Volume 57, Issue 5, May 2016
 ARTICLES

 Representation Theory and Algebraic Methods

Classification of linearly compact simple NambuPoisson algebras
View Description Hide DescriptionWe introduce the notion of a universal odd generalized Poisson superalgebra associated with an associative algebra A, by generalizing a construction made in the work of De Sole and Kac [Jpn. J. Math. 8, 1–145 (2013)]. By making use of this notion we give a complete classification of simple linearly compact (generalized) nNambuPoisson algebras over an algebraically closed field of characteristic zero.
 Relativistic Quantum Mechanics, Quantum Field Theory, Quantum Gravity, and String Theory

A Lie based 4–dimensional higher Chern–Simons theory
View Description Hide DescriptionWe present and study a model of 4–dimensional higher ChernSimons theory, special Chern–Simons (SCS) theory, instances of which have appeared in the string literature, whose symmetry is encoded in a skeletal semistrict Lie 2–algebra constructed from a compact Lie group with non discrete center. The field content of SCS theory consists of a Lie valued 2–connection coupled to a background closed 3–form. SCS theory enjoys a large gauge and gauge for gauge symmetry organized in an infinite dimensional strict Lie 2–group. The partition function of SCS theory is simply related to that of a topological gauge theory localizing on flat connections with degree 3 second characteristic class determined by the background 3–form. Finally, SCS theory is related to a 3–dimensional special gauge theory whose 2–connection space has a natural symplectic structure with respect to which the 1–gauge transformation action is Hamiltonian, the 2–curvature map acting as moment map.
 Dynamical Systems

Factorialtype Schur functions, orthogonal rational functions, and discrete dressing chains
View Description Hide DescriptionA relationship between orthogonal rational functions and discrete integrable systems is studied by an approach based on Schurtype symmetric functions. A system of orthogonal rational functions is constructed using a multiparameter deformation of the Schur functions. Spectral equations for the orthogonal rational functions are derived by using properties of the Schurtype symmetric functions. The compatibility condition of the spectral equations induces a discrete dressing chain which is a Todatype discrete integrable system describing dressing transformations for orthogonal rational functions.

Ergodicity of nonuniformly expanding transitive group (or semigroup) actions
View Description Hide DescriptionIn this paper, we prove that every nonuniformly expanding transitive group (or semigroup) action of C^{1+α} conformal local diffeomorphisms of a compact manifold is ergodic with respect to the Lebesgue measure.

BiHamiltonian structure of the bidimensional superintegrable nonlinear isotonic oscillator
View Description Hide DescriptionThe higherorder superintegrability of the twodimensional isotonic oscillator (noncentral oscillator with inversely quadratic nonlinearities also known as caged anisotropic oscillator) with rational ratio of frequencies is directly related with the existence of some complex functions with interesting Poisson bracket properties. First the properties of these functions are studied and then it is proved that these complex functions determine the existence of a biHamiltonian complex structure. In the second part several real symplectic structures are obtained and the properties of the recursion operators are studied.
 Methods of Mathematical Physics

On the canonical forms of the multidimensional averaged Poisson brackets
View Description Hide DescriptionWe consider here special Poisson brackets given by the “averaging” of local multidimensional Poisson brackets in the Whitham method. For the brackets of this kind it is natural to ask about their canonical forms, which can be obtained after transformations preserving the “physical meaning” of the field variables. We show here that the averaged bracket can always be written in the canonical form after a transformation of “Hydrodynamic Type” in the case of absence of annihilators of initial bracket. However, in general case the situation is more complicated. As we show here, in more general case the averaged bracket can be transformed to a “pseudocanonical” form under some special (“physical”) requirements on the initial bracket.

Optimal space of linear classical observables for Maxwell kforms via spacelike and timelike compact de Rham cohomologies
View Description Hide DescriptionBeing motivated by open questions in gauge field theories, we consider nonstandard de Rham cohomology groups for timelike compact and spacelike compact support systems. These cohomology groups are shown to be isomorphic respectively to the usual de Rham cohomology of a spacelike Cauchy surface and its counterpart with compact support. Furthermore, an analog of the usual Poincaré duality for de Rham cohomology is shown to hold for the case with nonstandard supports as well. We apply these results to find optimal spaces of linear observables for analogs of arbitrary degree k of both the vector potential and the Faraday tensor. The term optimal has to be intended in the following sense: The spaces of linear observables we consider distinguish between different configurations; in addition to that, there are no redundant observables. This last point in particular heavily relies on the analog of Poincaré duality for the new cohomology groups.