Jet modules for the centerless Virasoro-like algebra

In this paper, we studied the jet modules for the centerless Virasoro-like algebra which is the Lie algebra of the Lie group of the area-preserving diffeomorphisms of a 2-torus. The jet modules are certain natural modules over the Lie algebra of semi-direct product of the centerless Virasoro-like algebra and the Laurent polynomial algebra in two variables. We reduce the irreducible jet modules to the finite-dimensional irreducible modules over some infinite-dimensional Lie algebra and then characterize the irreducible jet modules with irreducible finite dimensional modules over sl(2). To determine the indecomposable jet modules, we use the technique of polynomial modules in the sense of [Irreducible representations for toroidal Lie algebras, J. Algebras 221 (1999) 188-231; Weight modules over exp-polynomial Lie algebras, J. Pure Appl. Algebra 191 (2004) 23-42]. Consequently, indecomposable jet modules are described using modules over the algebra B+, which is the "positive part" of a Block type algebra studied first by [Some infinite-dimensional simple Lie algebras in characteristic 0 related to those of Block, J. Pure Appl. Algebra 127(2) (1998) 153-165] and recently by [A Z(N)-graded generalization of the Witt-algebra, preprint; Classification of simple Lie algebras on a lattice, Proc. London Math. Soc. 106(3) (2013) 508-5641).