Integrable systems associated with elliptic algebras

We construct new integrable systems (IS), both classical and quantum, associated with elliptic algebras. Our constructions are based both on a construction of commuting families in skew fields and on properties of the elliptic algebras and their representations. We give some examples showing how these IS are related to previously studied systems

Integer solutions of integral inequalities and H-invariant Jacobian Poisson structures

We study the Jacobian Poisson structures in any dimension invariant with respect to the discrete Heisenberg group. The classification problem is related to the discrete volume of suitable solids. Particular attention is given to dimension 3 whose simplest example is the Artin-Schelter-Tate Poisson tensors

On the Higher Poisson Structures of the Camassa–Holm Hierarchy

We find a generating series for the higher Poisson structures of the nonlocal Camassa–Holm hierarchy, following the method used by Enriques, Orlov, and third author for the KdV case

Compatible Poisson brackets, quadratic Poisson algebras and classical r-matrices

We show that for a general quadratic Poisson bracket it is possible to define a lot of associated linear Poisson brackets: linearizations of the initial bracket in the neighborhood of special points. We prove that the constructed linear Poisson brackets are always compatible with the initial quadratic Poisson bracket. We apply the obtained results to the cases of the standard quadratic r-matrix bracket and to classical “twisted reflection algebra” brackets. In the first case we obtain that there exists only one non-equivalent linearization: the standard linear r-matrix bracket and recover well-known result that the standard quadratic and linear r-matrix brackets are compatible.We show that there are a lot of non-equivalent linearizations of the classical twisted Reflection Equation Algebra bracket and all of them are compatible with the initial quadratic bracket

An algebraic index theorem for Poisson manifolds

The formality theorem for Hochschild chains of the algebra of functions on a smooth manifold gives us a version of the trace density map from the zeroth Hochschild homology of a deformation quantization algebra to the zeroth Poisson homology. We propose a version of the algebraic index theorem for a Poisson manifold which is based on this trace density map

Dedication to Gerardus F. Helminck

International audienc

Algebraic properties of Manin matrices II: q-analogues and integrable systems

We study a natural q-analogue of a class of matrices with non-commutative entries, which were first considered by Yu.I. Manin in 1988 in relation with quantum group theory, (called Manin matrices in [5]). We call these q-analogues q-Manin matrices  . These matrices are defined, in the 2×22×2 case by the following relations among their matrix entries: M21M12=qM12M21, M22M12 = qM12M22 [M11,M22]=q-1M21M12-qM12M21 They were already considered in the literature, especially in connection with the q-MacMahon master theorem [10], and the q-Sylvester identities [22]. The main aim of the present paper is to give a full list and detailed proofs of the algebraic properties of q-Manin matrices known up to the moment and, in particular, to show that most of the basic theorems of linear algebras (e.g., Jacobi ratio theorems, Schur complement, the Cayley–Hamilton theorem and so on and so forth) have a straightforward counterpart for such a class of matrices. We also show how q-Manin matrices fit within the theory of quasideterminants of Gelfand–Retakh and collaborators (see, e.g., [11]). We frame our definitions within the tensorial approach to non-commutative matrices of the Leningrad school in the last sections. We finally discuss how the notion of q-Manin matrix is related to theory of Quantum Integrable Systems

Generalizations of Poisson Structures Related to Rational Gaudin Model

The Poisson structure arising in the Hamiltonian approach to the rational Gaudin model looks very similar to the so-called modified Reflection Equation Algebra.  Motivated by this analogy, we realize a braiding of the mentioned Poisson structure, i.e. we introduce a ”braided Poisson” algebra associated with an involutive solution to the quantum Yang-Baxter equation. Also, we exhibit another generalization of the Gaudin type Poisson structure by replacing the first derivative in the current parameter, entering the so-called local form of this structure, by a higher order derivative.  Finally, we introduce a structure, which combines both generalizations.  Some commutative families in the corresponding braided Poisson algebra are found

Classical elliptic current algebras. I

In this paper we discuss classical elliptic current algebras and show that there are two different choices of commutative test function algebras related to a complex torus leading totwo different elliptic current algebras. Quantization of these classical current algebras gives rise to two classes of quantized dynamical quasi-Hopf current algebras studied by Enriquez, Felder and Rubtsov and by Arnaudon, Buffenoir, Ragoucy, Roche, Jimbo, Konno, Odake and Shiraishi