### A note on the rotationally symmetric SO(4) Euler rigid body.

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Lax operator algebras constitute a new class of infinite dimensional Lie algebras of geometric origin. More precisely, they are algebras of matrices whose entries are meromorphic functions on a compact Riemann surface. They generalize classical current algebras and current algebras of Krichever-Novikov type. Lax operators for 𝔤𝔩(n), with the spectral parameter on a Riemann surface, were introduced by Krichever. In joint works of Krichever and Sheinman their algebraic structure was revealed and...

We show that the dimer model on a bipartite graph $\Gamma $ on a torus gives rise to a quantum integrable system of special type, which we call acluster integrable system. The phase space of the classical system contains, as an open dense subset, the moduli space ${\u0141}_{\Gamma}$ of line bundles with connections on the graph $\Gamma $. The sum of Hamiltonians is essentially the partition function of the dimer model. We say that two such graphs ${\Gamma}_{1}$ and ${\Gamma}_{2}$ areequivalentif the Newton polygons of the corresponding partition functions...

We consider multiply periodic functions, sometimes called Abelian functions, defined with respect to the period matrices associated with classes of algebraic curves. We realise them as generalisations of the Weierstraß ℘-function using two different approaches. These functions arise naturally as solutions to some of the important equations of mathematical physics and their differential equations, addition formulae, and applications have all been recent topics of study. The first approach discussed...

We use the methods that were developed by Adler and van Moerbeke to determine explicit equations for a certain moduli space, that was studied by Narasimhan and Ramanan. Stated briefly it is, for a fixed non-hyperelliptic Riemann surface $\Gamma $ of genus $3$, the moduli space of semi-stable rank two bundles with trivial determinant on $\Gamma $. They showed that it can be realized as a projective variety, more precisely as a quartic hypersurface of ${\mathbb{P}}^{7}$, whose singular locus is the Kummer variety of $\Gamma $. We first construct...

We apply the General Galois Theory of difference equations introduced in the first part to concrete examples. The General Galois Theory allows us to define a discrete dynamical system being infinitesimally solvable, which is a finer notion than being integrable. We determine all the infinitesimally solvable discrete dynamical systems on the compact Riemann surfaces.