B. G. Pusztai

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, G. 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 G with Lie algebra G, and its phase space is interpreted as a Hamiltonian reduction of an open submanifold of the cotangent bundle T∗G, using the symmetry arising from the adjoint action of G 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.

On dynamical r-matrices obtained from Dirac reduction and their generalizations to affine Lie algebras

According to Etingof and Varchenko, the classical dynamical Yang-Baxter equation is a guarantee for the consistency of the Poisson bracket on certain Poisson-Lie groupoids. Here it is noticed that Dirac reductions of these Poisson manifolds give rise to a mapping from dynamical r-matrices on a pair ⊂ to those on another pair ⊂, where ⊂⊂ is a chain of Lie algebras for which admits a reductive decomposition as = + . Several known dynamical r-matrices appear naturally in this setting, and its application provides new r-matrices, too. In particular, we exhibit a family of r-matrices for which the dynamical variable lies in the grade zero subalgebra of an extended affine Lie algebra obtained from a twisted loop algebra based on an arbitrary finite-dimensional self-dual Lie algebra.

The hyperbolic BCn Sutherland and the rational BCn Ruijsenaars–Schneider–van Diejen models: Lax matrices and duality

Abstract In this paper, we construct canonical action–angle variables for both the hyperbolic B C n Sutherland and the rational B C n Ruijsenaars–Schneider–van Diejen models with three independent coupling constants . As a byproduct of our symplectic reduction approach, we establish the action–angle duality between these many-particle systems. The presented dual reduction picture builds upon the construction of a Lax matrix for the B C n -type rational Ruijsenaars–Schneider–van Diejen model.

Spin Calogero models associated with Riemannian symmetric spaces of negative curvature

The Hamiltonian symmetry reduction of the geodesics system on a symmetric space of negative curvature by the maximal compact subgroup of the isometry group is investigated at an arbitrary value of the momentum map. Restricting to regular elements in the configuration space, the reduction generically yields a spin Calogero model with hyperbolic interaction potentials defined by the root system of the symmetric space. These models come equipped with Lax pairs and many constants of motion, and can be integrated by the projection method. The special values of the momentum map leading to spinless Calogero models are classified under some conditions, explaining why the BCn models with two independent coupling constants are associated with SU(n+1,n)/S(U(n+1)×U(n)) as found by Olshanetsky and Perelomov. In the zero curvature limit our models reproduce rational spin Calogero models studied previously and similar models correspond to other (affine) symmetric spaces, too. The construction works at the quantized level as well.

Action-angle duality between the Cn-type hyperbolic Sutherland and the rational Ruijsenaars–Schneider–van Diejen models

In a symplectic reduction framework we construct action-angle systems of canonical coordinates for both the hyperbolic Sutherland and the rational Ruijsenaars–Schneider–van Diejen integrable models associated with the Cn root system. The presented dual reduction picture permits us to establish the action-angle duality between these many-particle systems.

Hamiltonian reductions of free particles under polar actions of compact Lie groups

We investigate classical and quantum Hamiltonian reductions of free geodesic systems of complete Riemannian manifolds. We describe the reduced systems under the assumption that the underlying compact symmetry group acts in a polar manner in the sense that there exist regularly embedded, closed, connected submanifolds intersecting all orbits orthogonally in the configuration space. Hyperpolar actions on Lie groups and on symmetric spaces lead to families of integrable systems of the spin Calogero-Sutherland type.

Twisted spin Sutherland models from quantum Hamiltonian reduction

Recent general results on Hamiltonian reductions under polar group actions are applied to study some reductions of the free particle governed by the Laplace-Beltrami operator of a compact, connected, simple Lie group. The reduced systems associated with arbitrary finite-dimensional irreducible representations of the group by using the symmetry induced by twisted conjugations are described in detail. These systems generically yield integrable Sutherland-type many-body models with spin, which are called twisted spin Sutherland models if the underlying twisted conjugations are built on non-trivial Dynkin diagram automorphisms. The spectra of these models can be calculated, in principle, by solving certain Clebsch-Gordan problems, and the result is presented for the models associated with the symmetric tensorial powers of the defining representation of SU(N).

Generalizations of Felder's elliptic dynamical r-matrices associated with twisted loop algebras of self-dual Lie algebras

Abstract A dynamical r-matrix is associated with every self-dual Lie algebra A which is graded by finite-dimensional subspaces as A =⊕ n∈ Z A n , where A n is dual to A −n with respect to the invariant scalar product on A , and A 0 admits a nonempty open subset A 0 for which adκ is invertible on A n if n≠0 and κ∈ A 0 . Examples are furnished by taking A to be an affine Lie algebra obtained from the central extension of a twisted loop algebra l( G ,μ) of a finite-dimensional self-dual Lie algebra G . These r-matrices, R : A 0 → End ( A ) , yield generalizations of the basic trigonometric dynamical r-matrices that, according to Etingof and Varchenko, are associated with the Coxeter automorphisms of the simple Lie algebras, and are related to Felders elliptic r-matrices by evaluation homomorphisms of l( G ,μ) into G . The spectral-parameter-dependent dynamical r-matrix that corresponds analogously to an arbitrary scalar-product-preserving finite order automorphism of a self-dual Lie algebra is here calculated explicitly.

A note on a canonical dynamical r-matrix

It is well known that a classical dynamical r-matrix can be associated with every finite-dimensional self-dual Lie algebra by the definition R(?): = f(ad???), where ? and f is the holomorphic function given by f(z) = (1/2)coth (z/2)-1/z for z2?i*. We present a new, direct proof of the statement that this canonical r-matrix satisfies the modified classical dynamical Yang-Baxter equation on .

The non-dynamical r-matrices of the degenerate Calogero-Moser models

A complete description of the non-dynamical r-matrices of the degenerate Calogero-Moser models based on gln is presented. First, the most general momentum-independent r-matrices are given for the standard Lax representation of these systems and those r-matrices whose coordinate dependence can be gauged away are selected. Then the constant r-matrices resulting from the gauge transformation are determined and are related to well known r-matrices. In the hyperbolic/trigonometric case a non-dynamical r-matrix equivalent to a real/imaginary multiple of the Cremmer-Gervais classical r-matrix is found. In the rational case the constant r-matrix corresponds to the antisymmetric solution of the classical Yang-Baxter equation associated with the Frobenius subalgebra of gln consisting of the matrices with vanishing last row. These claims are consistent with previous results of Hasegawa and others, which imply that Belavins elliptic r-matrix and its degenerations appear in the Calogero-Moser models. The advantages of our analysis are that it is elementary and also clarifies the extent to which the constant r-matrix is unique in the degenerate cases.