Structure of symplectic invariant lie subalgebras of. Symplectic and contact lie algebras with anapplication to. The symplectic group acts by conjugation on itself and on its lie algebra. In this chapter, we present linear symplectic algebra. In the following we shall only consider lie groups consisting of matrices. With the help of this result, we construct from any nitedimensional lie algebra g a canonical 1parameter family of lie 2algebras gwhich reduces to g at 0. Hence the problem to find all the symplectic homogeneous spaces of a given lie group is to find all regular leftinvariant closed 2forms 0 on g. An abelian quotient of the symplectic derivation lie. An abelian quotient of the symplectic derivation lie algebra.
Associative algebras, lie algebras, and bilinear forms. Notes on lie algebras cornell department of mathematics. Here k is a maximal compact subgroup of a semisimple lie group g, and p the fixedpointsubspace of a cartan involution on the semisimple lie algebra g. It is possible to show that evey element of w is induced by an automorphism of g that carries h to itself. Unlike the orthogonal and unitary groups, the symplectic group sp2n,r is noncompact.
In cartans classification of the simple lie algebras, the lie algebra of the complex group sp2n, c is denoted c n, and spn is the compact real form of sp2n, c. The approach is worked out in detail for symplectic integrators based on rungekuttamunthekaas methods and crouchgrossman methods. Thanks for contributing an answer to mathematics stack exchange. In physics, one encounters not only finite or discrete symmetry groups, but. It turns out that all symplectic groups corresponding to symplectic. In chapter 3 the units in a nite dimensional algebra over r or c are studied as a source of matrix groups using the. One writes spn, r and spn, c for the symplectic lie algebras of r. In the case of symplectic groups, the associative algebra is a weyl algebra of polynomial coe cient di erential. Symplectic and contact lie algebras it is natural to consider leftinvariant structures on lie groups.
Onishchik, lie transformation groups encyclopaedia of mathematical sciences, volume 20, springer 1993. An introduction to lie groups and symplectic geometry aseriesofnine lectures on lie groups and symplectic geometry delivered at the regional geometry institute in park city, utah, 24 june20 july 1991. Quaternions, cli ord algebras and some associated groups 37 1. The special cases of su2 and sl 2c and their relationships with so3 and the lorentz group are studied in detail. Next, we study the symplectic group and its lie algebra and prove the symplectic eigenvalue theorem. The course note really was designed to be an introduction, aimed at an audience of students who were familiar with basic constructions in differential topology and rudimentary differential geometry, who wanted to get a feel for lie groups and symplectic geometry. The real rank of the corresponding lie algebra, and hence of the lie group sp2n, f, is n. Moreover, we give a system of representatives of all nonabelian metric symplectic lie algebras of dimension less than ten. Let gbe a compact lie group, g its lie algebra and g the dual of g.
But avoid asking for help, clarification, or responding to other answers. Introduce matrix lie group sp2 and its matrix lie algebra sp2 consider our goal. Using the connection between th e gradients of a function with. This symplectic lie group will be called the symplectic reduction of g. In both cases, we show that it is possible to obtain symplectic integrators of arbitrarily high order by this approach. Let n be the fundamental group of a closed oriented surface of genus 1 and denote by 9 the category of symplectic outrspaces as defined above. 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. Discussion with a view towards chernweil theory is in chapter iv in. A linear lie group, or matrix lie group, is a submanifold of mn. Then we consider the action of the authomorphism group of the lie algebra on the space of symplectic structures and we get the classi. This lie algebra is a quite fundamental object, that crops up at. We shall sometimes call the injection itself a lie subgroup. In this lecture we do for the symplectic lie algebras exactly what we did for the special linear ones in 15. The symplectic 2graded lie algebras can be defined over any field of characteristic zero and they are semisimple.
