Classification of simple Lie algebras on a lattice

Let Lambda=Z(n) for some n >= 1. The aim of the paper is to classify all simple Lambda-graded Lie algebras L=circle plus(lambda is an element of Lambda) L-lambda such that dim L-lambda=1 for all lambda. The classification involves two affine Lie algebras, namely A1(1) and A2(2), and a family (W-pi), parametrized by a dense open set of the space of all embeddings pi: Lambda ->(2). The family (W-l) of generalized Witt algebras, indexed by all embeddings l:Lambda ->, appears as a subfamily. In general, the algebras W-pi are described as Lie algebras of symbols of twisted pseudo-differential operators.