There arent very many geometrically flavored books on lie groups. However in the case of compact lie groups, answers to most of these ques tions are well. In this chapter we discuss elementary properties of lie groups, lie algebras and their. In this article we describe the structure and representation theory of compact lie groups. We have rigidly adhered to the analytic approach in establishing the relations between lie groups and lie algebras.
Signature quantization and representations of compact lie. Applications to classical and quantum mechanics dedicated to eldar straume on his 70th birthday. We may therefore view them as rigid analytic quantizations. Lie groups, lie algebras, representations and the eightfold way. Spherical twopiece property stpp let m be a compact surface embedded in sm. We discuss integration on a lie group, the lie algebra, and the exponential map from the lie algebra to the lie group. Classify compact connected lie groups classify all irreducible representations of such groups calculate the characters of these irreducible representations recall that the peterweyl theorem tells one that the matrix elements of irreducible representations form an orthonormal basis of l2g for g a compact lie group. We present some classical results on compact lie groups, such as the peterweyl theorem, on the completeness of the matrix entries of irreducible unitary representations of a. Of particular importance is the problem of the unitary dual. Special functions and linear representations of lie groups. Lie theory of representations of compact connected lie groups 15 20.
We will primarily focus on analytic and algebraic aspects of compact lie groups through a wide variety of examples. Lie s ideas played a central role in felix kleins grand erlangen program to classify all possible geometries using group theory. Todays lecture is basically going to be a mishmash. About 20 years ago i read in textbook that all irreducible representations of compact groups are finitedimensional, but me and the proof of this fact never met each other. We apply these objects to general results on representations. Relations between representations of compact lie groups and complex lie algebras 63 4. Representations of compact padic analytic groups 457 the subalgebras ou n. There are of course many kinds of textbooks treating lie groups, lie algebras, and representation theory.
Lectures on lie groups and geometry imperial college london. Representations of compact lie groups pdf free download epdf. We summarize the results of this chapter in the peterweyltheorem. For example, the group of all affine transformations of a connected manifold with an affine connection and the group of all isometries of a pseudoriemannian manifold are known to be lie groups in the compact open topology.
For noncompact lie groups and various choices of spaces of maps, one needs. We say m has the stpp if m \b is connected whenever b is a closed ball in sm. A great deal is known about such representations, a basic tool in. Ctsntctconsider the compact groups g and k defined above and let f be the factor space gk consisting of all cosets of the form xk xeg. In a nutshell, restricting a representation of a compact connected lie group to a maximal torus thereof does not, in principle, lose any information. The other series are somewhat more delicate, replacingelie cartans theory of. The general theory of linear representations of compact groups 9 14. Nilpotent or solvable lie groups dont have very interesting compact subgroups. In the case where gis a lie group, then we ask that the map g glv be a smooth map. Orthogonality about irreducible unitary representations of compact group. The topic of this course is not \ representation theory of reductive lie groups, but simply \ representation theory, or simply the study of lie groups. Chapter 6 representations of compact groups throughout this chapter, g denotes a compact group. The representations of compact groups are particularly well behaved, which we shall show in chapter 4.
This introduction to the representation theory of compact lie groups follows herman weyls original approach. Finally, this treatment of compact lie groups relies on the same ideas which, supported by additional machinery from homotopy theory, give structure theorems for pcompact groups. Representations of compact lie groups springerlink. The representation rin g of a compact lie group graeme segal introduction let g be a compact group. There will be a lot of circles oating around, and so we will distinguish these by subscripts. It culminates in the complete proof of the weyl character formula for representations of compact lie groups and the frobenius formula for characters of permutation groups. Lie algebras, though mentioned occasionally, are not used in an essential way. There is a theory of complex lie groups we mostly will not develop. Representation theory of compact groups and complex reductive. It introduces the reader to the representation theory of compact lie groups. We can combine these two operations to obtain four possible representations. The second edition of lie groups, lie algebras, and representations contains many substantial improvements and additions, among them.
R, the set of all n nreal matrices, is di eomorphic to rn2. Applications of representations of compact groups249 6. Let c v be the space of all continuous functions on f. Representations of compact lie groups graduate texts in. Direct sums of representations and complete reducibility 79 6.
Generalizing this idea, we argue that representations of the lie groups can be deduced from representations of the lie algebras, and more. Over the course of the years i realized that more than 90% of the most useful material in that book could be presented in less than 10% of the space. Unitary repns of noncompact nonabelian lie groups tend to be infinitedimensional. Lectures on lie groups and representations of locally. A compact group that is a finitedimensional real lie group. Although compact lie groups play a more prominent role in physics, noncompact ones have important applications too. Quasi unitary representations of noncompact lie groups. Assuming no prior knowledge of lie groups, this book covers the structure and representation theory of compact lie groups. Themainreferencesusedwere8forbanachalgebratheory,17forthespectral theorem and its application to schurs lemma, and 5 for locally compact groupsandrepresentationtheory.
Sections 417 deal with the general theory of lie groups. Lie groups arise in a natural way as transformation groups of geometric objects. Commutativity makes complex irreducible representations onedimensional. On irreducible representations of compact padic analytic groups. This very important special chapter of the representation theory of compact lie groups is key to further. Lie groups and lie algebras, together called lie theory, originated in the study of natural symmetries of solutions of di erential equations. Representations of semisimple lie groups anthony w. Lie groups, lie algebras, and their representations.
