Algebraic structures in integrable systems, moscow, december 37, 2012 december 3, 2012 mon. Denote by bun g bun gx the moduli stack of gbundles on x. A dynamical system is said to admit a lax representationwith spectral parameter. On the algebraic structure of rational discrete dynamical. Mark adler is a professor of mathematics at brandeis university. This introductory section revisits ideas met in the early part of analysis i and in linear algebra i, to set the scene and provide.
Algebraic completely integrable hamiltonian systems 123 1. In fact, the magnetic deformation of the ising model highlights its underlying e8 structure as well as the thermal deformation of the tim and of the tpm, which highlights respectively the e7 and the e6 structures of these models. How to solve actual mechanical integrable systems using this. We attempt to present this beautiful interplay between the abstract mathematical objects in one hand and the physical. A poisson structure on the dual of a lie algebra appendix 2. Therefore, many mathematicians and physicists are interested in. These are nonempty sets equipped with arbitrary operations and relations, ful. In particular, they are integrable systems in liouville sense, i. This article suggests a series of problems related to various algebraic and geometric aspects of integrability. Such quivers were completely classified by fordy and marsh, who characterised them in terms of the skewsymmetric matrix that defines the quiver. Boundary conditions and defects for these theories are described via algebraic constructions from these quantum groups or quantum affine algebras. Integrable hamiltonian systems on affine poisson varieties 43 3. Hamiltonian structures and their functional reductions are discussed in great detail in the examples of operator kortewegde vries and benneykaup dynamical systems.
After the introducing natural algebraic structures related to these examples we will pass to classical integrable systems. Let us take a closer look at nullary operations and clarify the term constants. Let x be a connected smooth projective curve over c of genus g1, ga semisimple group over c, g the lie algebra of g. Theme in recent years, there has been much progress and development in the field of algebraic geometry and integrable systems. The work is devoted to recent investigations of the laxsato compatible linear vector field equations, especially to the related lie algebraic structures and integrability properties of a very interesting class of nonlinear dynamical systems called the dispersionless heavenly type equations, which were initiated by plebanski and later analyzed in a series of articles. Lie algebraic and differential algebraic structures which proved to be algorithmically effective for studying lax integrability of nonlinear dynamical systems on func tional manifolds. We discuss selected topics on the topology of moduli spaces of curves and maps, emphasizing their relationwith gromov. Algebraic aspects of integrable systems springerlink. The starting point for the algebrogeometric integration method is lax representation. This is a set of notes i developed for an elearning course in algebraic structures offered by maseno, university in western kenya. After a short introductory chapter consisting mainly of reminders about such topics as functions, equivalence relations. Symmetry approach to integrability and nonassociative.
Mg which has a natural structure of an algebraic poisson v ariety. Symmetry approach to integrability and nonassociative algebraic structures. The integrability structure is shown to be geometrically encoded. Algebraic structures and discrete mathematics class notes for. In this paper, we discuss an interaction between complex geometry and integrable systems. The phase space of the simplest mechanical systems has the structure of a sym plectic manifold. In this case we say that this system possesses a bi.
This book deals with some aspects of this relation, the main focus being on the algebraic geometry of the level manifolds of integrable systems and the construction of integrable systems, starting from algebraic geometric data. The eigenvector mapping and linearising flows appendix 4. Constructions of integrable hamiltonian systems 53 3. Physical meaning of abstract mathematical notions like universal rmatrix, quantized algebras, sklyanin algebra, braided algebra, hopf algebra etc. Pdf the algebraic structures underlying integrability. We consider the general liealgebraic scheme of construction of integrable nonlinear dynamical systems on extended functional manifolds. Algebraic structures related to integrable differential. We are going to see other discrete structures, and learn to recognize them when the arise at the very heart of problems. We modify the algebraic structure of a poisson bialgebra by considering the deformed coproduct. In addition, we concentrate on the speed at which these functions grow, their smoothness and the strength of their convergence to zero. The theme of this conference is to survey various aspects of mathematical subjects in the field, such as minimal model theory in higher dimensional algebraic geometry, moduli spaces of connections and higgs bundles, character varieties, geometry of riemannhilbert. These modifications lead to the quantum groups and provide new classes of completely integrable hamiltonian systems. Discrete integrable systems also arise in algebraic combinatorics, as iterations of mutations in cluster algebras.
