Differential Geometry of Singular Spaces and Reduction of Symmetry

Śniatycki, Jędrzej

doi:10.1017/cbo9781139136990

Cited by 73 publications

(150 citation statements)

References 50 publications

Supporting

Mentioning

130

Contrasting

Order By: Relevance

“…One should apply the theory of singular Hamiltonian reduction [21,27,29] to uncover the global structure of the reduced system that emerges from the geodesic motion on G C R . Finally, it is an open problem if there is any relation between the results of this paper and the previous works [1,7] devoted to the bi-Hamiltonian structure of the (spinless) rational Calogero-Moser system.…”

Section: Resultsmentioning

confidence: 99%

Bi-Hamiltonian structure of a dynamical system introduced by Braden and Hone

Fehér

2019

Nonlinearity

View full text Add to dashboard Cite

We investigate the finite dimensional dynamical system derived by Braden and Hone in 1996 from the solitons of A n−1 affine Toda field theory. This system of evolution equations for an n × n Hermitian matrix L and a real diagonal matrix q 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 L 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 GL(n, C)/U(n). 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.

show abstract

Section: Resultsmentioning

confidence: 99%

Bi-Hamiltonian structure of a dynamical system introduced by Braden and Hone

Fehér

2019

Nonlinearity

View full text Add to dashboard Cite

show abstract

“…that generates the 'conjugation action' of G on M. A dense open subset of the reduced phase space belonging to the zero value of µ can be identified with the (stratified) symplectic space (see [49,50])…”

Section: )mentioning

confidence: 99%

Poisson-Lie analogues of spin Sutherland models

Fehér

2019

Nuclear Physics B

View full text Add to dashboard Cite

We present generalizations of the well-known trigonometric spin Sutherland models, which were derived by Hamiltonian reduction of 'free motion' on cotangent bundles of compact simple Lie groups based on the conjugation action. Our models result by reducing the corresponding Heisenberg doubles with the aid of a Poisson-Lie analogue of the conjugation action. We describe the reduced symplectic structure and show that the 'reduced main Hamiltonians' reproduce the spin Sutherland model by keeping only Integrable systems of particles moving in one dimension have been studied intensively for nearly 50 years, beginning with the pioneering papers of Calogero [5], Sutherland [51] and Moser [35]. Thanks to their fascinating mathematics and diverse applications [10,37,38,44,52], the interest in these models shows no sign of diminishing. New connections to mathematics and new applications are still coming to light in the current literature, see e.g. [6,7,22,24,46,53].The richness of these models is also due to their many generalizations and deformations. These are associated with different interaction potentials (from rational to elliptic), root systems and extensions with internal degrees of freedom. We call 'Sutherland models' the systems defined by trigonometric or hyperbolic potentials. For all these systems, classical and quantum mechanical versions are studied separately, and one needs to pay attention to the distinct features of the systems with real particle positions and their complexifications. The investigations of Ruijsenaars-Schneider (RS) type deformations [44,45] is motivated, for example, by relations to solitons, spin chains, special functions and double affine Hecke algebras.The internal degrees of freedom are colloquially called 'spin', and can be of two rather different kinds. First, the point particles can carry spins varying in a vector space, as is the case for the Gibbons-Hermsen models [20] and their RS type generalizations introduced by Krichever and Zabrodin [28]. Second, the models can involve a collective spin variable that typically belongs to a coadjoint orbit, and is not assigned separately to the particles. An example of this second type is the trigonometric spin Sutherland model defined classically by a Hamiltonian of the following form:

show abstract

“…This ensures that subcartesian spaces do not require the additional assumption that their differential structures are C ∞ -rings. In particular, this justifies integration of derivations of differential structures of subcartesian spaces studied in [5].…”

Section: Introductionmentioning

confidence: 85%

“…Proof. See the proof of proposition 2.1.5 in [5] A differential space S is subcartesian if its topology is Hausdorff and every point x ∈ S has a neighbourhood U diffeomorphic to a subset V of R n . It should be noted that V in the definition above may be an arbitrary subset of R n , and n may depend on x ∈ S. As in the theory of manifolds, diffeomorphisms of open subsets of S onto subsets of R n are called charts on S. The family of all charts is the complete atlas on S. Aronszajn [1] used the notion of a complete atlas on a Hausdorff topological space in his definition of subcartesian space.…”

Section: Differential Spacesmentioning

confidence: 99%