Irreducible Modules over Finite Simple Lie Pseudoalgebras II. Primitive Pseudoalgebras of Type K

TL;DR

This paper advances the understanding of irreducible modules over finite simple Lie pseudoalgebras of type K, extending previous classifications to include modules related to the contact pseudo de Rham complex, a concept from contact geometry.

Contribution

It establishes a classification of finite irreducible modules over simple Lie pseudoalgebras of type K, analogous to prior results for types W and S, using the contact pseudo de Rham complex.

Findings

Classification of irreducible modules over type K pseudoalgebras

Extension of previous results to contact geometry context

Identification of modules as kernels of differentials in the contact pseudo de Rham complex

Abstract

One of the algebraic structures that has emerged recently in the study of the operator product expansions of chiral fields in conformal field theory is that of a Lie conformal algebra. A Lie pseudoalgebra is a generalization of the notion of a Lie conformal algebra for which C[\partial] is replaced by the universal enveloping algebra H of a finite-dimensional Lie algebra. The finite (i.e., finitely generated over H) simple Lie pseudoalgebras were classified in our previous work. The present paper is the second in our series on representation theory of simple Lie pseudoalgebras. In the first paper we showed that any finite irreducible module over a simple Lie pseudoalgebra of type W or S is either an irreducible tensor module or the kernel of the differential in a member of the pseudo de Rham complex. In the present paper we establish a similar result for Lie pseudoalgebras of type K, with the pseudo de Rham complex replaced by a certain reduction called the contact pseudo de Rham complex. This reduction in the context of contact geometry was discovered by Rumin.