Related papers: On Lie Groups and Toda Lattices
Some generalizations of spin Sutherland models descend from `master integrable systems' living on Heisenberg doubles of compact semisimple Lie groups. The master systems represent Poisson--Lie counterparts of the systems of free motion…
In this paper we discuss the relation between the functions that give first integrals of full symmetric Toda system (an important Hamilton system on the space of traceless real symmetric matrices) and the vector fields on the group of…
It is shown that supersymmetric integrable models in two dimensions, both relativistic (i.e. super-Toda type theories) and non-relativistic (reductions of super-KP hierarchies) can be associated to general Poisson-brackets structures given…
In this paper we study the finitely generated algebras underlying $W$ algebras. These so called 'finite $W$ algebras' are constructed as Poisson reductions of Kirillov Poisson structures on simple Lie algebras. The inequivalent reductions…
The Lie algebra of pseudodifferential symbols on the circle has a nontrivial central extension (by the ``logarithmic'' 2-cocycle) generalizing the Virasoro algebra. The corresponding extended subalgebra of integral operators generates the…
Lie bialgebras occur as the principal objects in the infinitesimalisation of the theory of quantum groups - the semi-classical theory. Their relationship with the quantum theory has made available some new tools that we can apply to…
The purpose of this paper is to bring together various loose ends in the theory of integrable systems. For a semisimple Lie algebra $\mathfrak g$, we obtain several results on completeness of homogeneous Poisson-commutative subalgebras of…
The S-matrices for non-simply-laced affine Toda field theories are considered in the context of a generalised bootstrap principle. The S-matrices, and in particular their poles, depend on a parameter whose range lies between the Coxeter…
We study extensions of the classical Toda lattices at several different space-time scales. These extensions are from the classical tridiagonal phase spaces to the phase space of full Hessenberg matrices, referred to as the Full Kostant-Toda…
We develop a framework for Poisson geometry on loop spaces of low regularity, extending Mokhov's classical constructions from smooth loops to weak Sobolev spaces $W^{s,p}(\mathbb{S^1},\mathbb{R}^m)$ with $o < s \frac{1}{2}$ and $1 < p <…
Kontsevich's formality theorem states that the differential graded Lie algebra of multidifferential operators on a manifold M is L-infinity-quasi-isomorphic to its cohomology. The construction of the L-infinity map is given in terms of…
We construct dimer graphs for relativistic Toda chains associated with classical untwisted Lie algebras of A, B, C$_0$, C$_\pi$, D types and twisted A, D types. We show that the Seiberg-Witten curve of 5d $\mathcal{N}=1$ pure supersymmetric…
Lie theory for the integration of Lie algebroids to Lie groupoids, on the one hand, and of Poisson manifolds to symplectic groupoids, on the other, has undergone tremendous developements in the last decade, thanks to the work of…
The Kodaira-Thurston manifold is a quotient of a nilpotent Lie group by a cocompact lattice. We compute the family Gromov-Witten invariants which count pseudoholomorphic tori in the Kodaira-Thurston manifold. For a fixed symplectic form the…
Coadjoint orbits and multiplicity free spaces of compact Lie groups are important examples of symplectic manifolds with Hamiltonian groups actions. Constructing action-angle variables on these spaces is a challenging task. A fundamental…
This text presents some basic notions in symplectic geometry, Poisson geometry, Hamiltonian systems, Lie algebras and Lie groups actions on symplectic or Poisson manifolds, momentum maps and their use for the reduction of Hamiltonian…
In this paper we classify symplectic leaves of the regular part of the projectivization of the space of meromorphic endomorphisms of a stable vector bundle on an elliptic curve, using the study of shifted Poisson structures on the moduli of…
We consider abelian twisted loop Toda equations associated with the complex general linear groups. The Dodd--Bullough--Mikhailov equation is a simplest particular case of the equations under consideration. We construct new soliton solutions…
We present a new realization of scalar integrable hierarchies in terms of the Toda lattice hierarchy. In other words, we show on a large number of examples that an integrable hierarchy, defined by a pseudodifferential Lax operator, can be…
We consider a class of complex manifolds constructed as multiplicative quiver varieties associated with a cyclic quiver extended by an arbitrary number of arrows starting at a new vertex. Such varieties admit a Poisson structure, which is…