We prove that if g is a compact connected lie group and x is a compact connected hyperkahler manifold, then the l2 metric on the smooth locus of. The padic reflection groups play an important role in the theory of socalled pcompact groups, which constitute a homotopy theoretic analogue of compact lie groups. Lie groups form a class of topological groups, and the compact lie groups have a particularly welldeveloped theory. The theoretical results are illustrated by computations and the theory provides a commentary on the computational work. Finally we will apply this to the classi cation of compact lie groups. We have chosen a geometrical and analytical approach since we feel that this is the easiest way to motivate and.
Much of the material in this chapter is used only rarely in later chapters, so that it is not at all necessary for the reader to be familiar with all of the results presented here. Intrinsically and algebraically, a compact lie algebra is a real lie algebra whose killing form is negative definite. Cornell university 2014 let g be a compact, connected, and simplyconnected lie group, equipped with a lie group involution. Tolkien media text id 777c955e online pdf ebook epub library are described by its past plus probabilities for successive changes eg the price of a st lie groups distributed uniformly with respect to haar measure are generated using qr. The kernel of the covering map from the lie algebra of a maximal. In the first part, it is shown that these fall naturally into three types. Representation types and 2primary homotopy groups of certain compact lie groups davis, donald m. The second half deals with representation theory of groups, particularly compact groups. This book is based on several courses given by the authors since 1966. On the other hand, some compact groups or algebraic groups occur most naturally as disconnected groups with an interesting component group.
Autg is a holomorphic map on a compact set it must be constant. The book goes beyond the representation theory of compact lie groups, which is the basis of many texts, and provides a carefully chosen range of material to give the student the bigger picture. The book is organized to allow different paths through the material depending on ones interests. It introduces the reader to the representation theory of compact lie groups. Lectures on lie groups and representations of locally. Compact lie groups can be characterized as finitedimensional locally connected compact topological groups. It discusses all aspects of finitedimensional lie theory, consistently emphasizing the groups themselves. Thep compact groups seem to be the best available homotopical analogues of compact lie groups 10, 11, 12, but analytical objects like lie algebras are not available for them. In chapter 7, we study the lie algebras and duals thereof of the maximal tori of classical compact lie groups. Compact lie group an overview sciencedirect topics. Extrinsically and topologically, a compact lie algebra is the lie algebra of a compact lie group.
These notes are an introduction to lie algebras, algebraic groups, and lie groups in characteristic zero, emphasizing the relationships between these objects visible in their categories of representations. Haar measure on locally compact hausdor groups 16 5. 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. Representations of compact lie groups pdf free download epdf. First, the mathematical theory of tensor operators for a lie group is recast in a new way which is used to generalise the wigner eckart theorem to noncompact groups. Rm is called differentiable at x 0 if there is a linear map l. Representation theory of compact groups and complex reductive groups, winter 2011 joel kamnitzer march 30, 2011. A lattice is a discrete additive subgroup l v such that the set l spans the vector space v over r.
Compact lie groups and representation theory lecture notes. A compact group that is a finitedimensional real lie group. 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. Representations of compact lie groups springerlink.
All compact, complex, connected lie groups are abelian. The elementary geometric structure of compact lie groups. Many years ago i wrote the book lie groups, lie algebras, and some of their applications ny. Noncompact groups appear naturally if the spacetime is of lorentzian signature, but can also have an important role in the euclidean case, as will be shown. Naturally some care always has to be taken with the connectedness question, since finite groups might be regarded as compact lie groups or as algebraic groups.
In the mathematical field of lie theory, there are two definitions of a compact lie algebra. Section 2 introduces tori, which turn out to be exactly the connected. For g 2, the lie algebra of t is that of su3 but the action of w is extended by an inversion. Introduction a compact lie group gis a compact di erentiable manifold together with a. Contraction of compact semisimple lie groups via berezin quantization cahen, benjamin, illinois journal of mathematics, 2009. On isometry groups of pseudoriemannian compact lie groups. Consequently, we provide a symbolic descriptions of complex powers of such operators. Continuous representations of locally compact groups 29 1.
Compact lie groups are matrix groups 317 bibliography 319. 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. Global functional calculus for operators on compact lie groups abstract. Kronecker products for compact semisimple lie groups g r e black, r c king and b g wybourne branching rules for classical lie groups using tensor and spinor methods r c king standard young tableaux and weight multiplicities of the classical lie groups r c king and n g i elsharkaway models for su3 in terms of son,2 and so2n algebras. We will discuss the classi cation of semisimple lie algebras, root systems, the weyl group, and dynkin diagrams.
