex operator algebra is a tensor product of two vertex operator algebras. Moreover, associating to properties of semi-conformal vectors of the twisted Heisenberg-Virasoro vertex operator algebra, we charaterized twisted Heisenberg-Virasoro vertex operator algebras. This will be used to understand the classification problems of vertex operator algebras whose varieties of semi-conformal vectors are finite sets.     

A $$\mathbb{Z}_{2}$$-orbifold model of the symplectic fermionic vertex operator superalgebra

We give an example of an irrational C ₂-cofinite vertex operator algebra whose central charge is −2d for any positive integer d. This vertex operator algebra is given as the even part of the vertex operator superalgebra generated by d pairs of symplectic fermions, and it is just the realization of the c = −2-triplet algebra given by Kausch in the case d = 1. We also classify irreducible modules for this vertex operator algebra and determine its automorphism group. This research is supported in part by a grant from Japan Society for the Promotion of Science.     

The Notion of Vertex Operator Coalgebra and a Geometric Interpretation

The notion of vertex operator coalgebra is presented and motivated via the geometry of conformal field theory. Specifically, we describe the category of geometric vertex operator coalgebras, whose objects have comultiplicative structures meromorphically induced by conformal equivalence classes of worldsheets. We then show this category is isomorphic to the category of vertex operator coalgebras, which is defined in the language of formal algebra. The latter has several characteristics which give it the flavor of a coalgebra with respect to the structure of a vertex operator algebra and several characteristics that distinguish it from a standard dual—both of them will be highlighted.     

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 Quantized Tits-Kantor-Koecher Algebra

We propose a quantum analogue of a Tits-Kantor-Koecher algebra with a Jordan torus as an coordinated algebra by looking at the vertex operator construction over a Fock space.     

A Quantized Tits–Kantor–Koecher Algebra

We propose a quantum analogue of a Tits–Kantor–Koecher algebra with a Jordan torus as an coordinated algebra by looking at the vertex operator construction over a Fock space.     

On the concepts of intertwining operator and tensor product module in vertex operator algebra theory

We produce counterexamples to show that in the definition of the notion of intertwining operator for modules for a vertex operator algebra, the commutator formula cannot in general be used as a replacement axiom for the Jacobi identity. We further give a sufficient condition for the commutator formula to imply the Jacobi identity in this definition. Using these results we illuminate the crucial role of the condition called the “compatibility condition” in the construction of the tensor product module in vertex operator algebra theory, as carried out in work of Huang and Lepowsky. In particular, we prove by means of suitable counterexamples that the compatibility condition was indeed needed in this theory.     

A theory of tensor products for module categories for a vertex operator algebra, III

This is the third part in a series of papers developing a tensor product theory for modules for a vertex operator algebra. The goal of this theory is to construct a “vertex tensor category” structure on the category of modules for a suitable vertex operator algebra. The notion of vertex tensor category is essentially a “complex analogue” of the notion of symmetric tensor category, and in fact a vertex tensor category produces a braided tensor category in a natural way. In this paper, we focus on a particular element P(z) of a certain moduli space of three-punctured Riemann spheres; in general, every element of this moduli space will give rise to a notion of tensor product, and one must consider all these notions in order to construct a vertex tensor category. Here we present the fundamental properties of the P(z)-tensor product of two modules for a vertex operator algebra. We give two constructions of a P(z)-tensor product, using the results, established in Parts I and II of this series, for a certain other element of the moduli space. The definitions and results in Parts I and II are recalled.     

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.     

Logarithmic intertwining operators and associative algebras

We establish an isomorphism between the space of logarithmic intertwining operators among suitable generalized modules for a vertex operator algebra and the space of homomorphisms between suitable modules for a generalization of Zhu’s algebra given by Dong-Li-Mason.     

Generalized rationality and a 'Jacobi identity' for intertwining operator algebras

We prove a generalized rationality property and a new identity that we call the 'Jacobi identity' for intertwining operator algebras. Most of the main properties of genus-zero conformal field theories, including the main properties of vertex operator algebras, modules, intertwining operators, Verlinde algebras, and fusing and braiding matrices, are incorporated into this identity. Together with associativity and commutativity for intertwining operators proved by the author in [H4] and [H6], the results of the present paper solve completely the problem of finding a natural purely algebraic structure on the direct sum of all inequivalent irreducible modules for a suitable vertex operator algebra. Two equivalent definitions of intertwining operator algebra in terms of this Jacobi identity are given.     

