Date: Friday 4 March 2011, 4:30pm
Location: Hempfield High School, Room 213
 Speaker: Zhaohu Nie, Penn State University
   Title: Invariant functions and spin Calogero-Moser systems

Abstract.  This is joint work with Luen-Chau Li. Calogero-Moser
systems originated from the many-body problem on a line and have gone
through various generalizations over the time. Our topic will be a type
of spin Calogero-Moser systems associated to complex simple Lie algebras
and dynamical r-matrices, introduced by L.-C. Li and P. Xu.
 
We will discuss the Liouville integrability of such systems, that is,
the existence of a maximal set of Poisson commuting functionally
independent integrals. These integrals are obtained from evaluation and
expansion of the invairant functions of the Lie algebra through the Lax
operator. We will concentrate on the functional independence of these
functions. It turns out to be a purely Lie algebraic question, where we
need to show the functional independence of some expansions of the
classical Chevalley invariants. This is done by computing the Jacobian
of these functions in some variables and at some subspace chosen after
B. Kostant. The Jacobian matrix is lower triangular for a choice of
order and the determinants of the diagonal blocks can be expressed in
terms of the roots of the Lie algebra.