Besides halls book, there are standard texts by helgason, knapp, procesi, and others. The rotation group so3 is a compact lie group and thus every finitedimensional representation of so3 decomposes as a direct sum of irreducible representations. This is an overview article on compact lie groups and their representations, written for the encyclopedia of mathematical physics to be published by elsevier. Lie groups beyond an introduction, with emphasis on chapters iv, v, and ix. I can think of only this one and compact lie groups by sepanski. Principal series representations of direct limit groups. During the past few years several authors have studied unitary representations of a lie group or, more generally, of a locally compact group on a hubert space. A lattice is a discrete additive subgroup l v such that the set l spans the vector space v over r. We discuss integration on a lie group, the lie algebra, and general results on representations. This richness comes from the merging of two initially unrelated mathemat. Equivalently, a representation is a smooth homomorphism of the group into the group of invertible operators on the vector space. I shall use the word gmodule to mean a finitedimensional complex vector space m together with a continuous linear action of g on m.
However, unlike say the nite collection of symmetries of the hexagon, these symmetries occurred in continuous families, just as the rotational symmetries. Induced representations and frobenius reciprocity math g4344, spring 2012. R which is also a lie group, with group structure the matrix multiplication. Proofs, if any, are either sketched or given in simpler cases. These objects themselves and techniques used to study them are prevalent in several areas of mathematics. Theyre all conjugate inside g, so it doesnt matter which one we choose.
This is one of the main reasons we will mostly be restricting our attention to representations of compact lie groups this semester. Blending algebra, analysis, and topology, the study of compact lie groups is one. We have chosen a geometrical and analytical approach since we feel that this is the easiest way to motivate and establish the theory and to indicate relations to other branches of mathematics. A linear lie group, or matrix lie group, is a submanifold of mn. Lie groups, lie algebras, and representations springerlink. Chapter 6 revolves about the concept of maximal tori of a lie group, that is, maximal connected abelian lie subgroups. Compact lie groups can be characterized as finitedimensional locally connected compact topological groups.
Finally we will apply this to the classi cation of compact lie groups. Inside a general semisimple lie group there is a maximal compact subgroup, and the representation theory of such groups, developed largely by harishchandra, uses intensively the restriction of a representation to such a subgroup, and also the model of weyls character theory. The atlas of lie groups and representations is a project to make available information about representations of reductive lie groups. It discusses all aspects of finitedimensional lie theory, consistently emphasizing the. In chapter four we deal on compact lie groups, whose rich structure allows us to have a more accurate knowledge of them. Setting is that we have a compact connected lie group with a choice of maximal torus 1a g.
The haar measure and the regular representation of a locally compact group212 5. Most lie groups books fall into one of two categories. Throughout the article, g is a compact real lie group with lie algebra g. Lie groups, this book covers the structure and representation theory of compact. Representations of compact lie groups and their orbit spaces joint work with alexander lytchak claudio gorodski geometry and lie theory. The second half deals with representation theory of groups, particularly compact groups. If youre perfectly happy with a more algebraic treatment, read no further. We will discuss the classi cation of semisimple lie algebras, root systems, the weyl group, and dynkin diagrams. If m and n are gmodules, one can form their sum mn, and with respect to this operation the isomorphism classes.
Representation ring and algebraicity of compact lie groups. Representations of compact lie groups and their orbit. Representations play an important role in the study of continuous symmetry. Why all irreducible representations of compact groups are. Representation theory springer also various writings of atiyah, segal, bott, guillemin and. Norwegian university of science and technology, trondheim november 34, 2016. We present some classical results on compact lie groups, such as the peterweyl theorem, on. Blending algebra, analysis, and topology, the study of compact lie groups is one of the most beautiful areas of mathematics and a key stepping stone to the theory of general lie groups. Compact lie groups and representation theory lecture notes. Induced representations of compact groups 56 chapter 4. The group so3 has one irreducible representation in each odd dimension. Representations of compact lie groups this is to write down some facts which are needed to do the homework. Part i treats two basic topics in the analysis of compact lie groups.
The complete reducibility to finite dimensional representations result you read about is a consequence of or part of, depending on how you phrase it the peterweyl theorem. Since lie groups are real manifolds, g is a real lie algebra. Pdf structure theory of semisimpie lie groups researchgate. A representation of g consists of a finitedimensional complex vector. Lie groups beyond an introduction representations of. Representation theory of classical compact lie groups. Lecture 1 representations of reductive lie groups notes lecture 1 128 this is math 224.
This will lead into nitedimensional representations and the weyl character formula. Thepcompact groups seem to be the best available homotopical analogues of compact lie groups. A topological group is a topological space g with a group structure such that the multiplication map m. They can therefore be identi ed with the tate algebras khpnu 1pnu dias kbanach spaces. Compact lie groups and representation theory lecture notes lecturer.
Donaldson march 25, 2011 abstract these are the notes of the course given in autumn 2007 and spring 2011. Lectures on lie groups and representations of locally compact. Ametricapproachto representationsofcompactliegroups. Lie groups form a class of topological groups, and the compact lie groups have a particularly welldeveloped theory. In chapter 5 we study complex representations of connected abelian lie groups tori. Three other nice references are the springer graduate texts in mathematics representations of compact lie groups by brocker and tom dieck, representation theory by fulton and harris, and introduction to lie algebras and representation theory by humphreys. Characters and matrix coe cients for compact groups237 5. Signature quantization and representations of compact lie groups victor guillemin and etienne rassart department of mathematics, massachusetts institute of technology, cambridge, ma 029 contributed by victor guillemin, may 11, 2004 we discuss some applications of signature quantization to the representation theory of compact lie groups.
928 1496 1060 1394 1172 99 961 153 1028 273 452 1114 877 1364 1106 154 343 948 1456 1498 143 170 453 378 1281 877 978 752 1423 1382 475 946 230 519 1150 208 1166