20 research outputs found
Integrable spin Calogero-Moser systems
We introduce spin Calogero-Moser systems associated with root systems of
simple Lie algebras and give the associated Lax representations (with spectral
parameter) and fundamental Poisson bracket relations. The associated integrable
models (called integrable spin Calogero-Moser systems in the paper) and their
Lax pairs are then obtained via Poisson reduction and gauge transformations.
For Lie algebras of -type, this new class of integrable systems includes
the usual Calogero-Moser systems as subsystems. Our method is guided by a
general framework which we develop here using dynamical Lie algebroids.Comment: 30 pages, Latex fil
Spin Calogero models obtained from dynamical r-matrices and geodesic motion
We study classical integrable systems based on the Alekseev-Meinrenken
dynamical r-matrices corresponding to automorphisms of self-dual Lie algebras,
. We prove that these r-matrices are uniquely characterized by a
non-degeneracy property and apply a construction due to Li and Xu to associate
spin Calogero type models with them. The equation of motion of any model of
this type is found to be a projection of the natural geodesic equation on a Lie
group with Lie algebra , and its phase space is interpreted as a
Hamiltonian reduction of an open submanifold of the cotangent bundle ,
using the symmetry arising from the adjoint action of twisted by the
underlying automorphism. This shows the integrability of the resulting systems
and gives an algorithm to solve them. As illustrative examples we present new
models built on the involutive diagram automorphisms of the real split and
compact simple Lie algebras, and also explain that many further examples fit in
the dynamical r-matrix framework.Comment: 25 pages, with minor stylistic changes and updated references in v
Lie Groupoids and Lie algebroids in physics and noncommutative geometry
The aim of this review paper is to explain the relevance of Lie groupoids and
Lie algebroids to both physicists and noncommutative geometers. Groupoids
generalize groups, spaces, group actions, and equivalence relations. This last
aspect dominates in noncommutative geometry, where groupoids provide the basic
tool to desingularize pathological quotient spaces. In physics, however, the
main role of groupoids is to provide a unified description of internal and
external symmetries. What is shared by noncommutative geometry and physics is
the importance of Connes's idea of associating a C*-algebra C*(G) to a Lie
groupoid G: in noncommutative geometry C*(G) replaces a given singular quotient
space by an appropriate noncommutative space, whereas in physics it gives the
algebra of observables of a quantum system whose symmetries are encoded by G.
Moreover, Connes's map G -> C*(G) has a classical analogue G -> A*(G) in
symplectic geometry due to Weinstein, which defines the Poisson manifold of the
corresponding classical system as the dual of the so-called Lie algebroid A(G)
of the Lie groupoid G, an object generalizing both Lie algebras and tangent
bundles. This will also lead into symplectic groupoids and the conjectural
functoriality of quantization.Comment: 39 pages; to appear in special issue of J. Geom. Phy
Representations of Lie Groups and Supergroups
The workshop focussed on recent developments in the representation theory of group objects in several categories, mostly finite and infinite dimensional smooth manifolds and supermanifolds. The talks covered a broad range of topics, with a certain emphasis on benchmark problems and examples such as branching, limit behavior, and dual pairs. In many talks the relation to physics played an important role