Differential equations in vertex algebras and simple modules for the Lie algebra of vector fields on a torus

We study irreducible representations for the Lie algebra of vector fields on a 2-dimensional torus constructed using the generalized Verma modules. We show that for a certain choice of parameters these representations remain irreducible when restricted to a loop subalgebra in the Lie algebra of vector fields. We prove this result by studying vertex algebras associated with the Lie algebra of vector fields on a torus and solving non-commutative differential equations that we derive using the vertex algebra technique.     

Nonlocal vertex algebras generated by formal vertex operators

This is the first paper in a series to study vertex algebra-like objects arising from infinite-dimensional quantum groups (quantum affine algebras and Yangians). In this paper we lay the foundations for this study. For any vector space W, we study what we call quasi compatible subsets of Hom (W,W((x))) and we prove that any maximal quasi compatible subspace has a natural nonlocal (namely noncommutative) vertex algebra structure with W as a natural faithful quasi module in a certain sense, and that any quasi compatible subset generates a nonlocal vertex algebra with W as a quasi module. In particular, taking W to be a highest weight module for a quantum affine algebra we obtain a nonlocal vertex algebra with W as a quasi module. We also formulate and study a notion of quantum vertex algebra and we give general constructions of nonlocal vertex algebras, quantum vertex algebras and their modules.     

Finite vertex algebras and nilpotence

I show that simple finite vertex algebras are commutative, and that the Lie conformal algebra structure underlying a reduced (= without nilpotent elements) finite vertex algebra is nilpotent.     

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.     

Twisted modules for quantum vertex algebras

We study twisted modules for (weak) quantum vertex algebras and we give a conceptual construction of (weak) quantum vertex algebras and their twisted modules. As an application we construct and classify irreducible twisted modules for a certain family of quantum vertex algebras.     

On vertex algebras and their modules associated with even lattices

We study vertex algebras and their modules associated with possibly degenerate even lattices, using an approach somewhat different from others. Several known results are recovered and a number of new results are obtained. We also study modules for Heisenberg algebras and we classify irreducible modules satisfying certain conditions and obtain a complete reducibility theorem.     

On replacement axioms for the Jacobi identity for vertex algebras and their modules

We discuss the axioms for vertex algebras and their modules, using formal calculus. Following certain standard treatments, we take the Jacobi identity as our main axiom and we recall weak commutativity and weak associativity. We derive a third companion property that we call “weak skew-associativity”. This third property in some sense completes an S₃-symmetry of the axioms, which is related to the known S₃-symmetry of the Jacobi identity. We do not initially require a vacuum vector, which is analogous to not requiring an identity element in ring theory. In this more general setting, one still has a property, occasionally used in standard treatments, which is closely related to skew-symmetry, which we call “vacuum-free skew-symmetry”. We show how certain combinations of these properties are equivalent to the Jacobi identity for both vacuum-free vertex algebras and their modules. We then specialize to the case with a vacuum vector and obtain further replacement axioms. In particular, in the final section we derive our main result, which says that, in the presence of certain minor axioms, the Jacobi identity for a module is equivalent to either weak associativity or weak skew-associativity. The first part of this result has appeared previously and has been used to show the (nontrivial) equivalence of representations of and modules for a vertex algebra. Many but not all of our results appear in standard treatments; some of our arguments are different from the usual ones.     

Representations of a class of lattice type vertex algebras

In this paper we study the representation theory for certain “half lattice vertex algebras.” In particular we construct a large class of irreducible modules for these vertex algebras. We also discuss how the representation theory of these vertex algebras are related to the representation theory of some associative algebras.     

Invariant forms of the elementary unitary Lie superalgebra

In this paper we describe the invariant forms of toral K-graded Lie superalgebras and, in particular, of the elementary unitary Lie superalgebra over a superring K containing .     

Representations of centrally-extended Lie algebras over differential operators and vertex algebras

We construct irreducible modules of centrally-extended classical Lie algebras over left ideals of the algebra of differential operators on the circle, through certain irreducible modules of centrally-extended classical Lie algebras of infinite matrices with finite number of nonzero entries. The structures of vertex algebras associated with the vacuum representations of these algebras are determined. Moreover, we prove that under certain conditions, the highest-weight irreducible modules of centrally-extended classical Lie algebras of infinite matrices with finite number of nonzero entries naturally give rise to the irreducible modules of the simple quotients of these vertex algebras. From vertex algebra and its representation point of view, our results with positive integral central charge are high-order differential operator analogues of the well-known WZW models in conformal field theory associated with affine Kac-Moody algebras. Indeed, when the left ideals are the algebra of differential operators, our Lie algebras do contain affine Kac-Moody algebras as subalgebras and our results restricted on them are exactly the representation contents in WZW models. Similar results with negative central charge are also obtained.     

The Z2-orbifold of the universal affine vertex algebra

Let $\mathfrak{g}$ be a simple, finite-dimensional complex Lie algebra, and let $V^k(\mathfrak{g})$ denote the universal affine vertex algebra associated to $\mathfrak{g}$ at level k. The Cartan involution on $\mathfrak{g}$ lifts to an involution on $V^k(\mathfrak{g})$, and we denote by $V^k{(\mathfrak{g})}^{{\mathbb{Z}}_2}$ the orbifold, or fixed-point subalgebra, under this involution. Our main result is an explicit minimal strong finite generating set for $V^k{(\mathfrak{g})}^{{\mathbb{Z}}_2}$ for generic values of k.     

