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

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

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

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

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

The Dirac equation in Taub-NUT space

Using chiral supersymmetry, we show that the massless Dirac equation in the Taub-NUT gravitational instanton field is exactly soluble and explain the arisal and the use of the dynamical (super) symmetry.Comment: An importatn misprint in a reference is corrected. Plain Tex. 8 page