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

On the self-adjointness of certain reduced Laplace-Beltrami operators

The self-adjointness of the reduced Hamiltonian operators arising from the Laplace-Beltrami operator of a complete Riemannian manifold through quantum Hamiltonian reduction based on a compact isometry group is studied. A simple sufficient condition is provided that guarantees the inheritance of essential self-adjointness onto a certain class of restricted operators and allows us to conclude the self-adjointness of the reduced Laplace-Beltrami operators in a concise way. As a consequence, the self-adjointness of spin Calogero-Sutherland type reductions of `free' Hamiltonians under polar actions of compact Lie groups follows immediately.Comment: 9 pages, minor changes, updated references in v

Thom series of contact singularities

Thom polynomials measure how global topology forces singularities. The power of Thom polynomials predestine them to be a useful tool not only in differential topology, but also in algebraic geometry (enumerative geometry, moduli spaces) and algebraic combinatorics. The main obstacle of their widespread application is that only a few, sporadic Thom polynomials have been known explicitly. In this paper we develop a general method for calculating Thom polynomials of contact singularities. Along the way, relations with the equivariant geometry of (punctual, local) Hilbert schemes, and with iterated residue identities are revealed

On the spectra of the quantized action-variables of the compactified Ruijsenaars-Schneider system

A simple derivation of the spectra of the action-variables of the quantized compactified Ruijsenaars-Schneider system is presented. The spectra are obtained by combining Kahler quantization with the identification of the classical action-variables as a standard toric moment map on the complex projective space. The result is consistent with the Schrodinger quantization of the system worked out previously by van Diejen and Vinet.Comment: Based on talk at the workshop CQIS-2011 (Protvino, Russia, January 2011), 12 page

Adler-Kostant-Symes systems as Lagrangian gauge theories

It is well known that the integrable Hamiltonian systems defined by the Adler-Kostant-Symes construction correspond via Hamiltonian reduction to systems on cotangent bundles of Lie groups. Generalizing previous results on Toda systems, here a Lagrangian version of the reduction procedure is exhibited for those cases for which the underlying Lie algebra admits an invariant scalar product. This is achieved by constructing a Lagrangian with gauge symmetry in such a way that, by means of the Dirac algorithm, this Lagrangian reproduces the Adler-Kostant-Symes system whose Hamiltonian is the quadratic form associated with the scalar product on the Lie algebra.Comment: 10 pages, LaTeX2

W-Algebras of Generalized Toda Theories

The Hamiltonian reduction of Wess-Zumino-Novikov-Witten (WZNW) theories to conformally invariant Toda theories is reviewed. The relationship between the WZNW and the Lax pair approaches to Toda theories is clan fled. Extended conformal algebras associated to arbitrary embeddings of sl(2) into the simple Lie algebras, and generalized Toda theories possessing these W-algebras as their canonical symmetries are considered by using the WZNW framework

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

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

A dynamical $r$-matrix is associated with every self-dual Lie algebra \A which is graded by finite-dimensional subspaces as \A=\oplus_{n \in \cZ} \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 \check \A_0 for which \ad \kappa is invertible on \A_n if $n\neq 0$ and \kappa \in \check \A_0. Examples are furnished by taking \A to be an affine Lie algebra obtained from the central extension of a twisted loop algebra \ell(\G,\mu) of a finite-dimensional self-dual Lie algebra \G. These $r$-matrices, R: \check \A_0 \to \mathrm{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 Felder's elliptic $r$-matrices by evaluation homomorphisms of \ell(\G,\mu) 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.Comment: LaTeX2e, 22 pages. Added a reference and a remar