603 research outputs found
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
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
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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
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