Double Poisson Brackets Research Articles

Modified double brackets and a conjecture of S. Arthamonov

Around 20 years ago, M. Van den Bergh introduced double Poisson brackets as operations on associative algebras inducing Poisson brackets under the representation functor. Weaker versions of these operations, called modified double Poisson brackets, were later introduced by S. Arthamonov in order to induce a Poisson bracket on moduli spaces of representations of the corresponding associative algebras. Moreover, he defined two operations that he conjectured to be modified double Poisson brackets. The first case of this conjecture was recently proved by M. Goncharov and V. Gubarev motivated by the theory of Rota-Baxter operators of nonzero weight. We settle the conjecture by realising the second case as part of a new family of modified double Poisson brackets. These are obtained from mixed double Poisson algebras, a new class of algebraic structures that are introduced and studied in the present work.

Double Poisson brackets and involutive representation spaces

Double Poisson brackets and involutive representation spaces

Remarks on Yangian-Type Algebras and Double Poisson Brackets

Remarks on Yangian-Type Algebras and Double Poisson Brackets

Remarks on Yangian-type algebras and double Poisson brackets

В недавней работе одного из авторов была предложена конструкция ассоциативных алгебр, которые обладают рядом свойств янгианов серии A, но при этом более массивны. Мы показываем, что эта конструкция допускает обобщение, в результате которого выявляется прямая связь с обширным семейством двойных скобок Пуассона на свободных алгебрах, описанным в работе Пишеро и Ван де Вейера (2008).

Around Van den Bergh’s double brackets for different bimodule structures

A double Poisson bracket, in the sense of M. Van den Bergh, is an operation on an associative algebra A which induces a Poisson bracket on each representation space in an explicit way. In this note, we study the impact of changing the Leibniz rules underlying a double bracket. This change amounts to make a suitable choice of A-bimodule structure on . In the most important cases, we describe how the choice of A-bimodule structure fixes an analogue to Jacobi identity, and we obtain induced Poisson brackets on representation spaces. The present theory also encodes a formalization of the widespread tensor notation used to write Poisson brackets of matrices in mathematical physics. Communicated by P. Kolesnikov

Momentum Maps and Stochastic Clebsch Action Principles

We derive stochastic differential equations whose solutions follow the flow of a stochastic nonlinear Lie algebra operation on a configuration manifold. For this purpose, we develop a stochastic Clebsch action principle, in which the noise couples to the phase space variables through a momentum map. This special coupling simplifies the structure of the resulting stochastic Hamilton equations for the momentum map. In particular, these stochastic Hamilton equations collectivize for Hamiltonians that depend only on the momentum map variable. The Stratonovich equations are derived from the Clebsch variational principle and then converted into It\^o form. In comparing the Stratonovich and It\^o forms of the stochastic dynamical equations governing the components of the momentum map, we find that the It\^o contraction term turns out to be a double Poisson bracket. Finally, we present the stochastic Hamiltonian formulation of the collectivized momentum map dynamics and derive the corresponding Kolmogorov forward and backward equations.

Modified double Poisson brackets

We propose a non-skew-symmetric generalization of the original definition of double Poisson Bracket by M. Van den Bergh. It allows one to explicitly construct a more general class of H0-Poisson structures on finitely generated associative algebras. We show that modified double Poisson brackets inherit certain major properties of the double Poisson brackets.

Quantization and Dynamisation of Trace-Poisson Brackets

The quantization problem for the trace-bracket algebra, derived from double Poisson brackets, is discussed. We obtain a generalization of the boundary YBE (or so-called ABCD-algebra) for the quantization of quadratic trace-brackets. A dynamical deformation is proposed on the lines of Gervais–Neveu–Felder dynamical quantum algebras.

Noncommutative Inverse Scattering Method for the Kontsevich System

We formulate an analog of Inverse Scattering Method for integrable systems on noncommutative associative algebras. In particular we define Hamilton flows, Casimir elements and noncommutative analog of the Lax matrix. The noncommutative Lax element generates infinite family of commuting Hamilton flows on an associative algebra. The proposed approach to integrable systems on associative algebras satisfy certain universal property, in particular it incorporates both classical and quantum integrable systems as well as provides a basis for further generalization. We motivate our definition by explicit construction of noncommutative analog of Lax matrix for a system of differential equations on associative algebra recently proposed by Kontsevich. First we present these equations in the Hamilton form by defining a bracket of Loday type on the group algebra of the free group with two generators. To make the definition more constructive we utilize (with certain generalizations) the Van den Bergh approach to Loday brackets via double Poisson brackets. We show that there exists an infinite family of commuting flows generated by the noncommutative Lax element.

Normal coordinate in harmonic crystal obtained by virtue of the classical correspondence of the invariant eigen-operator

Noticing that the equation with double-Poisson bracket, where On is normal coordinate, Hc is classical Hamiltonian, is the classical correspondence of the invariant eigen-operator equation (2004 Phys. Lett. A. 321 75), we can find normal coordinates in harmonic crystal by virtue of the invariant eigen-operator method.

C 0-Rigidity of the Double Poisson Bracket

This paper is devoted to function theory on symplectic manifolds. We study a natural class of functionals involving the double Poisson brackets from the viewpoint of their robustness properties with respect to small perturbations in the uniform norm. We observe a hierarchy of such robustness properties. The methods involve Hofer's geometry on the symplectic side and Landau-Hadamard-Kolmogorov inequalities on the function-theoretic side.

Double Poisson Structures on Finite Dimensional Semi-Simple Algebras

We give a description of the bimodule of double derivations of a finite dimensional semi-simple algebra S and its double Schouten bracket in terms of a quiver. This description is used to determine which degree two monomials induce double Poisson brackets on S. In case S = ℂ⊕n , a criterion for any degree two element to give a double Poisson bracket is deduced. For S = ℂ⊕n and S′ = ℂ⊕m the induced Poisson bracket on the variety of isomorphism classes of semi-simple representations iss n (S * T) of the free product S * T is given.

Double Poisson cohomology of path algebras of quivers

In this note, we give a description of the graded Lie algebra of double derivations of a path algebra as a graded version of the necklace Lie algebra equipped with the Kontsevich bracket. Furthermore, we formally introduce the notion of double Poisson–Lichnerowicz cohomology for double Poisson algebras, and give some elementary properties. We introduce the notion of a linear double Poisson tensor on a quiver and show that it induces the structure of a finite-dimensional algebra on the vector spaces V v generated by the loops in the vertex v. We show that the Hochschild cohomology of the associative algebra can be recovered from the double Poisson cohomology. Then, we use the description of the graded necklace Lie algebra to determine the low-dimensional double Poisson–Lichnerowicz cohomology groups for three types of (linear and nonlinear) double Poisson brackets on the free algebra C 〈 x , y 〉 . This allows us to develop some useful techniques for the computation of the double Poisson–Lichnerowicz cohomology.

Stochastic Canonical Flows

There has been some opposition to the use of Markov processes in quantum mechanics based upon the fact that no quantum mechanical regression principle can be applied. It is argued here however that one may just as well conclude that there is no classical mechanical regression principal either. The dissipative component of a canonical stochastic flow is of the form of a double Poisson bracket 1/2 ll·, Fαr Fαr while the vector component of the noise is the Hamiltonian vector field l·, Fαr; thus the dissipative term is smoother than the noise coefficient. In contrast, the classical Langevin theory is based upon the Ornstein-Uhlenbeck process which has linear drift and constant noise coefficient. Preservation of the canonical structure is at odds with the approximation procedure by Ornstein-Uhlenbeck processes and this is the case in both classical and quantum situations.