Lie group elements are related to the lie algebra elements by. We start with a discussion of the elementary properties of symplectic vector spaces, the various types of their subspaces and linear symplectic reduction. Its submanifold tangent space at the identity is the symplectic lie algebra sp2n. The authors develop the structure theory of symplectic lie groups based on the study of their isotropic normal subgroups. Thus every nearidentity symplectic matrix can be obtained as the exponential of a hamiltonian matrix and corresponds to the. 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. Thanks for contributing an answer to mathematics stack exchange. 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. The overflow blog introducing dark mode beta for stack overflow. 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.
The real rank of the corresponding lie algebra, and hence of the lie group. But avoid asking for help, clarification, or responding to other answers. In order for a group to be isomorphic to spn, it must have the same invariants to preserve structure. Slicker computation of the lie algebra of the symplectic group and computing differentials of matrix equations of polynomials ask question asked 10 days ago.
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. 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. The members of the symplectic lie algebra sp2n, f are the hamiltonian matrices. 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. Notes on symplectic geometry 7 we now proceed to compute examples using the above theorem. Kurt bernardo wolf, the symplectic groups, their parametrization. An introduction to lie groups and symplectic geometry duke. The symplectic group of degree 2n over a field f, denoted sp2n, f, is the group of 2n. An introduction peter woit department of mathematics, columbia university. An introduction to lie groups and symplectic geometry. 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. Nonabelian lie groups are mostly discussed in the context of yangmills gauge theory.
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. 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. The special cases of su2 and sl 2c and their relationships with so3 and the lorentz group are studied in detail. The extension classified by this is the string lie 2algebra. Modular lie algebras pdf 74p this note covers the following topics. 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. Since all symplectic matrices have determinant 1, the symplectic group is a subgroup of the special linear group sl2n, f.
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. It is possible to show that evey element of w is induced by an automorphism of g that carries h to itself. 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.
The symplectic group acts by conjugation on itself and on its lie algebra. John lees book question about symplectic group math stack. 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. Classification of adjoint orbits for orthogonal and symplectic lie algebras.
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 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 lowestdimensional symplectic group sp2, of dimension n 3, is isomorphic to su2 and so3. 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.
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. It is easily checked that the exponential of a hamiltonian matrix g e. Jan 18, 2016 answer 1 of 2 for every even dimension 2n, the symplectic group sp2n is the group of 2n. We can construct an in nitesimal version of the action by associating to each a2k, a vector. 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. Symplectic techniques in physics victor guillemin, shlomo.
Slicker computation of the lie algebra of the symplectic. Cartan sub lie algebra, roots, weyl group, dynkin diagram. The lie algebra is g fj j 2r3g, found by deriving the identity gg i. This book is a true introduction to symplectic geometry, assuming only a general background in analysis and familiarity with linear algebra. The lie algebra k t ekis an in nitesimal version of the lie group k. Examples for given group g, identify all coadjoint orbits and express as a hamiltonian gspace. An introduction to symplectic geometry rolf berndt. 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. Notes on lie algebras cornell department of mathematics. Another excellent feature of the book is that many of the basic notions, properties and results are illustrated by a. What makes a group \ lie is that it has an associated vector algebra or lie algebra. 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. An introduction to symplectic geometry rolf berndt download.
A basis for each finitedimensional irreducible representation of the symplectic lie algebra sp2n is constructed. An introduction to lie groups and lie algebras by alexander. It has the advantage of giving the basic facts about lie algebra theory with enough arguments but skipping the tedious technical details of proofs. This algebra can be found by exploiting the continuous nature of a lie group and bestowing upon it the structure of a lie. Next so 3 and su 2 are introduced as examples of elementary lie groups and their relation to physics and angular momentum. What makes a group \lie is that it has an associated vector algebra or lie algebra. Unitarity imposes conditions on the gauge group, cf. In the fourth chapter, koszul describes what he calls a symplectic gspace. In this chapter we discuss elementary properties of lie groups, lie algebras. Orbits of a symplectic group on its lie algebra in the finite field case. The basis vectors are expressed in terms of the mickelsson lowering operators. Mat 4451196 complex symplectic lie algebras let n be an integer greater than or equal to 2. A particularly relevant group for lie optics is the metaplectic.
Special symplectic lie groups and hypersymplectic lie groups. The coresponding poisson tensor is the canonical 3cocycle. 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. Can be denoted sp2n, f, although other notations are also. 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. An introduction to matrix groups and their applications. Lie groups and lie algebras oi oxford university press. In mathematics, the name symplectic group can refer to two different, but closely related. Weyl thus avoided that this group connote the complex numbers. 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. It is thus a subject of interest to both mathematicians and physicists, though they have approached the subject from different view points. In this lecture we do for the symplectic lie algebras exactly what we did for the special linear ones in 15.
Suppose that g is the lie algebra of a lie group g. A series of nine lectures on lie groups and symplectic. One of the worlds foremost geometers, alan weinstein has made deep contributions to symplectic and differential geometry, lie theory, mechanics, and related fields. These groups are themselves infinitely connected, and possess a common universal cover sp2,r. 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. Chevalleys most important contribution to mathematics is certainly his work on group theory. Classification of adjoint orbits for orthogonal and. 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. Indeed, symplectic methods are key ingredients in the study of dynamical systems, differential equations, algebraic geometry, topology, mathematical physics and representations of lie groups. The book by magnus and winkler 20 is a classical reference. Free lie algebra books download ebooks online textbooks. Introduce matrix lie group sp2 and its matrix lie algebra sp2 consider our goal. This lie algebra is a quite fundamental object, that crops up at.
Symplectic group project gutenberg selfpublishing ebooks. 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. Next, we study the symplectic group and its lie algebra and prove the symplectic eigenvalue theorem. In this chapter, we present linear symplectic algebra. A basis for representations of symplectic lie algebras. This book remained the basic reference on lie groups for at least two decades. Whether matrix exponential from skewsymmetric 3x3 matrices to so3 is local homeomorphism. 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. A courant 2algebroid over the point if given by a semisimple lie algebra with the symplectic form being the killing form. 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. 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.
606 1267 1469 885 795 1073 1560 468 477 797 1174 1515 13 389 351 277 777 102 596 893 566 679 750 238 944 1505 310 1218 394 520 999 961 93 35 415 1390 183 755 986