The survey is devoted to algebraic structures related to integrable odes and evolution pdes. Depending on the degree a participant of the lecture algebraic structures is aiming at he will take this course in his. Kharlamov mikhail some integrable systems with algebraic separation of variables we consider integrable systems of mechanical origin with a compact configuration space. These are the notes of the secondyear course algebraic structures i at queen mary, university of london, as i taught it in the second semester 20052006. It is suitable for phd students and can serve as an introduction to classical. Liealgebraic structure of laxsato integrable heavenly. He was the principal organizer of the 2009 6months programme on discrete integrable systems dis at the isaac newton institute, and a royal society leverhulme trust senior research fellow in 2011. The author presents a unified approach to describe the algebraic properties of integrable systems, identifying an abstract and elementary operator structure underlying integrability and showing.
Maybe this reference integrable systems in the realm of algebraic geometry vanhaecke, 2001 may be of use. In turn, the combinatorics of grassmanians and their associated cluster algebra structure has led to a new understanding of the scattering of kp solitons. Algebraic structures and discrete mathematics class notes. The structure of the complex level manifolds tf4 83 3. New examples of integrable systems, which have been discovered, are based on the lax representation of the equations of motion.
A description of lax representations is given in terms of vector space decomposition of loop algebras into a direct sum of taylor series and a complementary subalgebra. If time permits, i will explain expected relations to representation theory of. By making precise the algebraic structures that for example the integers satisfy, we can figure out what makes our favourite facts about the integers true, and easily see where those same facts hold true. A graph is a good example of discrete object, or structure in opposition to a continuous object like a curve. The idea is to introduce the key concepts of algebraic structures without assuming much background in higher mathematics. Completely integrable hamiltonian systems concerned with semisimple lie algebras. Complex billiard hamiltonian systems and nonlinear waves. Some algebraic and analytic structures in integrable systems. Distinguished by a rich underlying lie algebraic structure these models have in recent years attracted much attention not only as test laboratories for nonperturbative methods in quantum field theory but also in the context of offcritical models. Lie algebraic methods in integrable systems research notes. Topological and algebraic structure of linear problems associated with completely integrable systems chudnovsky, d. Integrable systems and algebraic curves springerlink. For the high dimensional systems, fokas and santini, introduced a generalized recursion operator to construct the hierarchies of equations, by extending the notions of strong and hereditary symmetries, they have studied the algebraic properties of integrable systems where kp system as an example. In mathematics, an algebraic structure on a set a called the underlying set, carrier set or domain is a collection of operations on a of finite arity, together with a finite set of identities, called axioms of the structure that these operations must satisfy.
The author presents a unified approach to describe the algebraic properties of integrable systems, identifying an abstract and elementary operator structure underlying integrability and showing that well known and novel examples of integrable equations are generated by different concrete realisations of this structure. Algebraic structures abstract algebra is the study of algebraic structures. The fundamental problem in mathematical physics is. Abstract citations 1 references 9 coreads similar papers. Topological and algebraic structure of linear problems. Algebraic geometry and stability for integrable systems. Hidden symmetries of lax integrable nonlinear systems.
Introduction many of the algebraic aspects of rational discrete systems have already been investigated, especially in viewof their integrability. As is wellknown, many finitedimensional integrable systems can be explicitly solved by means of algebraic geometry. Cartans lemma associates the lie algebra valued 1form, the cartan connection, to the levi. Basic ideas behind the construction of integrable ultralocal and nonultralocal quantum models are explored by exploiting the underlying algebraic structures. As an important example of applying these recently devised techniques, a new generalized riemann type hy drodynamic system is studied by means of a novel com. Introduction to groups, rings and fields ht and tt 2011 h. Before poincare, there seemed to be a kind of general hope in the air that every autonomous hamiltonian system was integrable. Request pdf on jan 5, 2007, nicolai reshetikhin and others published some algebraic and analytic structures in integrable systems find, read and cite all the research you need on researchgate. Integrable hamiltonian systems on affine poisson varieties 17 1. Hamiltonian formulation of nonlinear integrable dynamical systems in multidimensions. The algebraic structures underlying integrability iopscience. Issues related to the quantization of this algebraic structure and the formulation of quantum complex sinegordon on those lines are introduced and discussed. A numberof these aspects are extensionstothe discrete. Compatible integrable hamittonian systems and multihamiltonian systems.
