Since their inception, the study of symplectic structures and the applications of symplectic techniques as well as their odddimensional contact geometric counterparts have benefited from a. Representation theory remains the method of choice for. On the homological algebra of relative symplectic geometry pomerleano, daniel, 2019. The junior hausdorff trimester program gives young mathematicians postdocs, junior faculty the opportunity to carry out collaborative research in symplectic geometry and representation theory. Symplectic approaches in geometric representation theory. For each with d 0 the equation 4 implies that the operator h is a differential on fock. By darbouxs theorem every symplectic form is locally di. Pdf we study the geometrical structure of the coadjoint orbits of an arbitrary complex or real lie. We will say that two symplectic spaces e and e00 are isomorphic if there exists a vector space isomorphism s. It covers the essentials of the analytical theory of punctured pseudoholomorphic curves, taking the opportunity to fill in gaps in the existing literature where necessary, and then gives detailed explanations of a few of the standard applications in contact. This thesis studies applications of algebraic microlocal geometry in the representation theory of padic groups and symplectic geometry.
Hamiltonian systems on linear symplectic spaces the lagrange and hamilton equations. Pdf structure of the coadjoint orbits of lie groups researchgate. Other examples come from geometric representation theory. The symplectic geometry of penrose rhombus tilings battaglia, fiammetta and prato, elisa, journal of symplectic geometry, 2008. A poisson algebra a is an associative algebra which is, at the same time. Recall that an embedding 6 is an immersion which is a homeomorphism onto its image. Roman bezrukavnikov, mit and higher school of economics moscow tom braden, u. To specify the trajectory of the object, one requires both the position q and the momentum p, which form a point p, q in the euclidean plane. An important bridge between symplectic geometry and representation theory is supplied by geometric quantization 8. Representation theory and geometry of symplectic resolutions. Request pdf orbits, symplectic structures and representation theory we introduce a general approach to unitary representations for all lie groups.
The eld grew from a sequence of signi cant discoveries of connections among dmodules, constructible sheaves, and representation theory, including the microlocal studies. Topology of symplectic torus actions with symplectic orbits. Then kfqa is a d altsymplectic aalgebra compatible with the commu tator, according to our differential formula for matrices over a field. Basic group and representation theory thomas willwacher february 27, 2014. Quantization, the orbit method, and unitary representations mit math. Physicists study the coulomb branch of the 3dimensional gauge theory associated with g,n, which is a hyperkaehler manifold, but have no rigorous mathematical definition. Ring theory from symplectic geometry sciencedirect. An introduction to symplectic geometry megumi harada, lecture notes by kaname hashimoto summer school, osaka, 2007 abstract the intent of this series of lectures is twofold. In case we denote v p xjej for the coordinates relative to this basis then v,g gets identi. Symplectic dynamics attheinstituteforadvancedstudy.
An introduction to the orbit method semantic scholar. Unitary representations, lie groups, coadjoint orbits, symplectic ge. Symplectic geometry 2nd semester 201220 symplectic geometry studies the geometry of manifolds equipped with a nondegenerate, closed 2form. For these reasons, we eschew the study of representations of lie algebras. Nearly all branches of mathematics are treated, including many parts of dynamical systems, representation theory, combinatorics, packing problems.
Newtons second law for a particle moving in euclidean threespace r3, under the in. Pdf a symplectic map between hyperbolic and complex. Introduction we fix once and for all a ground field f of characteristic zero. If you consult a major english dictionary, you are likely to. In section 1 we point out a fundamental bifurcation in the theory of poisson alge bras. Representation theory for symplectic 21graded lie algebras dragomir 2. In mathematical field of representation theory, a symplectic representation is a representation of a group or a lie algebra on a symplectic vector space v. Written for the 2015 algebraic geometry summer institute. Symplectic dynamics at the institute for advanced study. A symplectic structure is also called an almosthamiltonian structure, and if is closed, i. Dec 21, 2017 symplectic geometry and representation theory.
R3, fqvqisthe force, m is the mass of the particle, and a d2qdt2 is the acceleration assuming that we start in a postulated. If kuranishi structures worked, they belonged among the foundational techniques in symplectic geometry and would open up huge new areas of research. Commutative algebras the basic features of a theory of symplectic structures for commutative algebras are folklore. Symplectic geometry of frobenius structures 3 proof. Examples covered in these notes include distinguishing tight contact structures on the 3torus that are homotopic but not isomorphic lecture 10, and the nonexistence of symplectic. All talks will be in lecture room 204 of the maths building. Geometric representation theory lies in the crossroads of a number of di erent elds, including representation theory, algebraic geometry, number theory and symplectic geometry. Here x corresponds to the cotangent bundle tm in the. Symplectic theory of completely integrable hamiltonian systems. Letzterl journal of pure and applied algebra 125 1998 155190 157 definition.
Important developments in the early 1980s led to the emergence of new questions about these structures. Here, l is the centralizer of a subgroup contained in the cartan. We introduce a variant of the affine grassmannian steinberg variety, define convolution product on its. This is probably due to the fact that, unlike the lie groups u n and on, there does not exist a canonical orthogonal basis for the irreducible representations ofsp n. Our perspective on the theory originates from the work of sato. In this theory, one considers representations of the group algebra a cg of a. Enumerativegeometryandgeometricrepresentationtheory. Bulletin new series of the american mathematical society.