Notes for a talk on cohomology of compact lie groups 1 introduction. The goal is to take a fresh look at these theorems, prove some that are di cult to nd in the literature, and illustrate an approach to the theorems that can be imitated in the homotopy theoretic setting of p compact groups. This book is intended for a oneyear graduate course on lie groups and lie algebras. 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. Representations of compact lie groups by theodor brocker. 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 real ktheory of compact lie groups chikwong fok, ph.
We have rigidly adhered to the analytic approach in establishing the relations between lie groups and lie algebras. These become standard results in the algebraic topology of compact lie groups. This document consists of lectures notes from a course at. Pdf topological quantum field theories from compact lie. This will lead into nitedimensional representations and the weyl character formula. Lectures on lie groups and representations of locally compact. Representation theory of compact groups and complex. This report aims at giving a general overview on the classification of the maximal subgroups of compact lie groups not necessarily connected. Representations of compact lie groups graduate texts in. This introduction to the representation theory of compact lie groups follows herman weyls original approach. Most lie groups books fall into one of two categories. Introduction to compact lie groups series in pure mathematics. X is an inverse system of compact lie groups with x as inverse limit, and.
Much of the material in this chapter is used only rarely in later chapters, so that it is not at all. Representations of compact groups are unitary 32 3. In this thesis we compute atiyahs real ktheory of g in several contexts. Assuming no prior knowledge of lie groups, this book covers the structure and representation theory of compact lie groups. The pcompact groups seem to be the best available homotopical analogues of compact lie groups 10, 11, 12, but analytical objects like lie algebras are not available for them.
If youre perfectly happy with a more algebraic treatment, read no further. I can think of only this one and compact lie groups by sepanski. Chapter 6 revolves about the concept of maximal tori of a lie group, that is, maximal connected abelian lie subgroups. There arent very many geometrically flavored books on lie groups. Notes for a talk on cohomology of compact lie groups 1. Let 0 2 u g and e 2 v g small symmetric neighborhoods of 0 in g and resp. By definition, a pcompact group is a p complete topological space bx such that the homology h x.
A topological group is a topological space g with a group structure such that the multiplication map m. The p compact groups seem to be the best available homotopical analogues of compact lie groups 10, 11, 12, but analytical objects like lie algebras are not available for them. There is a clear, selfcontained classification of compact, connected lie groups in lie groups. In general, the lie algebra of a compact lie group decomposes as the lie algebra direct sum of a commutative summand for which the corresponding subgroup is a torus and a summand on which the killing form is. Pdf maximal subgroups of compact lie groups semantic. In fact, readers with a scant background would probably be wise to. G and viewed as a gspace via the conjugation action. If the killing form of a lie algebra is negative definite, then the lie algebra is the lie algebra of a compact semisimple lie group. Non compact groups appear naturally if the spacetime is of lorentzian signature, but can also have an important role in the euclidean case, as will be shown. For the computation of the cohomology groups of compact lie groups, we demonstrate. The great appeal of this book is the blending of these two approaches. In a nutshell, restricting a representation of a compact. Eventually these notes will consist of three chapters, each about 100 pages long, and a short appendix.
Coset spaces and quasiinvariant measures 21 chapter 2. A representation a is of degree 1 with infinite image, if and only if ax is a circle group. On the balmer spectrum for compact lie groups compositio. Notes for a talk on cohomology of compact lie groups 3 linear dependence.
Thepcompact groups seem to be the best available homotopical analogues of compact lie groups 10, 11, 12, but analytical objects like lie algebras are not available for them. Pdf maximal subgroups of compact lie groups semantic scholar. Greenlees, markus hausmann skip to main content accessibility help we use cookies to distinguish you from other users and to provide you with a better experience on our websites. It discusses all aspects of finitedimensional lie theory, consistently emphasizing the read more. There are a lot of classical examples arising from the study of matrix groups, as we will see.
Holomorphic discrete series and harmonic series unitary. Compact lie groups and representation theory lecture notes lecturer. Translationinvariant function algebras 1097 let x be a compact connected group. Representation theory of classical compact lie groups. Topological quantum field theories from compact lie groups. On the balmer spectrum for compact lie groups volume 156 issue 1 tobias barthel, j. Compact lie groups are ubiquitous in topology, algebra, and analysis. First, the mathematical theory of tensor operators for a lie group is recast in a new way which is used to generalise the wigner eckart theorem to non compact groups. The next theorem asserts the existence of a translation invariant measure on every compact lie group. We prove that if g is a compact connected lie group and x is a compact connected hyperkahler manifold, then the l2 metric on the smooth locus of the moduli space of flat gbundles on x is a.
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