Banach Poisson-Lie groups and integrable systems

Wider research context: One of ever-present ideas in mathematics is the idea of infinity. At each step one encounters things which are infinitely many starting from numbers, sets, functions and so on. Having an infinite amount of possible numbers which are needed to describe a real world problem is a typical situation. One says that a variable can assume infinitely many values. This setting is sufficient to describe a position of a particle on a line or a plane pendulum. In order to have more freedom of motion though one needs to take several variables of that type. It allows us for example to describe a motion of a planet in the Solar System (with 6 variables) or a rigid body (with 12 variables). These problems possess usually extra structures, for example geometrical or differential structures which are crucial in the process of finding solutions and describing their behavior. The typical mathematical concept needed in this framework is a smooth n-dimensional manifold. It can be seen as a generalization of the notion of a surface in space to arbitrary dimension. Additionally one uses the notion of a Lie group to describe symmetries of the problem. This approach however is not sufficient to formulate more complicated problems present in contemporary science. For example problems in quantum mechanics or hydrodynamics require an infinite number of variables and new structures are needed. We say that spaces where these systems live are infinite dimensional. Instead of geometry, one usually employs functional analysis which is a branch of mathematics dealing with such spaces. Approaches: In the last years there is a trend to apply the methods of functional analysis to the geometry in order to create a rigorous setting for Hamiltonian mechanics on infinite dimensional manifolds. The aim is to create a mathematically consistent framework in which both quantum mechanics and integrable systems can be studied. One of the topic of research is related to the so-called Poisson structures on infinite dimensional manifolds which are a tool which allows an elegant construction of equations and integrals of motion for a system. However taking a tool out of the world of finite dimensional geometry and attempting to apply it in the world of infinite dimensional geometry is often very tricky. Straightforward approach usually fails and unexpected problems present themselves. Understanding the possible pathologies occurring in the infinite- dimensional context is a challenge, and finding good non-trivial examples and counter-examples to the expected situation we are used to in the finite-dimensional context is a big step forward. Hypotheses/Research/Objectives: One of the first systems to benefit from geometrical approach was the Kortewegde Vries equation describing solitary waves traveling without dissipation in the shallow water (so-called solitons). The geometrical object used by Segal and Wilson in 1985 for the description of this system is an infinite- dimensional manifold called the restricted Grassmannian. Understanding the Hamiltonian structure of this infinite-dimensional manifold is at the heart of our research. Hamiltonian mechanics is part of Poisson geometry. The natural action of Lie groups on phase spaces of classical systems leads to the notion of Poisson--Lie groups. For systems with an infinite number of degree of freedom, it is natural to study the concept of Poisson--Lie groups in the framework of Banach geometry. Structures related to the restricted Grassmannian are key examples in the understanding of this theory. Originality: The theory of Banach Poisson--Lie groups that we intend to explore is a new concept in the context of infinite-dimensional geometry. Extension to the Fréchet context will allow to study Hamiltonian systems coming from gauge theories. The analysis of Poisson structures and new integrable systems associated to the restricted Grassmannian using modern geometrical tools is part of the project. This study will allow to increase the understanding of Poisson geometry in the infinite-dimensional setting, and find new links to other known problems. Primary researchers involved: Alice Barbora Tumpach (WPI, Vienna) and Tomasz Golinski (University of Bialystok, Poland)

Wider research context : In the last years there is a trend to apply the methods of functional analysis to the geometry in order to create a rigorous setting for Hamiltonian mechanics on infinite dimensional manifolds. The aim is to create a mathematically consistent framework in which both quantum mechanics and integrable systems can be studied. Approaches : The research lies in the field of mathematical physics and it employs both methods of dier- ential geometry and functional analysis. The focus is on infinite-dimensional counterparts of geometric structures which lie at the foundation of classical mechanics. Hypotheses/Research/Objectives : Hamiltonian mechanics is part of Poisson geometry. The natural action of Lie groups on phase spaces of classical systems leads to the notion of Poisson-Lie groups. For systems with an infinite number of degree of freedom, it is natural to study the concept of Poisson-Lie groups in the framework Banach geometry. Structures related to the restricted Grassmannian are key examples in the understanding of this theory. Key Results: We developped the theorey of Banach Poisson-Lie groups and their link to integrable systems. Banach version of notions related to the theory of R-matrices, Rota-Baxter algebras and Nijenhuis operators, in particular in relation with Banach Poisson-Lie groups, are at the core of this research. The notion of Banach Lie-Poisson space with respect to an arbitrary duality pairing is crucial for the equations of motion to make sense. In the presence of an invariant non-degenerate pairing on a Banach Lie algebra, these equations of motion can be written as Lax equations. We prove a version of the Adler- Kostant-Symes theorem adapted to R-matrices on infinite-dimensional Banach algebras. This theorem is then applied to Manin triples of Banach Lie algebras in Schatten classes related to Iwasawa decompositions of the corresponding groups. The semi-infinite Toda lattice is also included in this Banach theory.