Pol Vanhaecke
Téléport 2  BP 30179 Boulevard Marie et Pierre Curie 86962 Futuroscope Chasseneuil Cedex
Laboratoire de Mathématiques et Application, CNRSUniversity of Poitiers
Publications:
Evripidou C., Kassotakis P., Vanhaecke P.
Integrable Deformations of the Bogoyavlenskij–Itoh Lotka–Volterra Systems
2017, vol. 22, no. 6, pp. 721–739
Abstract
We construct a family of integrable deformations of the Bogoyavlenskij–Itoh systems and construct a Lax operator with spectral parameter for it. Our approach is based on the construction of a family of compatible Poisson structures for the undeformed systems, whose Casimirs are shown to yield a generating function for the integrals in involution of the deformed systems.We show how these deformations are related to the Veselov–Shabat systems.

Damianou P. A., Sabourin H., Vanhaecke P.
Intermediate Toda Systems
2015, vol. 20, no. 3, pp. 277292
Abstract
We construct a large family of Hamiltonian systems which interpolate between the classical Kostant–Toda lattice and the full Kostant–Toda lattice and we discuss their integrability. There is one such system for every nilpotent ideal $\mathcal{I}$ in a Borel subalgebra $\mathfrak{b}_+$ of an arbitrary simple Lie algebra $\mathfrak{g}$. The classical Kostant–Toda lattice corresponds to the case of $\mathcal{I}=[\mathfrak{n}_+, \mathfrak{n}_+]$, where $\mathfrak{n}_+$ is the unipotent ideal of $\mathfrak{b}_+$, while the full Kostant–Toda lattice corresponds to $\mathcal{I}=\{0\}$. We mainly focus on the case $\mathcal{I}=[[\mathfrak{n}_+, \mathfrak{n}_+], \mathfrak{n}_+]$. In this case, using the theory of root systems of simple Lie algebras, we compute the rank of the underlying Poisson manifolds and construct a set of (rational) functions in involution, large enough to ensure Liouville integrability. These functions are restrictions of wellchosen integrals of the full Kostant–Toda lattice, except for the case of the Lie algebras of type $C$ and $D$ where a different function (Casimir) is needed. The latter fact, and other ones listed in the paper, point at the Liouville integrability of all the systems we construct, but also at the nontriviality of obtaining the result in full generality.

Damianou P. A., Vanhaecke P.
Foreword
2011, vol. 16, no. 34, pp. 185186
Abstract

Pedroni M., Vanhaecke P.
A Lie algebraic generalization of the Mumford system, its symmetries and its multihamiltonian structure
1998, vol. 3, no. 3, pp. 132160
Abstract
In this paper we generalize the Mumford system which describes for any fixed $g$ all linear flows on all hyperelliptic Jacobians of dimension $g$. The phase space of the Mumford system consists of triples of polynomials, subject to certain degree constraints, and is naturally seen as an affine subspace of the loop algebra of $\mathfrak{sl}(2)$. In our generalizations to an arbitrary simple Lie algebra $\mathfrak{g}$ the phase space consists of $\mathrm{dim}\,\mathfrak{g}$ polynomials, again subject to certain degree constraints. This phase space and its multiHamiltonian structure is obtained by a Poisson reduction along a subvariety $N$ of the loop algebra $\mathfrak{g}((\lambda1))$ of $\mathfrak{g}$. Since $N$ is not a Poisson subvariety for the whole multiHamiltonian structure we prove an algebraic. Poisson reduction theorem for reduction along arbitrary subvarieties of an affine Poisson variety; this theorem is similar in spirit to the Marsden–Ratiu reduction theorem. We also give a different perspective on the multiHamiltonian structure of the Mumford system (and its generalizations) by introducing a master symmetry; this master symmetry can be described on the loop algebra $\mathfrak{g}((\lambda1))$ as the derivative in the direction of $\lambda$ and is shown to survive the Poisson reduction. When acting (as a Lie derivative) on one of the Poisson structures of the system it produces a next one, similarly when acting on one of the Hamiltonians (in involution) or their (commuting) vector fields it produces a next one. In this way we arrive at several multiHamiltonian hierarchies, built up by a master symmetry.
