4,401 research outputs found
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 -matrix can be associated with
every finite-dimensional self-dual Lie algebra \G by the definition
, where \omega\in \G and is the
holomorphic function given by for
z\in \C\setminus 2\pi i \Z^*. We present a new, direct proof of the statement
that this canonical -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
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
Generalizations of Felder's elliptic dynamical r-matrices associated with twisted loop algebras of self-dual Lie algebras
A dynamical -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 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 -matrices, R: \check
\A_0 \to \mathrm{End}(\A), yield generalizations of the basic trigonometric
dynamical -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 -matrices by evaluation homomorphisms of \ell(\G,\mu)
into \G. The spectral-parameter-dependent dynamical -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
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
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