It it known that to get the usual Hamiltonian formulation of lattice Yang-Mills theory in the temporal gauge A 0 = 0 one should place on every link the cotangent bundle of a Lie group. The cotangent bundle may be considered as a limiting case of a so called Heisenberg double of a Lie group which is one of the basic objects in the theory of Lie-Poisson and quantum groups. It is shown in the pape...

This manuscript introduces various notions of k-locality of stabilizer codes inherited from the associated stabilizer groups. A choice of generators for the group leads to a Hamiltonian with the code in its groundspace, while a Hamiltonian holding the code in its groundspace might be called effective if its locality is less than that of a natural choice of generators (or any choice). This paper...

We show that a large class of physical theories which has been under intensive investigation recently, share the same geometric features in their Hamiltonian formulation. These dynamical systems range from harmonic oscillations to WZW-like models and to the KdV dynamics on DiffoS. To the same class belong also the Hamiltonian systems on groups of maps. The common feature of these models are the...

The main theorem of this paper asserts that the inclusion of the space of projective Lagrangian planes into the space of Lagrangian submanifolds of complex projective space induces an injective homomorphism of fundamental groups. We introduce three invariants of exact loops of Lagrangian submanifolds that are modelled on invariants introduced by Polterovich for loops of Hamiltonian symplectomor...

In the present article we show how to produce new examples of integrable dynamical systems of differential geometry origin. This is based on a construction of a canonical Hamiltonian structure for the geodesic flows of Carnot–Carathéodory metrics ([7, 17]) via the Pontryagin maximum principle. This Hamiltonian structure is achieved by introducing Lagrange multipliers bundles being the phase spa...

These are notes for a lecture course given in Jussieu, Paris, July 2001. They discuss groups of symplectomorphisms of closed symplectic manifolds (M, ω) from various points of view. The first two lectures provide an overview of their algebraic, geometric and homotopy theoretic properties. Much of the recent progress in understanding the geometry and topology of these groups has come from studyi...

Let M be a compact smooth manifold. Its cotangent bundle T ∗M carries a natural symplectic structure associated to a Liouville form θ = pdq. For a given compactly supported Hamiltonian function H : T ∗M → R and a closed submanifold N ⊂ M Oh [30, 27] defined a symplectic invariants of certain Lagrangian submanifolds in T ∗M in a following way. Let ν∗N ⊂ T ∗M be a conormal bundle of N . Denote by...

In this paper we establish for the first time the bi-Hamiltonian nature of the Toda lattices corresponding to classical simple Lie groups. These are systems that generalize the usual finite, non-periodic Toda lattice (which corresponds to a root system of type An). This generalization is due to Bogoyavlensky [2]. These systems were studied extensively in [10] where the solution of the system wa...