In this work a large number of irreducible representations with finite dimensional weight spaces are constructed for some toroidal Lie algebras. To accomplish this we develop a general theory of ‫ޚ‬ n -graded Lie algebras with polynomial multiplication. We construct modules by the standard inducing procedure and study their irreducible quotients using the vertex operator technics.ᮊ 1999 Academic Press

We solve a long standing problem of the classification of all simple modules with finite-dimensional weight spaces over Lie algebra of vector fields on

Introduction.Toroidal Lie algebras are very natural multi-variable generalizations of affine Kac-Moody algebras. The theory of affine Lie algebras is rich and beautiful, having connections with diverse areas of mathematics and physics. Toroidal Lie algebras are also proving themselves to be useful for the applications. Frenkel, Jing and Wang [FJW] used representations of toroidal Lie algebras to construct a new form of the McKay correspondence. Inami et al., studied toroidal symmetry in the context of a 4-dimensional conformal field theory [IKUX], [IKU]. There are also applications of toroidal Lie algebras to soliton theory. Using representations of the toroidal algebras one can construct hierarchies of non-linear PDEs [B2], [ISW]. In particular, the toroidal extension of the Korteweg-de Vries hierarchy contains the Bogoyavlensky's equation, which is not in the classical KdV hierarchy [IT]. One can use the vertex operator realizations to construct n-soliton solutions for the PDEs in these hierarchies. We hope that further development of the representation theory of toroidal Lie algebras will help to find new applications of this interesting class of algebras.The construction of a toroidal Lie algebra is totally parallel to the well-known construction of an (untwisted) affine Kac-Moody algebra [K1]. One starts with a finite-dimensional simple Lie algebraġ and considers Fourier polynomial maps from an N + 1-dimensional torus intȯ g. Setting t k = e ix k , we may identify the algebra of Fourier polynomials on a torus with the Laurent polynomial algebra R = C[t Just as for the affine algebras, the next step is to build the universal central extension (R ⊗ġ) ⊕ K of R ⊗ġ. However unlike the affine case, the center K is infinite-dimensional when N ≥ 1. The infinite-dimensional center makes this Lie algebra highly degenerate. One can show, for example, that in an irreducible bounded weight module, most of the center should act trivially. To eliminate this degeneracy, we add the Lie algebra of vector fields on a torus, D = Der (R) to (R ⊗ġ) ⊕ K. The resulting algebra,is called the full toroidal Lie algebra (see Section 1 for details). The action of D on K is nontrivial, making the center of the toroidal Lie algebra g finite-dimensional. This enlarged algebra will have a much better representation theory. 1The most important class of modules for the affine Lie algebras are the highest weight modules, and one would certainly want to construct their toroidal analogs. The first problem that arises here is that one needs a triangular decomposition for the Lie algebra in order to introduce the notion of the highest weight module. Toroidal Lie algebras are graded by Z N+1 , and for N > 0, there is no canonical way of dividing this lattice into positive and negative parts. This difficulty is not present for the affine Lie algebras, which are graded by Z, and for Z such a splitting is natural.One way to split Z N+1 is to cut it with a hyperplane that intersects with the lattice only at zero. The corresponding class of the highe...

Abstract. In this paper we classify indecomposable modules for the Lie algebra of vector fields on a torus that admit a compatible action of the algebra of functions. An important family of such modules is given by spaces of jets of tensor fields.

Abstract. We describe the structure of the irreducible highest weight modules for the twisted Heisenberg-Virasoro Lie algebra at level zero. We prove that such a module is either isomorphic to a Verma module or to a quotient of two Verma modules.