On the structure of graded transitive Lie algebras

We study finite-dimensional Lie algebras ${\mathfrak L}$ of polynomial vector fields in $n$ variables that contain the vector fields $\dfrac{\partial}{\partial x_i} \; (i=1,\ldots, n)$ and $x_1\dfrac{\partial}{\partial x_1}+ \dots + x_n\dfrac{\partial}{\partial x_n}$. We show that the maximal ones always contain a semi-simple subalgebra $\bar{{\mathfrak g}}$, such that $\dfrac{\partial}{\partial x_i}\in \bar{{\mathfrak g}} \; (i=1,\ldots, m)$ for an $m$ with $1 \leq m \leq n$. Moreover a maximal algebra has no trivial $\bar{{\mathfrak g}}$-module in the space spanned by $\dfrac{\partial}{\partial x_i} (i=m+1,\ldots, n)$. The possible algebras $\bar{{\mathfrak g}}$ are described in detail, as well as all $\bar{{\mathfrak g}}$-modules that constitute such maximal ${\mathfrak L}$. All maximal ${\mathfrak L}$ are described explicitly for $n\leq 3$

The AKNS-hierarchy

We present here an overview for the Encyclopaedia of Mathematics of the various forms and properties of this system of equations together with its geometric and Lie algebraic background

On the structure of transitively differential algebras

We study finite-dimensional Lie algebras of polynomial vector fields in $n$ variables that contain the vector fields ${\partial}/{\partial x_i} \; (i=1,\ldots, n)$ and $x_1{\partial}/{\partial x_1}+ \dots + x_n{\partial}/{\partial x_n}$. We derive some general results on the structure of such Lie algebras, and provide the complete classification in the cases $n=2$ and $n=3$. Finally we describe a certain construction in high dimensions

Spaces of boundary values related to a multipoint version of the KP-hierarchy

In this paper one considers a finite number of points in the complex plane and various spaces of boundary values on circles surrounding these points. To this geometric configuration one associates a Grassmann manifolds that is shown to yield solutions of a multipoint version of the linearization of the $KP$-hierarchy. These Grassmann manifolds are built in such a way that the determinant line bundle and its dual over them still make sense. The same holds for the so-called $\tau$-functions, determinants of certain Fredholm operators that measure the failure of equivariance at lifting the commuting flows of ths hierarchy to these bundles. Solutions of the linearization are described by wave functions of a certain type. They are perturbations of the trivial solution with the leading term of the perturbation determining the type. One concludes with showing that, if a plane in the Grassmann manifold yields wave functions of different types, they are connected by a differential operator in the coordinates of the flows

An irreducible smooth non-admissible representation

It is shown for the group of k-rational points of an affine algebraic group G with k a finite extension of Qp that the topological irreducibility of unitary representations of G does not have to be equivalent to the algebraic irreducibility of the representation on the smooth vectors. We give for a specific G an example of an irreducible smooth representation, which is not admissible

Where do experiments end?

types: Editorial CommentCopyright © 2010 Elsevier. NOTICE: This is the author’s version of a work accepted for publication by Elsevier. Changes resulting from the publishing process, including peer review, editing, corrections, structural formatting and other quality control mechanisms, may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Geoforum, 2010, Vol. 41, Issue 5 pp. 667 – 670 DOI: 10.1016/j.geoforum.2010.05.003Editoria

Loop-Erasure of Plane Brownian Motion

We use the coupling technique to prove that there exists a loop-erasure of a plane Brownian motion stopped on exiting a simply connected domain, and the loop-erased curve is the reversal of a radial SLE$_2$ curve.Comment: 10 page