[Author]
 A. M. Vershik

[Title]
 Graded Lie algebras and dynamical systems

[AMS Subj-class]
 37A55 Relations with the theory of $C^*$-algebras
 58B99 Infinite-dimensional manifolds

[Abstract]
 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 two-cocycle; the second description
 is the specification of the construction of the graded Lie algebras
 with continuum root system 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. We also suggest
 a new examples of such type algebras appeared from arithmetics:
 adding of 1 in the additive group $Z_p$ as a transformation of the
 group of $p$-adic integers. The set of positive simple roots in this
 case is $Z_p$; Cartan subalgebra is the algebra of continuous
 functions on the group $Z_p$ 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 $A^1_{p^n}$.

[Comments]
 10 pp., LaTeX, English

[Contact e-mail]
 vershik@pdmi.ras.ru