Generalized Drinfeld-Sokolov Hierarchies II: The Hamiltonian Structures

In this paper we examine the bi-Hamiltonian structure of the generalized KdV-hierarchies. We verify that both Hamiltonian structures take the form of Kirillov brackets on the Kac-Moody algebra, and that they define a coordinated system. Classical extended conformal algebras are obtained from the second Poisson bracket. In particular, we construct the $W_n^l$ algebras, first discussed for the case $n=3$ and $l=2$ by A. Polyakov and M. Bershadsky.Comment: 41 page

A note on the appearance of self-dual Yang-Mills fields in integrable hierarchies

A family of mappings from the solution spaces of certain generalized Drinfeld-Sokolov hierarchies to the self-dual Yang-Mills system on R^{2,2} is described. This provides an extension of the well-known relationship between self-dual connections and integrable hierarchies of AKNS and Drinfeld-Sokolov type

Field-dependent Mossbauer relaxation study of domain walls in the quasi 1-d antiferromagnet RbFeCl3.2H2O

Quantum Matter and Optic

In Situ X-ray Absorption Spectroscopy of LaFeO₃ and LaFeO₃/LaNiO₃ Thin Films in the Electrocatalytic Oxygen Evolution Reaction

We study the electrocatalytic oxygen evolution reaction using in situ X-ray absorption spectroscopy (XAS) to track the dynamics of the valence state and the covalence of the metal ions of LaFeO3 and LaFeO3/LaNiO3 thin films. The active materials are 8 unit cells grown epitaxially on 100 nm conductive La0.67Sr0.33MnO3 layers using pulsed laser deposition (PLD). The perovskite layers are supported on monolayer Ca2Nb3O10 nanosheet-buffered 100 nm SiNx membranes. The in situ Fe and Ni K-edges XAS spectra were measured from the backside of the SiNx membrane using fluorescence yield detection under electrocatalytic reaction conditions. The XAS spectra show significant spectral changes, which indicate that (1) the metal (co)valencies increase, and (2) the number of 3d electrons remains constant with applied potential. We find that the whole 8 unit cells react to the potential changes, including the buried LaNiO3 film.</p

Extensions of the matrix Gelfand-Dickey hierarchy from generalized Drinfeld-Sokolov reduction

The $p\times p$ matrix version of the $r$-KdV hierarchy has been recently treated as the reduced system arising in a Drinfeld-Sokolov type Hamiltonian symmetry reduction applied to a Poisson submanifold in the dual of the Lie algebra $\widehat{gl}_{pr}\otimes {\Complex}[\lambda, \lambda^{-1}]$. Here a series of extensions of this matrix Gelfand-Dickey system is derived by means of a generalized Drinfeld-Sokolov reduction defined for the Lie algebra $\widehat{gl}_{pr+s}\otimes {\Complex}[\lambda,\lambda^{-1}]$ using the natural embedding $gl_{pr}\subset gl_{pr+s}$ for $s$ any positive integer. The hierarchies obtained admit a description in terms of a $p\times p$ matrix pseudo-differential operator comprising an $r$-KdV type positive part and a non-trivial negative part. This system has been investigated previously in the $p=1$ case as a constrained KP system. In this paper the previous results are considerably extended and a systematic study is presented on the basis of the Drinfeld-Sokolov approach that has the advantage that it leads to local Poisson brackets and makes clear the conformal ($\cal W$-algebra) structures related to the KdV type hierarchies. Discrete reductions and modified versions of the extended $r$-KdV hierarchies are also discussed.Comment: 60 pages, plain TE

Susy Hierarchies and Affine Algebras

We review some basic features of the Lie-algebraic classification of W-algebras and related integrable hierarchies in 1+1 dimensions, pointing out the role of affine Lie algebras. We emphasize that the supersymmetric extensions of the above construction possibly lead, though some questions are still opened, to the classification of supersymmetric hierarchies based on ``generic'' supersymmetric affine Lie algebras. Here the word generic is used to make clear that well-known procedures, as those introduced by Inami and Kanno, are too restricted and do not lead to the full spectrum of supersymmetric integrable hierarchies one can construct. A particular attention is devoted to the large-N supersymmetric extensions (here N=4). The attention paid by large-N theories being due to the fact that they arise as dimensional reduction of N=1 models, and moreover that they realize an ``unification'' of known hierarchies.Comment: 11 pages, LaTex, uses lamuphys.sty: talk given at the UIC ``Supersymmetry and Integrable Systems Workshop'', Chicago, June 12-14 199