The lie algebra k t ekis an in nitesimal version of the lie group k. Mat 4451196 complex symplectic lie algebras let n be an integer greater than or equal to 2. Suppose that g is the lie algebra of a lie group g. Jan 18, 2016 answer 1 of 2 for every even dimension 2n, the symplectic group sp2n is the group of 2n. For an overview with references, based partly on steinbergs tata lectures, see chapter 3 of my 1995 ams book conjugacy classes in semisimple algebraic groups msn, book, and chapter. Since all symplectic matrices have determinant 1, the symplectic group is a subgroup of the special linear group sl2n, f more abstractly, the symplectic group can be. A basis for each finitedimensional irreducible representation of the symplectic lie algebra sp2n is constructed. These groups are themselves infinitely connected, and possess a common universal cover sp2,r. John lees book question about symplectic group math stack. Symplectic group project gutenberg selfpublishing ebooks. 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. Indeed, symplectic methods are key ingredients in the study of dynamical systems, differential equations, algebraic geometry, topology, mathematical physics and representations of lie groups.
Induction and reduction for systems with symmetry, symplectic geometry and topology, geometric quantization, the weinstein conjecture, poisson algebra and geometry, dirac structures, deformations for lie group actions, kahler geometry of moduli spaces, theory and. In the first part we show that every symplectic lie group admits a sequence of subsequent symplectic reductions to a unique irreducible symplectic lie group. A 2 2symplectic manifold encodes and is encoded by the structure of a courant algebroid a courant 2algebroid over the point if given by a semisimple lie algebra with the symplectic form being the killing form. The lie algebra is g fj j 2r3g, found by deriving the identity gg i. A basis for representations of symplectic lie algebras. Lie group elements are related to the lie algebra elements by. In the fourth chapter, koszul describes what he calls a symplectic gspace. Weyl thus avoided that this group connote the complex numbers. Cartan sub lie algebra, roots, weyl group, dynkin diagram. In mathematics, the name symplectic group can refer to two different, but closely related. It is thus a subject of interest to both mathematicians and physicists, though they have approached the subject from different view points. Can be denoted sp2n, f, although other notations are also. The basis vectors are expressed in terms of the mickelsson lowering operators. Since all symplectic matrices have determinant 1, the symplectic group is a subgroup of the special linear group sl2n, f.
Next so 3 and su 2 are introduced as examples of elementary lie groups and their relation to physics and angular momentum. Chevalleys most important contribution to mathematics is certainly his work on group theory. The real rank of the corresponding lie algebra, and hence of the lie group. A courant 2algebroid over the point if given by a semisimple lie algebra with the symplectic form being the killing form. We develop the structure theory of symplectic lie groups based on the study of their isotropic normal subgroups. Symplectic geometry is very useful for clearly and concisely formulating problems in classical physics and also for understanding the link between classical problems and their quantum counterparts. Free algebras, universal enveloping algebras, p th powers, uniqueness of restricted structures, existence of restricted structures, schemes, differential geometry of schemes, generalised witt algebra, filtrations, witt algebras are generalised witt algebra, differentials on a scheme, lie algebras of cartan type, root.
Thus every nearidentity symplectic matrix can be obtained as the exponential of a hamiltonian matrix and corresponds to the time \t\map of a linear. Nonabelian lie groups are mostly discussed in the context of yangmills gauge theory. The symplectic group of degree 2n over a field f, denoted sp2n, f, is the group of 2n. This lie algebra is a quite fundamental object, that crops up at. Classification of adjoint orbits for orthogonal and. 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. Next so3 and su2 are introduced as examples of elementary lie groups and their relation to physics and angular momentum. A particularly relevant group for lie optics is the metaplectic. Since all symplectic matrices have determinant 1, the symplectic group is a subgroup of the special linear group sl2n, f more abstractly, the symplectic group can be defined as the set of. An introduction to lie groups and lie algebras by alexander. Orbits of a symplectic group on its lie algebra in the finite field case. In this chapter we discuss elementary properties of lie groups, lie algebras. When comparing spn to other matrix groups with the same rank for some rank 4, dim u lie groups lie algebras symplectic geometry symplectic linear algebra or ask your own question. Lie groups and lie algebras oi oxford university press.
Modular lie algebras pdf 74p this note covers the following topics. An introduction to matrix groups and their applications. Slicker computation of the lie algebra of the symplectic. Theory of lie groups was the first systematic exposition of the foundations of lie group theory consistently adopting the global viewpoint, based on the notion of analytic manifold. Unitarity imposes conditions on the gauge group, cf.
Whether matrix exponential from skewsymmetric 3x3 matrices to so3 is local homeomorphism. The book by magnus and winkler 20 is a classical reference. Special symplectic lie groups and hypersymplectic lie groups. Introduce matrix lie group sp2 and its matrix lie algebra sp2 consider our goal. An introduction to lie groups and symplectic geometry duke. We start with a discussion of the elementary properties of symplectic vector spaces, the various types of their subspaces and linear symplectic reduction.
