Related papers: Dynamical r-matrices and Poisson-Lie symmetries in…
The precise relationship between the arbitrary monodromy dependent 2-form appearing in the chiral WZNW symplectic form and the `exchange r-matrix' that governs the corresponding Poisson brackets is established. Generalizing earlier results…
The chiral WZNW symplectic form $\Omega^{\rho}_{chir}$ is inverted in the general case. Thereby a precise relationship between the arbitrary monodromy dependent 2-form appearing in $\Omega^{\rho}_{chir}$ and the exchange r-matrix that…
This is a review of recent work on the chiral extensions of the WZNW phase space describing both the extensions based on fields with generic monodromy as well as those using Bloch waves with diagonal monodromy. The symplectic form on the…
The dynamical generalization of the classical Yang-Baxter equation that governs the possible Poisson structures on the space of chiral WZNW fields with generic monodromy is reviewed. It is explained that for particular choices of the chiral…
In this note we complement recent results on the exchange $r$-matrices appearing in the chiral WZNW model by providing a direct, purely finite-dimensional description of the relationship between the monodromy dependent 2-form that enters…
The Dirac reduction technique used previously to obtain solutions of the classical dynamical Yang-Baxter equation on the dual of a Lie algebra is extended to the Poisson-Lie case and is shown to yield naturally certain dynamical r-matrices…
According to Etingof and Varchenko, the classical dynamical Yang-Baxter equation is a guarantee for the consistency of the Poisson bracket on certain Poisson-Lie groupoids. Here it is noticed that Dirac reductions of these Poisson manifolds…
Poisson-Lie (PL) dynamical r-matrices are generalizations of dynamical r-matrices, where the base is a Poisson-Lie group. We prove analogues of basic results for these r-matrices, namely constructions of (quasi)Poisson groupoids and of…
The gauge action of the Lie group $G$ on the chiral WZNW phase space ${\cal M}_{\check G}$ of quasiperiodic fields with $\check G$-valued monodromy, where $\check G\subset G$ is an open submanifold, is known to be a Poisson-Lie (PL) action…
We study the Poisson bracket algebra of the $N=2$ supersymmetric chiral WZNW model in superspace. It involves two classical r-matrices, one of which comes from the geometrical constraints implied by $N=2$ supersymmetry. The phase space…
We construct the classical Poisson structure and $r$-matrix for some finite dimensional integrable Hamiltonian systems obtained by constraining the flows of soliton equations in a certain way. This approach allows one to produce new kinds…
We give a physicist oriented survey of Poisson-Lie symmetries of classical systems. We consider finite dimensional geometric actions and the chiral WZNW model as examples for the general construction. An essential point is that quadratic…
We derive a generalization of the classical dynamical Yang-Baxter equation (CDYBE) on a self-dual Lie algebra $\cal G$ by replacing the cotangent bundle T^*G in a geometric interpretation of this equation by its Poisson-Lie (PL) analogue…
Let $\g$ be a finite dimensional complex Lie algebra and $\l\subset \g$ a Lie subalgebra equipped with the structure of a factorizable quasitriangular Lie bialgebra. Consider the Lie group $\Exp \l$ with the Semenov-Tjan-Shansky Poisson…
We address the question of duality for the dynamical Poisson groupoids of Etingof and Varchenko over a contractible base. We also give an explicit description for the coboundary case associated with the solutions of the classical dynamical…
We construct a Poisson isomorphism between the formal Poisson manifolds g^* and G^*, where g is a finite dimensional quasitriangular Lie bialgebra. Here g^* is equipped with its Lie-Poisson (or Kostant-Kirillov-Souriau) structure, and G^*…
We give a detailed description of a dynamical system which enjoys a Poisson-Lie symmetry with two non-isomorphic dual groups. The system is obtained by taking the $q\to\infty$ limit of the q-deformed WZW model and the understanding of its…
We review the notion of (anomalous) Poisson-Lie symmetry of a dynamical system and we outline the Poisson-Lie symmetric deformation of the standard WZW model from the vantage point of the twisted Heisenberg double.
Fundamental representations of real simple Poisson Lie groups are Poisson actions with a suitable choice of the Poisson structure on the underlying (real) vector space. We study these (mostly quadratic) Poisson structures and corresponding…
We study the relationship between general dynamical Poisson groupoids and Lie quasi-bialgebras. For a class of Lie quasi-bialgebras naturally compatible with a reductive decomposition, we extend the description of the moduli space of…