Graded Lie Algebras and Dynamical Systems

We consider a class of infinite-dimensional Lie algebras which is associated to dynamical systems with invariant measures. There are two constructions of the algebras – one based on the associative cross product algebra which considered as Lie algebra and then extended with nontrivial scalar twococycle; the second description is the specification of the construction of the graded Lie algebras with continuum root system in spirit of the papers of Saveliev-Vershik [SV1, SV2, V] which is a generalization of the definition of classical Cartan finite-dimensional algebras as well as Kac–Moody algebras. In the last paragraph we present the third construction for the special case of dynamical systems with discrete spectrum. The first example of such algebras was so called sine-algebras which was discovered independently in [SV1] and [FFZ] and had been studied later in [GKL] from point of view Kac–Moody Lie algebras. In the last paragraph of this paper we also suggest a new examples of such type algebras appeared from arithmetics: adding of 1 in the additive group Zp as a transformation of the group of p-adic integers. The set of positive simple roots in this case is Zp; Cartan subalgebra is the algebra of continuous functions on the group Zp and Weyl group of this Lie algebra contains the infinite symmetric group. Remarkably this algebra is the inductive limit of Kac–Moody affine algebras of type Apn .