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.     

Whittaker Modules for Graded Lie Algebras

In this paper, we study Whittaker modules for graded Lie algebras over ℂ. We define Whittaker modules for a class of graded Lie algebras and obtain a bijective correspondence between the set of isomorphism classes of Whittaker modules and the set of ideals of a polynomial ring, parallel to a result from the classical setting and the case of the Virasoro algebra. As a consequence of this, we obtain a classification of simple Whittaker modules for such algebras. Also, we discuss some concrete algebras as examples.     

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.     

Hom-Lie-Rinehart algebras

We introduce hom-Lie-Rinehart algebras as an algebraic analogue of hom-Lie algebroids, and systematically describe a cohomology complex by considering coefficient modules. We define the notion of extensions for hom-Lie-Rinehart algebras. In the sequel, we deduce a characterization of low dimensional cohomology spaces in terms of the group of automorphisms of certain abelian extension and the equivalence classes of those abelian extensions in the category of hom-Lie-Rinehart algebras, respectively. We also construct a canonical example of hom-Lie-Rinehart algebra associated to a given Poisson algebra and an automorphism.     

Symbolic powers of monomial ideals and vertex cover algebras

We introduce and study vertex cover algebras of weighted simplicial complexes. These algebras are special classes of symbolic Rees algebras. We show that symbolic Rees algebras of monomial ideals are finitely generated and that such an algebra is normal and Cohen-Macaulay if the monomial ideal is squarefree. For a simple graph, the vertex cover algebra is generated by elements of degree 2, and it is standard graded if and only if the graph is bipartite. We also give a general upper bound for the maximal degree of the generators of vertex cover algebras.     

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.     

Geometric structures encoded in the lie structure of an Atiyah algebroid

We investigate Atiyah algebroids, i.e. the infinitesimal objects of principal bundles, from the viewpoint of the Lie algebraic approach to space. First we show that if the Lie algebras of smooth sections of two Atiyah algebroids are isomorphic, then the corresponding base manifolds are necessarily diffeomorphic. Further, we give two characterizations of the isomorphisms of the Lie algebras of sections for Atiyah algebroids associated to principal bundles with semisimple structure groups. For instance we prove that in the semisimple case the Lie algebras of sections are isomorphic if and only if the corresponding Lie algebroids are, or, as well, if and only if the integrating principal bundles are locally isomorphic. Finally, we apply these results to describe the isomorphisms of sections in the case of reductive structure groups—surprisingly enough they are no longer determined by vector bundle isomorphisms and involve dive rgences on the base manifolds.     

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.     

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.     

MC elements in pronilpotent DG Lie algebras

Consider a pronilpotent DG (differential graded) Lie algebra over a field of characteristic 0. In the first part of the paper we introduce the reduced Deligne groupoid associated to this DG Lie algebra. We prove that a DG Lie quasi-isomorphism between two such algebras induces an equivalence between the corresponding reduced Deligne groupoids. This extends the famous result of Goldman–Millson (attributed to Deligne) to the unbounded pronilpotent case.In the second part of the paper we consider the Deligne 2-groupoid. We show it exists under more relaxed assumptions than known before (the DG Lie algebra is either nilpotent or of quasi quantum type). We prove that a DG Lie quasi-isomorphism between such DG Lie algebras induces a weak equivalence between the corresponding Deligne 2-groupoids.In the third part of the paper we prove that an L-infinity quasi-isomorphism between pronilpotent DG Lie algebras induces a bijection between the sets of gauge equivalence classes of Maurer–Cartan elements. This extends a result of Kontsevich and others to the pronilpotent case.     

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.     

Existence of Gradings on Associative Algebras

In this article, we study the existence of gradings on finite dimensional associative algebras. We prove that a connected algebra A does not have a nontrivial grading if and only if A is basic, its quiver has one vertex, and its group of outer automorphisms is unipotent. We apply this result to prove that up to graded Morita equivalence there do not exist nontrivial gradings on the blocks of group algebras with quaternion defect groups and one isomorphism class of simple modules.     

A remark on simplicity of vertex algebras and Lie conformal algebras

I give a short proof of the following algebraic statement: in a simple vertex algebra, the underlying Lie conformal algebra is either abelian, or it is an irreducible central extension of a simple Lie conformal algebra. This provides many examples of non-finite simple Lie conformal algebras, and should prove useful for classification purposes.     

Representations of nongraded Lie algebras of block type

From his classification of quadratic conformal algebras corresponding to certain Hamiltonian pairs in integrable systems, Xu found a family of simple Lie algebras related to pairs of locally-finite derivations on certain commutative associative algebras. In this paper, we construct a large family of irreducible modules with four parameters for Xu's two-devivation algebras via the corresponding algebras of Weyl type. When the derivations are graded operators, we obtain a large family of uniformly-bounded irreducible weight modules for the Block algebras.     

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.     

Metric Lie algebras with maximal isotropic centre

A metric Lie algebra is a Lie algebra equipped with an invariant non-degenerate symmetric bilinear form. It is called indecomposable if it is not the direct sum of two metric Lie algebras. We are interested in describing the isomorphism classes of indecomposable metric Lie algebras. In the present paper we restrict ourselves to a certain class of solvable metric Lie algebras which includes all indecomposable metric Lie algebras with maximal isotropic centre. We will see that each metric Lie algebra belonging to this class is a twofold extension associated with an orthogonal representation of an abelian Lie algebra. We will describe equivalence classes of such extensions by a certain cohomology set. In particular we obtain a classification scheme for indecomposable metric Lie algebras with maximal isotropic centre and the classification of metric Lie algebras of index 2.     

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.     

Some General Results on Conformal Embeddings of Affine Vertex Operator Algebras

We give a general criterion for conformal embeddings of vertex operator algebras associated to affine Lie algebras at arbitrary levels. Using that criterion, we construct new conformal embeddings at admissible rational and negative integer levels. In particular, we construct all remaining conformal embeddings associated to automorphisms of Dynkin diagrams of simple Lie algebras. The semisimplicity of the corresponding decompositions is obtained by using the concept of fusion rules for vertex operator algebras.