Related papers: Dynamical $r$-matrices for Hitchin's systems on Sc…
One investigates the Hitchin systems over "large limit" curves.
The generalized Cremmer-Gervais R-matrix being a twist of the standard R-matrix of $SL_q(3)$, depends on two extra parameters. Properties of this R-matrix are discussed and two dynamical systems, the quantum group covariant $q$-oscillator…
We describe a class of spectral curves and find explicit formulas for Darboux coordinates for hyperelliptic Hitchin systems corresponding to classical simple Lie groups. We consider in detail the systems with classical rank 2 gauge groups…
We define graftable curves on real projective surfaces. In particular, we construct graftable ones in Hitchin case and show that real projective structures with the same Hitchin holonomy, carrying the same weight type, are related to each…
We adapt Hitchin's integrable systems to the case of a punctured curve. In the case of $\CC P^{1}$ and $SL_{n}$-bundles, they are equivalent to systems studied by Garnier. The corresponding quantum systems were identified by B. Feigin, E.…
We discuss geometrical aspects of different dualities in the integrable systems of the Hitchin type and its various generalizations. It is shown that T duality known in the string theory context is related to the separation of variables…
The dynamics of a linear dynamical system over a finite field can be described by using the elementary divisors of the corresponding matrix. It is natural to extend the investigation to a general finite commutative ring. In a previous…
This paper is devoted to the study of symplectic manifolds and their connection with Hamiltonian dynamical systems. We review some properties and operations on these manifolds and see how they intervene when studying the complete…
The classical r-matrix structure for the generic elliptic Ruijsenaars-Schneider model is presented. It makes the integrability of this model as well as of its discrete-time version that was constructed in a recent paper manifest.
In this work, the description of the moduli space of principal $G$-bundles as double quotient of loop groups is used to construct an \'etale local $r$-matrix for the Hitchin integrable system.
The Krichever construction in one variable, that is, for spectral curves, linearizes the KdV-hierarchy on the jacobian of the curve. We carry out an appropriate generalization of the Krichever construction for an arbitrary projective…
A generalized Hitchin equation was proposed as the BPS equation for a large class of four dimensional N=1 theories engineered using M5 branes. In this paper, we show how to write down the spectral curve for the moduli space of generalized…
We present a derivation of the dynamical r-matrices of the Calogero-Moser models using the Hamiltonian reduction procedure to get general formulae. We describe the dynamical r-matrices thus found for spin Calogero-Moser models and…
The Hamiltonian theory of isomonodromy equations for meromorphic connections with irregular singularities on algebraic curves is constructed. An explicit formula for the symplectic structure on the space of monodromy and Stokes matrices is…
The integrability of two symplectic maps, that can be considered as discrete-time analogs of the Garnier and Neumann systems is established in the framework of the $r$-matrix approach, starting from their Lax representation. In contrast…
Explicit solutions of the non-constant semi-dynamical reflection equation are constructed, together with suitable parametrizations of their structure matrices. Considering the semi-dynamical reflection equation with rational non-constant…
Some particular examples of classical and quantum systems on the lattice are solved with the help of orthogonal polynomials and its connection to continuous models are explored.
A generalization of the Marshal-Olkin parametrization scheme is developed and stochastic models related to it are discussed here.
The dynamical systems of identical particles admitting quadratic integrals of motion are classified. The relevant integrals are explicitly constructed and their relation to separation of variables in H-J equation is clarified.
Motivated by work of Kucharczyk and Scholze, we use sheafified rational Witt vectors to attach a new ringed space $W_{\mathrm{rat}} (X)$ to every scheme $X$. We also define $R$-valued points $W_{\mathrm{rat}} (X) (R)$ of $W_{\mathrm{rat}}…