arXiv subjects

Pol Vanhaecke

Publications and source records attributed to Pol Vanhaecke.

21 records · Page 2Linked to original sources

arXiv2001

Hyperelliptic Prym Varieties and Integrable Systems

We introduce two algebraic completely integrable analogues of the Mumford systems which we call hyperelliptic Prym systems, because every hyperelliptic Prym variety appears as a fiber of their momentum map. As an application we show that the generic fiber of the momentum map of the periodic Volterra lattice $$\dot a_i=a_i(a_{i-1}-a_{i+1}), \qquad i=1,...,n,\quad a_{n+1}=a_1,$$ is an affine part of a hyperelliptic Prym variety, obtained by removing $n$ translates of the theta divisor, and we conclude that this integrable system is algebraic completely integrable.

math-ph↗

arXiv1998

Deformation Quantization of Polynomial Poisson Algebras

This paper discusses the notion of a deformation quantization for an arbitrary polynomial Poisson algebra A. We examine the Hochschild cohomology group H^3(A) and find that if a deformation of A exists it can be given by bidifferential operators. We then compute an explicit third order deformation quantization of A and show that it comes from a quantized enveloping algebra. We show that the deformation extends to a fourth order deformation if and only if the quantized enveloping algebra gives a fourth order deformation; moreover we give an example where the deformation does not extend. A correction term to the third order quantization given by the enveloping algebra is computed, which precisely cancels the obstruction.

math.QA↗

arXiv1994

Integrable systems and symmetric products of curves

We show how there is associated to each non-constant polynomial $F(x,y)$ a completely integrable system with polynomial invariants on $\Rd$ and on $\C{2d}$ for each $d\geq1$; in fact the invariants are not only in involution for one Poisson bracket, but for a large class of polynomial Poisson brackets, indexed by the family of polynomials in two variables. We show that the complex invariant manifolds are isomorphic to affine parts of $d$-fold symmetric products of a deformation of the algebraic curve $F(x,y)=0$, and derive the structure of the real invariant manifolds from it. We also exhibit Lax equations for the hyperelliptic case (i.e., when $F(x,y)$ is of the form $y^2+f(x)$) and we show that in this case the invariant manifolds are affine parts of distinguished (non-linear) subvarieties of the Jacobians of the curves. As an application the geometry of the Hénon-Heiles hierarchy --- a family of superimposable integrable polynomial potentials on the plane --- is revealed and Lax equations for the hierarchy are given.

solv-int↗