An introduction to symplectic geometry rolf berndt. Another excellent feature of the book is that many of the basic notions, properties and results are illustrated by a. It is true that symplectic geometry plays a pivotal role in the hamiltonian formulation, but this aspect is not really explored in srednickis book in any depth. This book remained the basic reference on lie groups for at least two decades. Oct 21, 2011 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. The book starts with the definition of basic concepts such as group, vector space, algebra, lie group, lie algebra, simple and semisimple groups, compact and noncompact groups. 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. Notes on lie algebras cornell department of mathematics.
Thus every nearidentity symplectic matrix can be obtained as the exponential of a hamiltonian matrix and corresponds to the. What makes a group \ lie is that it has an associated vector algebra or lie algebra. It is easily checked that the exponential of a hamiltonian matrix g e. This algebra can be found by exploiting the continuous nature of a lie group and bestowing upon it the structure of a lie.
Thanks for contributing an answer to mathematics stack exchange. An introduction to lie groups and symplectic geometry. The coresponding poisson tensor is the canonical 3cocycle. In this lecture we do for the symplectic lie algebras exactly what we did for the special linear ones in 15. The symplectic group acts by conjugation on itself and on its lie algebra. A series of nine lectures on lie groups and symplectic. Next, we study the symplectic group and its lie algebra and prove the symplectic eigenvalue theorem. The extension classified by this is the string lie 2algebra. It has the advantage of giving the basic facts about lie algebra theory with enough arguments but skipping the tedious technical details of proofs. One of the worlds foremost geometers, alan weinstein has made deep contributions to symplectic and differential geometry, lie theory, mechanics, and related fields. Examples for given group g, identify all coadjoint orbits and express as a hamiltonian gspace. Another excellent feature of the book is that many of the basic notions, properties and results are illustrated by a great number of exercises and examples. While the general focus has been on developing a picture of the collection of all classes or orbits as some kind of quotient, quite a few special features of.
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. This book is a true introduction to symplectic geometry, assuming only a general background in analysis and familiarity with linear algebra. Written in his honor, the invited papers in this volume reflect the active and vibrant research in these areas and are a tribute to weinsteins ongoing influence. 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. 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. Finding the lie algebra of the symplectic lie group. 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.
But avoid asking for help, clarification, or responding to other answers. 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. 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 is, t is the. We can construct an in nitesimal version of the action by associating to each a2k, a vector. 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. This book aims to be both a graduate text and a study resource for lie groups. In order for a group to be isomorphic to spn, it must have the same invariants to preserve structure.
Notes on symplectic geometry 7 we now proceed to compute examples using the above theorem. Kurt bernardo wolf, the symplectic groups, their parametrization. Jacobsons book contains a fairly extensive list of references and some historical. An introduction peter woit department of mathematics, columbia university. The course really was designed to be an introduction, aimed at an audience of stu. The special cases of su2 and sl 2c and their relationships with so3 and the lorentz group are studied in detail. It is possible to show that evey element of w is induced by an automorphism of g that carries h to itself. The authors develop the structure theory of symplectic lie groups based on the study of their isotropic normal subgroups.
Free algebras, universal enveloping algebras, p th powers, uniqueness of restricted structures, existence of restricted structures, schemes, differential geometry of schemes, generalised witt algebra, filtrations, witt algebras are generalised witt algebra, differentials on a scheme, lie algebras of cartan type. Symplectic techniques in physics victor guillemin, shlomo. The members of the symplectic lie algebra sp2n, f are the hamiltonian matrices. Slicker computation of the lie algebra of the symplectic group and computing differentials of matrix equations of polynomials ask question asked 10 days ago. The wellrecognized contributors to this text cover a broad range of topics. A heisenberg group can be defined for any symplectic vector space, and this is the typical way that heisenberg groups arise a vector space can be thought of as a commutative lie group under addition, or equivalently as a commutative lie algebra, meaning with trivial lie bracket. In this chapter, we present linear symplectic algebra. 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.
The first three of the book s six chapters introduce the algebraic background of the symplectic form, the basic topology of symplectic manifolds, and the structure of cotangent bundles. The lowestdimensional symplectic group sp2, of dimension n 3, is isomorphic to su2 and so3. Free lie algebra books download ebooks online textbooks. Its submanifold tangent space at the identity is the symplectic lie algebra sp2n. What makes a group \lie is that it has an associated vector algebra or lie algebra. An introduction to symplectic geometry rolf berndt download. Classification of adjoint orbits for orthogonal and symplectic lie algebras. It is proved in this paper that for a given simply connected lie group g with lie algebra g, every leftinvariant closed 2form induces a symplectic homogeneous space.