Topology of symplectic torus actions with symplectic orbits j. As a curiosity, note that two centuries ago the name symplectic geometry did not exist. Symplectic manifold is manifold m with lie algebra structure. Dec 03, 2016 this is the preliminary manuscript of a book on symplectic field theory based on a lecture course for phd students given in 201516.
Symplectic geometry has its origins in the hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic manifold. Representation theory philosophy the representation theory of g is secretly just the geometry of the springer resolution tgb n. Their study usually combines ideas from several areas of mathematics, notably partial differential equations, microlocal analysis, lie theory, symplectic geometry and representation theory. G itself is the group of symplectic automorphisms commuting with s g is lie algebra of global hamiltonian vector. That is symplectic deformations of symplectic structures do not produce new symplectic structures. Symplectic dynamics attheinstituteforadvancedstudy alvaropelayo,memberofschoolofmathematics. R3, fqvqisthe force, m is the mass of the particle. Since their inception, the study of symplectic structures and the applications of symplectic techniques as well as their odddimensional contact geometric counterparts have benefited from a strong extraneous motivation. Symplectic representation theory university of glasgow scotland. The proof is similar to the proof of the existence of an orthonormal basis in a euclidean.
Suppose mis an almost complex manifold of dimension 2nwhere n 2, and moreover, when mis closed, there exists a cohomology class a 2h2m such that 0 6 an 2h2nm. In representation theory, indindg h converts an hrepresentation to a grepresentation. Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds. Representation theory of classical compact lie groups.
Algebras and representation theory features carefully refereed papers relating, in its broadest sense, to the structure and representation theory of algebras, including lie algebras and superalgebras, rings of differential operators, group rings and algebras, calgebras and hopf algebras, with particular emphasis on quantum groups. Symplectic geometry is a branch of differential geometry studying symplectic manifolds and some generalizations. Article pdf available in journal of lie theory 221 july 2010 with 194 reads. A symplectic embedding is an embedding that preserves the symplectic structures, i. Orbits, symplectic structures and representation theory. We may well be witnessing the advent of a symplectic revolution in fundamental science. Publishes high quality papers on all aspects of symplectic geometry, with its deep roots in mathematics, going back to huygens study of optics and to the hamilton jacobi formulation of mechanics. Furthermore symplectic structures are local in time. While there appears to be great diversity among poisson structures which occur for commutative algebras, we prove that if a prime poisson algebra is not commu. The conference will concentrate on the representation theoretic and geometric questions connected to symplectic singularities and their resolutions. Ben webster northeastern representation theory of symplectic singularities march 22, 20 10 20. An introduction to symplectic topology through sheaf theory. Related aspects of the geometric and categorical representation theory will also be discussed.
The gromov width of coadjoint orbits of the symplectic group. In this talk we discuss how many such varieties exist. These structures find application in global analytical mechanics, since the cotangent bundle of any smooth manifold admits a canonical. Microlocal geometry, in a rough sense, provides is with analytic objects that arise as solutions to a very broad class of operator equations. The nondegeneracy of the symplectic form can be restated as follows is nondegenerate is a monomorphism e. This is the preliminary manuscript of a book on symplectic field theory based on a lecture course for phd students given in 201516.
The orbit method in representation theory seeks to attach to a coadjoint orbit x. In particular, we supply details of some simple results from the axiomatic theory, including a severalline proof of the genus 0 virasoro constraints, but merely quote and refer to the. Instead the technique languished amid uncertainty in the mathematical community about whether fukayas approach worked as completely as he said it did. A basic example of a symplectic vector space is v u.
The role of orthogonal and symplectic clifford algebras in quantum field theory matej pavsic j. Let n be a representation of a complex reductive group g. Conical symplectic varieties now play an important role in algebraic geometry and geometric representation theory. An underlying feature is a study of sympletic manifolds x2n i. Symplectic geometry of rationally connected threefolds tian, zhiyu, duke mathematical journal, 2012. We introduce a general approach to unitary representations for all lie groups. Wongs equations in poisson geometry maspfuhl, oliver, journal of symplectic geometry, 2004.
In particular symplectic vector spaces have even dimension. Journal of symplectic geometry international press. Applications of algebraic microlocal analysis in symplectic. Then kfqa is a d alt symplectic aalgebra compatible with the commu tator, according to our differential formula for matrices over a field.
Symplectic structures in geometry, algebra and dynamics. Mosers lemma and local triviality of symplectic differential geometry 17 2. Pelayoyz abstract we give a concise overview of the classi. The given cartan subalgebra h is always equal to the space x. Symplectic geometry arose from the study of classical mechanics and an example of a symplectic structure is the motion of an object in one dimension. Symplectic structures have their origin in the study of classical mechanics. Here is a fundamental question in symplectic topology. We try to keep the text introductory and nontechnical. In this paper we carry out this idea on the semisimple coadjoint orbit m gl and e glss. The polar decomposition allows one to study many structural problems about. Representation theory complex algebraic geometry fourier theory. To equip the manifold gb with a symplectic structure, fix.
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