The symplectic nature of fundamental groups of surfaces. Associative algebras, lie algebras, and bilinear forms november 4, 2015 1 introduction the most basic and important example of a lie group is the group gln. The standard topology gives this group the di erentiable structure of s3. Suppose that g is the lie algebra of a lie group g. An introduction to matrix groups and their applications. These are the lecture notes for a short course entitled introduction to lie groups and symplectic geometry that i gave at the 1991 regional geometry institute at park city, utah starting on 24 june and ending on 11 july. This material is fairly standard and explained in many places, two references with many more details are 1 and 2. This is essential to show that the connected lie subgroup hg corresponding to the lie subalgebra ijg is closed in g. If k k is a field and \mathcalc svect is the category of super vector spaces over k k, then a lie algebra object is a super lie algebra. We study the structure of the symplectic invariant part h g, 1 sp of the lie algebra h g, 1 consisting of symplectic derivations of the free lie algebra generated by the rational homology group of a closed oriented surface.
Any compact simple lie group ghas a canonical 2plectic structure, so it is natural to wonder what lie 2algebra this example yields. In mathematics, the name symplectic group can refer to two different, but closely related. Anintroductiontoliegroups andsymplecticgeometry a series of nine lectures on lie groups and symplectic geometry delivered at the regional geometry institute in park city, utah, 24 june20 july 1991. Lie group integrators on a lie group g is presented. The authors develop the structure theory of symplectic lie groups based on the study of their isotropic normal subgroups. In 4 we use the graded lie algebra associated to give a new and very simple construction of the metaplectic representation of the 2sheeted covering group mpm. R which is also a lie group, with group structure the matrix multiplication. The symplectic group \sp2n\ symplectic matrices form a lie group called the symplectic group \sp2n\,\ whose lie algebra is the set of hamiltonian matrices, matrices of the form \ js\ where \s\ is symmetric. We start with a discussion of the elementary properties of symplectic vector spaces, the various types of their subspaces and linear symplectic reduction.
A series of nine lectures on lie groups and symplectic geometry delivered at the regional geometry institute in. The lie algebra su2 is the real span of the three tracefree antihermitian matrices x 1 p 1. An introduction to lie groups and symplectic geometry. Every symplectic matrix has determinant 1, and the 2n. This is essential to show that the connected lie subgroup hg corresponding to the lie sub algebra ijg is closed in g. An introduction peter woit department of mathematics, columbia university. The indefinite special orthogonal groups are the groups.
Symplectic representation theory and the weyl algebra in. Oct 29, 2016 the symplectic derivation lie algebra h g,1. Jan 18, 2016 answer 1 of 2 for every even dimension 2n, the symplectic group sp2n is the group of 2n. We will make use of the following two facts about algebraic groups including the symplectic group. This group is very closely related to the associative algebra mn. In fact, if g is a complex lie group with lie algebra g, and t is. In the first part, the authors show that every symplectic lie group admits a sequence of subsequent symplectic reductions to a unique irreducible symplectic lie group. General abstract perspective lie algebras are equivalently groups in infinitesimal geometry. Eckhard meinrenken, lie groups and lie algebas, lecture notes 2010. Symplectic geometry is a branch of differential geometry studying symplectic manifolds and some generalizations. In chapter 3 the units in a nite dimensional algebra over r or c are studied as a source of matrix groups using the reduced regular representation. Special symplectic lie groups and hypersymplectic lie groups. Thus every nearidentity symplectic matrix can be obtained as the exponential of a hamiltonian matrix and corresponds to the.
The basic example of a lie algebra arises from using the commutator in an associative algebra, so x. Note that when we refer to the compact symplectic group it is implied that we are talking about the collection of compact symplectic groups, indexed by their dimension n. Lie groups, convexity and symplectic structure science. Mat 4451196 complex symplectic lie algebras let n be an. We will rst give a rather naive intuitive idea, which we will later make precise. Finding the lie algebra of the symplectic lie group.
The course really was designed to be an introduction, aimed at an audience of stu. R, the set of all n nreal matrices, is di eomorphic to rn2. If a m nk, then the space of n nmatrices has the structure of a lie algebra with lie bracket x. The momentum map, symplectic reduction and an introduction to. In fact, if g is a complex lie group with lie algebra g, and t is the cartan subgroup of g that corresponds to h that. Yes, i would say this picture is very well understood. Can be denoted sp2n, f, although other notations are also. Gis a subgroup that is also an embedded submanifold. Every element in the symplectic group sp2n can be written in the form x x sx u, where x sis semisimple, x uis unipotent, and x sand x ucommute. For in this case they may be treated as structures on lie algebras.