1,699 research outputs found
Poisson-Lie dynamical r-matrices from Dirac reduction
The Dirac reduction technique used previously to obtain solutions of the
classical dynamical Yang-Baxter equation on the dual of a Lie algebra is
extended to the Poisson-Lie case and is shown to yield naturally certain
dynamical r-matrices on the duals of Poisson-Lie groups found by Etingof,
Enriquez and Marshall in math.QA/0403283.Comment: 10 pages, v2: minor stylistic changes, v3: corrected eq. (4.3
Bi-Hamiltonian structure of a dynamical system introduced by Braden and Hone
We investigate the finite dimensional dynamical system derived by Braden and
Hone in 1996 from the solitons of affine Toda field theory. This
system of evolution equations for an Hermitian matrix and a
real diagonal matrix with distinct eigenvalues was interpreted as a special
case of the spin Ruijsenaars--Schneider models due to Krichever and Zabrodin. A
decade later, L.-C. Li re-derived the model from a general framework built on
coboundary dynamical Poisson groupoids. This led to a Hamiltonian description
of the gauge invariant content of the model, where the gauge transformations
act as conjugations of by diagonal unitary matrices. Here, we point out
that the same dynamics can be interpreted also as a special case of the spin
Sutherland systems obtained by reducing the free geodesic motion on symmetric
spaces, studied by Pusztai and the author in 2006; the relevant symmetric space
being . This construction provides an
alternative Hamiltonian interpretation of the Braden--Hone dynamics. We prove
that the two Poisson brackets are compatible and yield a bi-Hamiltonian
description of the standard commuting flows of the model.Comment: 18 pages, references and some explanations added in v
On the Lagrangian Realization of the WZNW Reductions
We develop a phase space path-integral approach for deriving the Lagrangian
realization of the models defined by Hamiltonian reduction of the WZNW theory.
We illustrate the uses of the approach by applying it to the models of
non-Abelian chiral bosons, -algebras and the GKO coset construction, and
show that the well-known Sonnenschein's action, the generalized Toda action and
the gauged WZNW model are precisely the Lagrangian realizations of those
models, respectively.Comment: 15 pages, DIAS-STP-92-09/UdeM-LPN-TH-92-9
The Ruijsenaars self-duality map as a mapping class symplectomorphism
This is a brief review of the main results of our paper arXiv:1101.1759 that
contains a complete global treatment of the compactified trigonometric
Ruijsenaars-Schneider system by quasi-Hamiltonian reduction. Confirming
previous conjectures of Gorsky and collaborators, we have rigorously
established the interpretation of the system in terms of flat SU(n) connections
on the one-holed torus and demonstrated that its self-duality symplectomorphism
represents the natural action of the standard mapping class generator S on the
phase space. The pertinent quasi-Hamiltonian reduced phase space turned out to
be symplectomorphic to the complex projective space equipped with a multiple of
the Fubini-Study symplectic form and two toric moment maps playing the roles of
particle-positions and action-variables that are exchanged by the duality map.
Open problems and possible directions for future work are also discussed.Comment: Contribution to the proceedings of the workshop `Lie Theory and its
Applications in Physics IX' (Varna, June 2011), 13 page
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