Set Λ=ZN.

We consider Λ=ZN-graded Lie algebras =⊕λ∈Λλ such that each homogenous component λ is one dimensional. Moreover, we assume that  is graded simple.

Around 1978, V.G. Kac conjectured the classification of such Lie algebras in the case N=1 (proved by the author in 1983). In the 90's, I.M. Gelfand and A. Kirillov raise the question of the classification of such Lie algebras in the case N≥2, but without providing an explicit list.

In this talk, we will explain how to solve Gelfand-Kirillov question. Although the setting of the question is very abstract, it turns out that the Lie algebras occuring in the classification are very concrete. Most of them are connected with the Lie algebras of symbols of twisted PDO on the circle. The remaining Lie algebras are classified by using Jordan algebras theory.

The proofs appear in two long papers, in Proc. London Math. Soc. and Math. Z.