有限维非退化可解李代数的顶点算子代数

构造相应于非退化可解李代数g的顶点算子代数分两步进行,首先构造顶点代数.本文是在已经得到的相应于非退化可解李代数g的顶点代数(Vg(l,0),Y(V,1)上构造顶点算子代数.定义了非退化可解李代数g的Casimir算子Ω,给出了在伴随表示下Ω作用在g上是0及相关性质,并应用Ω定义出Vg(l,0)中元素ω,证明了Vg(l,0)关于ω的顶点算子YV(ω,x)的系数构成一个Virasoro代数-模,还证明了ω满足顶点算子代数定义中Virasoro-向量的所有公理.从而证得(Vg(l,0),Yv,1,ω)是一个顶点算子代数.     

Vertex coalgebras, comodules, cocommutativity and coassociativity

We introduce the notion of vertex coalgebra, a generalization of vertex operator coalgebras. Next we investigate forms of cocommutativity, coassociativity, skew-symmetry, and an endomorphism, D^∗, which hold on vertex coalgebras. The former two properties require grading. We then discuss comodule structure. We conclude by discussing instances where graded vertex coalgebras appear, particularly as related to Primc’s vertex Lie algebra and (universal) enveloping vertex algebras.     

The varieties of Heisenberg vertex operator algebras

For a vertex operator algebra V with conformal vector ω,we consider a class of vertex operator subalgebras and their conformal vectors.They are called semi-conformal vertex operator subalgebras and semiconformal vectors of(V,ω),respectively,and were used to study duality theory of vertex operator algebras via coset constructions.Using these objects attached to(V,ω),we shall understand the structure of the vertex operator algebra(V,ω).At first,we define the set Sc(V,ω)of semi-conformal vectors of V, then we prove that Sc(V,ω)is an affine algebraic variety with a partial ordering and an involution map.Corresponding to each semi-conformal vector,there is a unique maximal semi-conformal vertex operator subalgebra containing it.The properties of these subalgebras are invariants of vertex operator algebras.As an example,we describe the corresponding varieties of semi-conformal vectors for Heisenberg vertex operator algebras.As an application,we give two characterizations of Heisenberg vertex operator algebras using the properties of these varieties.     

On voa associated with special jordan algebras

For a given simple Jordan algebra A of type A, B or C over C, we construct a vertex operator algebra V such that the weight two space V ₂ ? A by using the structure of Heisenberg algebras. In addition, we compute the automorphism groups of these vertex operator algebras.     

Quasi-particle fermionic formulas for (k, 3)-admissible configurations

We construct new monomial quasi-particle bases of Feigin-Stoyanovsky type subspaces for the affine Lie algebra sl(3;ℂ)∧ from which the known fermionic-type formulas for (k, 3)-admissible configurations follow naturally. In the proof we use vertex operator algebra relations for standard modules and coefficients of intertwining operators.     

A theory of tensor products for module categories for a vertex operator algebra,II

This is the second part in a series of papers presenting a theory of tensor products for module categories for a vertex operator algebra. In Part I, the notions ofP(z)- andQ(z)-tensor product of two modules for a vertex operator algebra were introduced and under a certain hypothesis, two constructions of aQ(z)-tensor product were given, using certain results stated without proof. In Part II, the proofs of those results are supplied.     

Commutative quantum operator algebras

A key notion bridging the gap between quantum operator algebras [26] and vertex operator algebras [4, 9] is the definition of the commutativity of a pair of quantum operators (see Section 2). This is not commutativity in any ordinary sense, but it is clearly the correct generalization to the quantum context. In [26] we give a definition of a commutative quantum operator algebra. We show in [26] that a vertex operator algebra gives rise to a special case of a CQOA. The main purpose of the current paper is to further develop the foundations for a complete mathematical theory of CQOAs. We give proofs of most of the relevant results announced in [26], and we carry out some calculations with sufficient detail to enable the interested reader to become proficient with the algebra of commuting quantum operators.     

On vertex operator realizations of Jack functions

On the vertex operator algebra associated with a rank one lattice we derive a general formula for products of vertex operators in terms of generalized homogeneous symmetric functions. As an application we realize Jack symmetric functions of rectangular shapes as well as marked rectangular shapes.