The main algebraic tools in the theory of dynamical systems was notion of the crossed products of commutative algebras of functions on the phase space and group algebras of the acting groups, for example group Z. This approach arised to the early papers in 30-th by von Neumann, Gel’fand and their followers. But there is another possibilty which has appeared recently - to construct some Lie alegebra which is canoically corresponded to the measure preserving action of the group Z. This is Z-graded algebra and it has remarkable central extension. For the simpliest case - rotation of the circle this Lie algebra is generalization of Cartan-Kac-Moody Lie algebras but with infinite dimensional Cartan subalgebras. From other side it corresponds to the quantum torus. More interesting example of such class is Lie algebra generated by the shift on p-adic number. Representation theory for general type of those Lie algebras is not well developed upto now but there are representations with the highest wieghts for the examples above. I will tell about the construction of the central extension and consider a special examples.

Reference

1. A.Vershik. Lie algebrs generated by dynamical systems. Preprint ESI No 1076. 2001. (Aplicandae Math. 2002)

2. M.Saveliev, A.Vershik. Comm.Math.Phys.121(1989).

3. M.Kutuzova, D.Lebedev. Func. Anal. and its Appl. v.26, No 2, 1993.