An equivalence of two constructions of permutation-twisted modules for lattice vertex operator algebras

The problem of constructing twisted modules for a vertex operator algebra and an automorphism has been solved in particular in two contexts. One of these two constructions is that initiated by the third author in the case of a lattice vertex operator algebra and an automorphism arising from an arbitrary lattice isometry. This construction, from a physical point of view, is related to the space–time geometry associated with the lattice in the sense of string theory. The other construction is due to the first author, jointly with C. Dong and G. Mason, in the case of a multifold tensor product of a given vertex operator algebra with itself and a permutation automorphism of the tensor factors. The latter construction is based on a certain change of variables in the worldsheet geometry in the sense of string theory. In the case of a lattice that is the orthogonal direct sum of copies of a given lattice, these two very different constructions can both be carried out, and must produce isomorphic twisted modules, by a theorem of the first author jointly with Dong and Mason. In this paper, we explicitly construct an isomorphism, thereby providing, from both mathematical and physical points of view, a direct link between space–time geometry and worldsheet geometry in this setting.     

Higher level Zhu algebras and modules for vertex operator algebras

Motivated by the study of indecomposable, nonsimple modules for a vertex operator algebra V, we study the relationship between various types of V-modules and modules for the higher level Zhu algebras for V, denoted $A_n(V)$, for $n\in \mathbb{N}$, first introduced by Dong, Li, and Mason in 1998. We resolve some issues that arise in a few theorems previously presented when these algebras were first introduced, and give examples illustrating the need for certain modifications of the statements of those theorems. We establish that whether or not $A_{n?1}(V)$ is isomorphic to a direct summand of $A_n(V)$ affects the types of indecomposable V-modules which can be constructed by inducing from an $A_n(V)$-module, and in particular whether there are V-modules induced from $A_n(V)$-modules that were not already induced by $A_0(V)$. We give some characterizations of the V-modules that can be constructed from such inducings, in particular as regards their singular vectors. To illustrate these results, we discuss two examples of $A_1(V)$: when V is the vertex operator algebra associated to either the Heisenberg algebra or the Virasoro algebra. For these two examples, we show how the structure of $A_1(V)$ in relationship to $A_0(V)$ determines what types of indecomposable V-modules can be induced from a module for the level zero versus level one Zhu algebras. We construct a family of indecomposable modules for the Virasoro vertex operator algebra that are logarithmic modules and are not highest weight modules.     

Differential conformal superalgebras and their forms

We introduce the formalism of differential conformal superalgebras, which we show leads to the “correct” automorphism group functor and accompanying descent theory in the conformal setting. As an application, we classify forms of N=2 and N=4 conformal superalgebras by means of Galois cohomology.     

Symmetric invariant bilinear forms on vertex operator algebras

We give a criterion for determining the existence of nonzero symmetric invariant bilinear forms on vertex operator algebras and we establish an analogue of the Cartan criterion for semi-simplicity.     

The algebraic structure of relative twisted vertex operators

Twisted vertex operators based on rational lattices have had many applications in vertex operator algebra theory and conformal field theory. In this paper, “relativized” twisted vertex operators are constructed in a general context based on isometries of rational lattices, and a generalized twisted Jacobi identity is established for them. This result generalizes many previous results. Relatived untwisted vertex operators had been studied in a monograph by the authors. The present paper includes as a special case the proof of the main relations among twisted vertex operators based on even lattices announced some time ago by the second author.     

On the derived algebra of a centraliser

Let g be a classical Lie algebra, e∈g a nilpotent element and ge⊂g the centraliser of e. We prove that ge=[ge,ge] if and only if e is rigid. It is also shown that if e∈[ge,ge], then the nilpotent radical of ge coincides with [ge(1),ge], where ge(1)⊂ge is an eigenspace of a characteristic of e corresponding to the eigenvalue 1.     

Invariant symmetric bilinear forms for reflection groups

In this paper we describe a connection between Vinberg's criterion for the existence of an invariant symmetric bilinear form for a geometric representation of a Coxeter groups and other criteria which are formulated in terms of conjugation invariant sets of reflections generating a given group. Similar methods lead to the result that every non-symmetrizable Kac--Moody Lie algebra contains a non-symmetrizable subalgebra of rank 3. Finally we explain how the results for symmetric bilinear forms can also be obtained for skew-symmetric forms. Received 3 March 2000.     

Invariant polynomials on truncated multicurrent algebras

We construct invariant polynomials on truncated multicurrent algebras, which are Lie algebras of the form $\mathfrak{g}{?}_{\mathbb{F}}\mathbb{F}[t_1,\dots ,t_{?}]/I$, where $\mathfrak{g}$ is a finite-dimensional Lie algebra over a field $\mathbb{F}$ of characteristic zero, and I is a finite-codimensional ideal of $\mathbb{F}[t_1,\dots ,t_{?}]$ generated by monomials. In particular, when $\mathfrak{g}$ is semisimple and $\mathbb{F}$ is algebraically closed, we construct a set of algebraically independent generators for the algebra of invariant polynomials. In addition, we describe a transversal slice to the space of regular orbits in $\mathfrak{g}{?}_{\mathbb{F}}\mathbb{F}[t_1,\dots ,t_{?}]/I$. As an application of our main result, we show that the center of the universal enveloping algebra of $\mathfrak{g}{?}_{\mathbb{F}}\mathbb{F}[t_1,\dots ,t_{?}]/I$ acts trivially on all irreducible finite-dimensional representations provided I has codimension at least two.