Local and nonlocal poisson vertex algebras and applications to the theory of integrable systems lecture 1 a. Basic examples are heisenberg spin chain, quantum bose gas with interaction, and the toda system. Topological and conformal field theories and integrable systems can be described by the algebraic structures of quantum groups and quantum affine algebras. Such a system is called irreducible if it does not have any continuous symmetry group and therefore does not reduce, globally, to a family of systems with two degrees of freedom. With the above theorem in hand, the algebraic structure of. After a short introductory chapter consisting mainly of reminders about such topics as functions, equivalence relations, matrices, polynomials and permuta. On the other hand, the most fundamental exactly integrable systems often have applications in theoretical physics. Local and nonlocal poisson vertex algebras and applications to the theory of integrable systems lecture 1. Algebrogeometric concepts which have become useful in this theory, namely algebraic curves, moduli spaces of algebraic curves and jacobians. Lax pairs and the p operator consider a classical dynamical system with a finite number of degrees of freedom such as the toda systems 1. Such a structure consists of a set together with one or more binary operations, which are required to satisfy certain axioms. We are also going to see how to deal with such structures. A description of lax representations is given in terms of vector space decomposition of loop algebras into direct sum of taylor series and a complementary subalgebras.
Algebraic geometry and integrable systems, kobe 2016. The associated nonlinear recurrences are equivalent to birational maps, and we. Liealgebraic structure of integrable nonlinear dynamical. Pdf on algebraic structures of dynamical systems researchgate. Due to this the audience of the course is rather inhomogeneous. Integrable systems are related to algebraic geometry in many different ways. Lie algebraic structures in integrable models, affine toda. In addition, the operations may be required to satisfy a set of equations identities. Integrable systems in symplectic geometry vrije universiteit. Since a rigorous approach requires some sort of introduction, we.
In the context of universal algebra, the set a with this structure is called an algebra, while, in other contexts, it is somewhat. Symmetry approach to integrability and nonassociative algebraic. Integrable hamiltonian systems on other spaces 60 4. Finally, we apply such a structure to the coupled integrable. Integrable hamiltonian systems and symmetric products of curves 6. Integrable systems and the algebraicgeometric spectral theory of linear. The sign shift pattern of this deformation has the signature of the twisted boundary dynamical algebra. Algebraic structures in integrable systems, moscow, december 37, 2012. The algebraic structures of isospectral lax operators and applications to integrable equations article pdf available in journal of physics a general physics 2520. These systems possess a structure like p but are not completely integrable. Hamiltonian methods and algebraic structures for integrable.
Integrable systems in the realm of algebraic geometry. Lie algebraic methods in integrable systems chapman. They reflect some recent developments in the theory of finitedimensional integrable systems such as bipoisson linear algebra, jordankronecker invariants of finite dimensional lie algebras, the interplay between singularities of lagrangian fibrations and compatible poisson. Completely integrable systems, euclidean lie algebras, and. In this text, we focus on operations of arity 2, 1, and 0. We consider nonlinear recurrences generated from cluster mutations applied to quivers that have the property of being cluster mutationperiodic with period 1. An algebra or an algebraic structure is a set a, the carrier, together with a set of operations on a. Its also an area where most of the real payoff comes later. We modify the algebraic structure of a poisson bialgebra by considering the deformed coproduct and deformations of an algebra e2. Complex curves, real curves and their jacobians appendix. We consider the general lie algebraic scheme of construction of integrable nonlinear dynamical systems on extended functional manifolds.
Discrete integrable systems and poisson algebras from. We obtain an explicit expression for consistent poisson structures and write explicitly nonlinear equations generated by the spectrum of a periodic problem for an operator of laxtype representation. Modified algebraic structures for some completely integrable. Integrable systems and complex geometry springerlink. On the algebraic structure of rational discrete dynamical systems cm. Content the lectures will start from examples of quantum integrable systems.