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 \G by the definition $R(\omega):= f(\mathrm{ad} \omega)$, where \omega\in \G and $f$ is the holomorphic function given by $f(z)={1/2}\coth \frac{z}{2}-\frac{1}{z}$ for z\in \C\setminus 2\pi i \Z^*. We present a new, direct proof of the statement that this canonical $r$-matrix satisfies the modified classical dynamical Yang-Baxter equation on \G.Comment: 17 pages, LaTeX2

Conserved quantities in non-abelian monopole fields

Van Holten's covariant Hamiltonian framework is used to find conserved quantities for an isospin-carrying particle in a non-Abelian monopole-like field. For a Wu-Yang monopole we find the most general scalar potential such that the combined system admits a conserved Runge-Lenz vector. It generalizes the fine-tuned inverse-square plus Coulomb potential, found before by McIntosh and Cisneros, and by Zwanziger, for a charged particle in the field of a Dirac monopole. Following Feh\'er, the result is interpreted as describing motion in the asymptotic field of a self-dual Prasad-Sommerfield monopole. In the effective non-Abelian field for nuclear motion in a diatomic molecule due to Moody, Shapere and Wilczek, a conserved angular momentum is constructed, despite the non-conservation of the electric charge. No Runge-Lenz vector has been found.Comment: 8 pages, RevTex no figures. An error corrected and a new Section adde

Performance analysis and optimization of the JOREK code for many-core CPUs

This report investigates the performance of the JOREK code on the Intel Knights Landing and Skylake processor architectures. The OpenMP scaling of the matrix construction part of the code was analyzed and improved synchronization methods were implemented. A new switch was implemented to control the number of threads used for the linear equation solver independently from other parts of the code. The matrix construction subroutine was vectorized, and the data locality was also improved. These steps led to a factor of two speedup for the matrix construction

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 \L\subset \A to those on another pair \K\subset \A, where \K\subset \L\subset \A is a chain of Lie algebras for which \L admits a reductive decomposition as \L=\K+\M. 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.Comment: 19 pages, LaTeX, added a reference and a footnote and removed some typo

Quantization of a relativistic particle on the SL(2,R) manifold based on Hamiltonian reduction

A quantum theory is constructed for the system of a relativistic particle with mass m moving freely on the SL(2,R) group manifold. Applied to the cotangent bundle of SL(2,R), the method of Hamiltonian reduction allows us to split the reduced system into two coadjoint orbits of the group. We find that the Hilbert space consists of states given by the discrete series of the unitary irreducible representations of SL(2,R), and with a positive-definite, discrete spectrum.Comment: 12 pages, INS-Rep.-104

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

Classical and quantum Hamiltonian reductions of free geodesic systems of complete Riemannian manifolds are investigated. The reduced systems are described 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 meeting all orbits orthogonally in the configuration space. Hyperpolar actions on Lie groups and on symmetric spaces lead to families of integrable systems of spin Calogero-Sutherland type.Comment: 15 pages, minor correction and updated references in v

Extended matrix Gelfand-Dickey hierarchies: reduction to classical Lie algebras

The Drinfeld-Sokolov reduction method has been used to associate with $gl_n$ extensions of the matrix r-KdV system. Reductions of these systems to the fixed point sets of involutive Poisson maps, implementing reduction of $gl_n$ to classical Lie algebras of type $B, C, D$, are here presented. Modifications corresponding, in the first place to factorisation of the Lax operator, and then to Wakimoto realisations of the current algebra components of the factorisation, are also described.Comment: plain TeX, 12 page