Related papers: Elliptic Calogero-Moser Systems and Isomonodromic …
We construct integrable generalizations of the elliptic Calogero-Sutherland-Moser model of particles with spin, involving noncommutative spin interactions. The spin coupling potential is a modular function and, generically, breaks the…
We develop a bundle picture for the case that the configuration manifold has only a single isotropy type, and give a formula for the reduced symplectic form in this setting. Furthermore, as an application of this bundle picture we consider…
We study a second-order linear differential equation known as the deformed cubic oscillator, whose isomonodromic deformations are controlled by the first Painlev{\'e} equation. We use the generalised monodromy map for this equation to give…
The discrete-time rational Calogero's goldfish system is obtained from the Ansatz Lax pair. The discrete-time Lagrangians of the system possess the discrete-time 1-form structure as those in the discrete-time Calogero-Moser system and…
We construct twisted Calogero-Moser (CM) systems with spins as the Hitchin systems derived from the Higgs bundles over elliptic curves, where transitions operators are defined by an arbitrary finite order automorphisms of the underlying Lie…
The Lax pair representation in Fourier space is used to obtain a list of integrable scalar evolutionary equations with quadratic nonlinearity. The famous systems of this type such as KdV, intermediate long-wave equation (ILW), Camassa-Holm…
We construct complex root spaces remaining invariant under antilinear involutions related to all Coxeter groups. We provide two alternative constructions: One is based on deformations of factors of the Coxeter element and the other based on…
Let $(X,\check{X})$ be a mirror pair of a complex torus $X$ and its mirror partner $\check{X}$. This mirror pair is described as the trivial special Lagrangian torus fibrations $X\rightarrow B$ and $\check{X}\rightarrow B$ on the same base…
We describe a variational framework for non-commuting flows, extending the theories of Lagrangian multiforms and pluri-Lagrangian systems, which have gained prominence in recent years as a variational description of integrable systems in…
Let $X$ be a toric variety. Rationally Borel-Moore homology of $X$ is isomorphic to the homology of the Koszul complex $A^T_*(X)\otimes \Lambda^\x M$, where $A^T_*(X)$ is the equivariant Chow group and $M$ is the character group of $T$.…
For an integrable Hamiltonian system we construct a representation of the phase space symmetry algebra over the space of functions on a Lagrangian manifold. The representation is a result of the canonical quantization of the integrable…
Calogero-Moser systems can be generalized for any root system (including the non-crystallographic cases). The algebraic linearization of the generalized Calogero-Moser systems and of their quadratic (resp. quartic) perturbations are…
A method to construct integrable deformations of Hamiltonian systems of ODEs endowed with Lie-Poisson symmetries is proposed by considering Poisson-Lie groups as deformations of Lie-Poisson (co)algebras. Moreover, the underlying Lie-Poisson…
The article presents a new perspective on the isomorphism problem for non-ergodic measure-preserving dynamical systems with discrete spectrum which is based on the connection between ergodic theory and topological dynamics constituted by…
The integrability of the classical and quantum rational Calogero-Moser systems is verified explicitly via the Lax pair method for the case $n=3$. We provide an extensive survey of reflection groups and root systems. The…
We construct a quantum integrable model which is an $R$-matrix generalization of the Calogero-Moser system, based on the Baxter-Belavin elliptic $R$-matrix. This is achieved by introducing $R$-matrix Dunkl operators so that commuting…
We consider a family of classical elliptic integrable systems including (relativistic) tops and their matrix extensions of different types. These models can be obtained from the "off-shell" Lax pairs, which do not satisfy the Lax equations…
We describe classical top-like integrable systems arising from the quantum exchange relations and corresponding Sklyanin algebras. The Lax operator is expressed in terms of the quantum non-dynamical $R$-matrix even at the classical level,…
We propose to take a look at a new approach to the study of integral polyhedra. The main idea is to give an integral representation, or matrix model representation, for the key combinatorial characteristics of integral polytopes. Based on…
We analyze the integrability of the ${\cal N}$-extended supersymmetric Calogero-Moser model. We explicitly construct the Lax pair $\{L,A\}$ for this system, which properly reproduces all equations of motion. After adding a supersymmetric…