[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