Jet Modules.

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. 0. Introduction. In recent years there was a substantial progress in representation theory of infinitedimensional Lie algebras of rank n > 1, toroidal Lie algebras in particular. In this paper we turn our attention to another Lie algebra of rank n, the Lie algebra Wn of vector fields on an n-dimensional torus T. An important class of irreducible representations for Wn has its origin in differential geometry – these are the modules of tensor fields on a torus. In addition to being modules for the Lie algebra of vector fields, tensor fields also admit multiplication by functions. For the torus, which is a flat manifold, the spaces of tensor fields are free modules of a finite rank over the commutative algebra of functions F(T). We formalize this property in the definition of a category J of Wn-modules (cf., [R2]). We also discuss another class of Wn-modules of a geometric nature – the modules of jets of tensor fields [S]. Jets of functions are used as a tool for the symmetry analysis for partial differential equations [O]. The action on a space of jets of the Lie algebra of vector fields, known under the term “prolongation of vector fields”, plays a key role in that theory. From the algebraic point of view, jet modules are typically not irreducible, but are often indecomposable. The goal of the present paper is to classify indecomposable modules in category J . Let us state our result in case n = 1, for the sake of simplicity of notations. Theorem. There is a 1-1 correspondence between indecomposable W1-modules J in category J and pairs (λ, U), where λ ∈ C/Z and U is a finite-dimensional indecomposable module for a Lie algebra W 1 = Span 〈 z d dz ∣∣∣∣ m ≥ 1 〉 . Such a correspondence is given by the tensor product decomposition J = F(T)⊗ U, where W1 acts according to the formula ( 1 2πi e d dx )( e ⊗ u ) = (m+ λ)e ⊗ u+ ∑ b≥1 s b! e ⊗ ρ ( z d dz ) u. AMS subject classification: 17B66, 58A20.