Related papers: Toda systems for Takiff algebras

We consider in C^n the class of symmetric homogeneous quadratic dynamical systems. We introduce the notion of algebraic integrability for this class. We present a class of symmetric quadratic dynamical systems that are algebraically…

An integrable system is introduced, which is a generalization of the $\mathfrak{sl}(2)$ quantum affine Gaudin model. Among other things, the Hamiltonians are constructed and their spectrum is calculated within the ODE/IQFT approach. The…

The standard generators of tridiagonal algebras, recently introduced by Terwilliger, are shown to generate a new (in)finite family of mutually commuting operators which extends the Dolan-Grady construction. The involution property relies on…

We define the periodic Full Kostant-Toda on every simple Lie algebra, and show its Liouville integrability. More precisely we show that this lattice is given by a Hamiltonian vector field, associated to a Poisson bracket which results from…

We provide geometric quantization of a completely integrable Hamiltonian system in the action-angle variables around an invariant torus with respect to polarization spanned by almost-Hamiltonian vector fields of angle variables. The…

For an arbitrary semisimple Frobenius manifold we construct Hodge integrable hierarchy of Hamiltonian partial differential equations. In the particular case of quantum cohomology the tau-function of a solution to the hierarchy generates the…

By generalizing the Drinfeld-Sokolov reduction a large class of $W$ algebras can be constructed. We introduce 'finite' versions of these algebras by Poisson reducing Kirillov Poisson structures on simple Lie algebras. A closed and…

We introduce a new infinite class of superintegrable quantum systems in the plane. Their Hamiltonians involve reflection operators. The associated Schr\"odinger equations admit separation of variables in polar coordinates and are exactly…

The algebraic-geometric approach is extended to study solutions of N-component systems associated with the energy dependent Schrodinger operators having potentials with poles in the spectral parameter, in connection with Hamiltonian flows…

This work is intended to contextualize and enhance certain well-studied relationships between Hessenberg varieties and the Toda lattice, thereby building on the results of Kostant, Peterson, and others. One such relationship is the fact…

It is shown that the integrability conditions of the equations satisfied by the local Frenet frame associated with a holomorphic curve in a complex Grassmann manifold coincide with a special class of nonabelian Toda equations. A local…

We associate to an arbitrary $\mathbb Z$-gradation of the Lie algebra of a Lie group a system of Riccati-type first order differential equations. The particular cases under consideration are the ordinary Riccati and the matrix Riccati…

The Darboux-Halphen system of equations have common or individual additive terms depending on the matrices defining Yang-Mills gauge potential fields. Tod (Phys. Lett. A 190 (1994) 221-224), described a conserved quantity for the classical…

This paper is the first in a forthcoming series of works where the authors study the global asymptotic behavior of the radial solutions of the 2D periodic Toda equation of type $A_n$. The principal issue is the connection formulae between…

We use factorized $L$ operator to construct an integrable model with open boundary conditions. By taking trigonometic limit($\tau \to \sqrt{-1}\infty$) and scaling limit($\omega \to 0$), we get a Hamiltonian of a classical integrable…

There are different methods of discretizing integrable systems. We consider semi-discrete analog of two-dimensional Toda lattices associated to the Cartan matrices of simple Lie algebras that was proposed by Habibullin in 2011. This…

In this paper we derive new two-component integrable differential difference and partial difference systems by applying a Lax-Darboux scheme to an operator formed from an ${\mathfrak{sl}}_3({\mathbb{C}})$-based automorphic Lie algebra. The…

We introduce the notion of a real form of a Hamiltonian dynamical system in analogy with the notion of real forms for simple Lie algebras. This is done by restricting the complexified initial dynamical system to the fixed point set of a…

We prove that Toeplitz operators are norm dense in the Toeplitz algebra $\mathfrak{T}(L^\infty)$ over the weighted Bergman space $\mathcal{A}^2_\nu(\Omega)$ of a bounded symmetric domain $\Omega\subset\mathbb{C}^n$. Our methods use…

Our previous work on a hidden integrable structure of the melting crystal model (the U(1) Nekrasov function) is extended to a modified crystal model. As in the previous case, "shift symmetries" of a quantum torus algebra plays a central…