We show that having any planar (cyclic or acyclic) directed network on a disc with the only condition that all $n_1+m$ sources are separated from all $n_2+m$ sinks, we can construct a cluster-algebra realization of elements of an affine Lie--Poisson algebra $R(λ,μ)T^{1}(λ)T^{2}(μ)=T^{2}(μ)T^{1}(λ)R(λ,μ)$ with $(n_1\times n_2)$-matrices $T(λ)$ corresponding to a planar directed network on an annulus. Upon satisfaction of some invertibility conditions, we can extend this construction to realizations of a quantum loop algebra. Having the quantum loop algebra we can also construct a realization of the twisted Yangian algebra, or that of the quantum reflection equation. Every such planar network therefore corresponds to a symplectic leaf of the corresponding infinite-dimensional algebra.
This paper develops the theory of affine Lie-Poisson reduction and applies this process to Yang-Mills and Hall magnetohydrodynamics for fluids and superfluids. As a consequence of this approach, the associated Poisson brackets are obtained by reduction of a canonical cotangent bundle. A Kelvin-Noether circulation theorem is presented and is applied to these examples.
The notions of vertex Lie algebra and vertex Poisson algebra are presented and connections among vertex Lie algebras, vertex Poisson algebras and vertex algebras are discussed.
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 symmetry structure results in uncovering an interesting duality of its exchange relations.
A Poisson-Lie group acting by the coadjoint action on the dual of its Lie algebra induces on it a non-trivial class of quadratic Poisson structures extending the linear Poisson bracket on the coadjoint orbits.
We consider the integrability of a two-parameter deformation of the Wess-Zumino-Witten model, previously introduced in relation with Poisson-Lie T-duality. The resulting family of Poisson-Lie dual models is shown to be integrable by using the Maillet r/s formalism.
Lie bialgebra structures on the extended affine Lie algebra $\widetilde{sl_2(\mathbb{C}_q)}$ are investigated. In particular, all Lie bialgebra structures on $\widetilde{sl_2(\mathbb{C}_q)}$ are shown to be triangular coboundary. This result is obtained by employing some techniques, which may also work for more general extended affine Lie algebras, to prove the triviality of the first cohomology group of $\widetilde{sl_2(\mathbb{C}_q)}$ with coefficients in the tensor product of its adjoint module, namely, $H^1(\widetilde{sl_2(\mathbb{C}_q)},\widetilde{sl_2(\mathbb{C}_q)}\otimes\widetilde{sl_2…
We give an introduction to the Mathematica packages "MasterPVA" and "MasterPVAmulti used to compute lambda-brackets in Poisson vertex algebras, which play an important role in the theory of infinite-dimensional Hamiltonian systems. As an application, we give an introduction to the Mathematica package "WAlg" aimed to compute the lambda-brackets among the generators of classical affine W-algebras. The use of these packages is shown by providing some explicit examples.
Based on ideas of W. M. Tulczyjew, a geometric framework for a frame-independent formulation of different problems in analytical mechanics is developed. In this approach affine bundles replace vector bundles of the standard description and functions are replaced by sections of certain affine line bundles called AV-bundles. Categorial constructions for affine and special affine bundles as well as natural analogs of Lie algebroid structures on affine bundles (Lie affgebroids) are investigated. One discovers certain Lie algebroids and Lie affgebroids canonically associated with an AV-bundle which…
We introduce a dual notion of the Poisson algebra by exchanging the roles of the two binary operations in the Leibniz rule defining the Poisson algebra. We show that the transposed Poisson algebra thus defined not only shares common properties of the Poisson algebra, including the closure under taking tensor products and the Koszul self-duality as an operad, but also admits a rich class of identities. More significantly, a transposed Poisson algebra naturally arises from a Novikov-Poisson algebra by taking the commutator Lie algebra of the Novikov algebra. Consequently, the classic constructio…
Covering Algebras of extended affine Lie algebras(EALA's) relative to finite order automorphisms are studied. Conditions are given for when the resulting algebra is again an EALA. This paper deals with affinizations of EALA's relative to finite order automorphisms and these algebras are generalizations of the affine Kac-Moody Lie algebras.
It was proposed the Lie group such that symplectic structure of orbits of co-adjoint representation of the group is revealed symplectic structure of a rigid body dynamics in quaternion variables. It is shown that Poisson brackets of corresponding Lie-Poisson structure coincide with canonical Poisson brackets on cotangent bundle of group unit quaternions.
Classical r-matrices of the three-dimensional real Lie bialgebras are obtained. In this way all three-dimensional real coboundary Lie bialgebras and their types (triangular, quasitriangular or factorizable) are classified. Then, by using the Sklyanin bracket, the Poisson structures on the related Poisson-Lie groups are obtained.
We obtain the classical r-matrices of two and three dimensional Lie super-bialgebras. We thus classify all two and three dimensional coboundary Lie super-bialgebras and their types (triangular, quasi-triangular, or factorable). Using the Sklyanin superbracket, we then obtain the super Poisson structures on the related Poisson-Lie supergroups.
We introduce new examples of mappings defining noncommutative root space generalizations for the Witt, Ricci flow, and Poisson bracket continual Lie algebras.
The study of denominator identities for Lie superalgebras was recently developed by M. Gorelik, V.G. Kac, P.Moseneder Frajria, I. Musson, P. Papi, M. Wakimoto and the author. In this paper we generalize these identities to the twisted affine case, completing the result to all affine Lie superalgebras.
