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

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

Interferometric view of the circumstellar envelopes of northern FU Orionis-type stars

FU Orionis-type objects are young, low-mass stars with large outbursts in visible light that last for several years or decades. They are thought to represent an evolutionary phase during the life of every young star when accretion from the circumstellar disk is enhanced during recurring time periods. These outbursts are able to rapidly build up the star while affecting the circumstellar disk and thus the ongoing or future planet formation. In many models infall from a circumstellar envelope seems to be necessary to trigger the outbursts. We observed the J=1$-$0 rotational transition of $^{13}$CO and C$^{18}$O towards eight northern FU Orionis-type stars (V1057 Cyg, V1515 Cyg, V2492 Cyg, V2493 Cyg, V1735 Cyg, V733 Cep, RNO 1B and RNO 1C) and derive temperatures and envelope masses and discuss the morphology and kinematics of the circumstellar material. We detected extended CO emission associated with all our targets. Smaller scale CO clumps were found to be associated with five objects with radii of 2000$-$5000 AU and masses of 0.02$-$0.5 $M_{\odot}$; these are clearly heated by the central stars. Three of these envelopes are also strongly detected in the 2.7 mm continuum. No central CO clumps were detected around V733 Cep and V710 Cas but there are many other clumps in their environments. Traces of outflow activity were observed towards V1735 Cyg, V733 Cep and V710 Cas. The diversity of the observed envelopes enables us to set up an evolutionary sequence between the objects. We find their evolutionary state to range from early, embedded Class I stage to late, Class II-type objects with very low-mass circumstellar material. The results reinforce the idea of FU Orionis-type stars as representatives of a transitory stage between embedded Class I young stellar objects and classical T-Tauri stars.Comment: 17 pages, 11 figures; accepted for publication in A&

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

On the Completeness of the Set of Classical W-Algebras Obtained from DS Reductions

We clarify the notion of the DS --- generalized Drinfeld-Sokolov --- reduction approach to classical ${\cal W}$-algebras. We first strengthen an earlier theorem which showed that an $sl(2)$ embedding ${\cal S}\subset {\cal G}$ can be associated to every DS reduction. We then use the fact that a \W-algebra must have a quasi-primary basis to derive severe restrictions on the possible reductions corresponding to a given $sl(2)$ embedding. In the known DS reductions found to date, for which the \W-algebras are denoted by ${\cal W}_{\cal S}^{\cal G}$-algebras and are called canonical, the quasi-primary basis corresponds to the highest weights of the $sl(2)$. Here we find some examples of noncanonical DS reductions leading to \W-algebras which are direct products of ${\cal W}_{\cal S}^{\cal G}$-algebras and `free field' algebras with conformal weights $\Delta \in \{0, {1\over 2}, 1\}$. We also show that if the conformal weights of the generators of a ${\cal W}$-algebra obtained from DS reduction are nonnegative $\Delta \geq 0$ (which isComment: 48 pages, plain TeX, BONN-HE-93-14, DIAS-STP-93-0

Isopin-dependent o(4,2) symmetry of self-dual Wu-Yang monopoles

A spinless particle in an SU(2) self-dual Wu-Yang monopole field is shown to admit an o(4,2) dynamical symmetry with isospin dependent generators found previously by Barut and Bornzin. This same symmetry arises for a spinless particle with anomalous charge studied by D’Hoker and Vinet, which we relate to a (spinless) ‘nucleon’ in the self-dual Wu-Yang monopole’s field

Rational vs Polynomial Character of Wnl_n^l-Algebras

The constraints proposed recently by Bershadsky to produce $W^l_n$ algebras are a mixture of first and second class constraints and are degenerate. We show that they admit a first-class subsystem from which they can be recovered by gauge-fixing, and that the non-degenerate constraints can be handled by previous methods. The degenerate constraints present a new situation in which the natural primary field basis for the gauge-invariants is rational rather than polynomial. We give an algorithm for constructing the rational basis and converting the base elements to polynomials.Comment: 18 page

Rational W W algebras from composite operators

Factoring out the spin $1$ subalgebra of a $W$ algebra leads to a new $W$ structure which can be seen either as a rational finitely generated $W$ algebra or as a polynomial non-linear $W_\infty$ realization.Comment: 11 pages, LATEX, preprint ENSLAPP-AL-429/93 and NORDITA-93/47-