As a continuation of previous papers, we study the concept of a Lie algebroid structure on an affine bundle by means of the canonical immersion of the affine bundle into its bidual. We pay particular attention to the prolongation and various lifting procedures, and to the geometrical construction of Lagrangian-type dynamics on an affine Lie algebroid.
In this paper we extend the almost complex Poisson structures from almost complex manifolds to almost complex Lie algebroids. Examples of such structures are also given and the almost complex Poisson morphisms of almost complex Lie algebroids are studied.
We present the classical Poisson-Lichnerowicz cohomology for the Poisson algebra of polynomials $\mathbb{C}[X_{1},..., X_{n}]$ using exterior calculus. After presenting some non homogeneous Poisson brackets on this algebra, we compute Poisson cohomological spaces when the Poisson structure corresponds to a bracket of a rigid Lie algebra.
We introduce and study a class of Lie algebroids associated to faithful modules which is motivated by the notion of cotangent Lie algebroids of Poisson manifolds. We also give a classification of transitive Lie algebroids and describe Poisson algebras by using the notions of algebroid and Lie connections.
In this letter, first we give a decomposition for any Lie-Poisson structure $π_g$ associated to the modular vector. In particular, $π_g$ splits into two compatible Lie-Poisson structures if $dim{g} \leq 3$. As an application, we classified quadratic deformations of Lie-Poisson structures on $\mathbb R^3$ up to linear diffeomorphisms.
In this paper we construct two free field realizations of the elliptic affine Lie algebra sl(2,R) + Ω_R/dR, where R=C[t,t^{-1},u|u^2=t^3 - 2b t^2 + t]. The first realization gives an analogue of Wakimoto's construction for Affine Kac-Moody algebras, but in the setting of the elliptic affine Lie algebra. The second realization gives new type of representations analogous to Imaginary Verma modules in the Affine setting.
The first part of this paper is devoted to the theory of Poisson-Lie groups in the Banach setting. Our starting point is the straightforward adaptation of the notion of Manin triples to the Banach context. The difference with the finite-dimensional case lies in the fact that a duality pairing between two non-reflexive Banach spaces is necessary weak (as opposed to a strong pairing where one Banach space can be identified with the dual space of the other). The notion of generalized Banach Poisson manifolds introduced in this paper is compatible with weak duality pairings between the tangent spa…
We briefly review our results on the Lie theory underlying vector bundles over Lie groupoids and Lie algebroids, pointing out the role of Poisson geometry in extending these results to double Lie algebroids and LA-groupoids.
A Vogan diagram is a Dynkin diagram with a Cartan involution of twisted affine superlagebras based on maximally compact Cartan subalgebras. This article construct the Vogan diagrams of twisted affine superalgebras. It is a part of completion of classification of vogan diagrams to superalgebras cases.
In this paper, we introduce right-invariant (similarly, left-invariant) Poisson-Nijenhuis Structures on Lie groupoids and their infinitesimal counterparts as called $(Λ, \mathbf{n})-$structures. We present a mutual correspondence between $(Λ,\mathbf{n})-$structures on Lie algebroids with Poisson-Nijenhuis structures $(Π, \mathbf{N})$ on their Lie groupoids under some conditions. Also, we will construct Poisson-Nijenhuis pair groupoids as an example from the invariance's point of view.
We introduce the notion of a Lie algebroid structure on an affine bundle whose base manifold is fibred over the real numbers. It is argued that this is the framework which one needs for coming to a time-dependent generalization of the theory of Lagrangian systems on Lie algebroids. An extensive discussion is given of a way one can think of forms acting on sections of the affine bundle. It is further shown that the affine Lie algebroid structure gives rise to a coboundary operator on such forms. The concept of admissible curves and dynamical systems whose integral curves are admissible, brings …
We show how to reduce, under certain regularities conditions, a Poisson-Nijenhuis Lie algebroid to a symplectic-Nijenhuis Lie algebroid with nondegenerate Nijenhuis tensor. We generalize the work done by Magri and Morosi for the reduction of Poisson-Nijenhuis manifolds. The choice of the more general framework of Lie algebroids is motivated by the geometrical study of some reduced bi-Hamiltonian systems. An explicit example of reduction of a Poisson-Nijenhuis Lie algebroid is also provided.
In this paper, we discuss several relations between the existence of invariant volume forms for Hamiltonian systems on Poisson-Lie groups and the unimodularity of the Poisson-Lie structure. In particular, we prove that Hamiltonian vector fields on a Lie group endowed with a unimodular Poisson-Lie structure preserve a multiple of any left-invariant volume on the group. Conversely, we also prove that if there exists a Hamiltonian function such that the identity element of the Lie group is a nondegenerate singularity and the associated Hamiltonian vector field preserves a volume form, then the Po…
First, we derive an explicit formula for the Poisson bracket of the classical finite W-algebra W^{fin}(g,f), the algebra of polynomial functions on the Slodowy slice associated to a simple Lie algebra g and its nilpotent element f. On the other hand, we produce an explicit set of generators and we derive an explicit formula for the Poisson vertex algebra structure of the classical affine W-algebra W(g,f). As an immediate consequence, we obtain a Poisson algebra isomorphism between W^{fin}(g,f) and the Zhu algebra of W(g,f). We also study the generalized Miura map for classical W-algebras.
