Thus, most theorems in this book apply both to real and complex lie groups. We give a construction of the compact real form of the lie. A compact real form of an algebra 9 is one for which the killing form is negative. The notion of the complexification of a lie algebra is closely related to that of a real form of a complex lie algebra. Clifford algebras and spin groups columbia university.
In mathematics, the notion of a real form relates objects defined over the field of real and complex numbers. In the case of the complex special linear group sl n,c, the compact real form is the. On the compact real form of the lie algebra 2 school of. Let g be the complexi cation of g0, and choose a compact real form u0 of g. Our study of real semisimple lie groups and algebras is based on the theory of complex semisimple lie groups. It is easy to see that each l t is a cartan subalgebra, so that l has rank 2. This involution has the property that if l h p is corresponding eigenspace. Compact lie groups and representation theory lecture notes. We do this in the first section, giving a list ofthe simple classical real lie algebras and saying a few words about the corresponding groups and their complex representations. Many years ago i wrote the book lie groups, lie algebras, and some of their applications ny. The third chapter brings the theory of the finite dimensional representations of a semisimple lie alge bra, with the highest or extreme weight as central notion. Fubinigriffithsharris rigidity and lie algebra cohomology landsberg, joseph m. Show that spn is a compact real lie group, and compute its tangent. Xn is a real basis for 9 we can form a real lie algebra t by taking all real linear.
The book is aimed at students in lie groups, lie algebras and their. The book by kostrikin and tiep 4 is devoted to such decomposi. This page contains list of freely available e books, online textbooks and tutorials in lie algebra. Everyg ingl n canbefactoreduniquelyas k swith unand apositivede. In chapter 1, lie algebras, the structure theory of semisimple lie algebras in characteristic zero is presented, following the ideas of killing and cartan. Compact lie groups and representation theory lecture notes lecturer. An elementary introduction graduate texts in mathematics book 222 kindle edition by hall, brian. Part i treats two basic topics in the analysis of compact lie groups. In general, the construction of the compact real form uses structure theory of semisimple lie algebras. The classification of simple complex lie algebras 3 let j 0 i n i n 0. R is the group of nby n orthogonal matrices the group preserving the standard inner product on rn. The lie bracket of sl nr and in general5 acknowledgements9 reference 9 1. V \mathfrakglv of v v is the lie algebra whose elements are linear endomorphisms v v v \to v and whose lie bracket is given by the commutator of endomorphisms.
Modern reference for maximal connected subgroups of compact. If x2gis a strong real form, x intx is an involution of g. Lie groups and lie algebras, together called lie theory, originated in the study of natural symme. Is every connected complex semisimple lie group the. An introduction to lie groups and lie algebras stony brook. Note that one can start more generally with a nondegenerate quadratic form qover r. The symplectic group sp 2nr fx 2gl 2nr j xtjx jgrepresents automorphisms of r2npreserving the nondegenerate skew symmetric innerproduct represented by the matrix j.
For v v a vector space, the general linear lie algebra or endomorphism lie algebra. On the compact real forms of the lie algebras of type e 6 and f 4. We have shown that the killing form is nonsingular, which implies that the lie algebra l is semisimple. The notion of the complexification of a lie algebra is closely related to that of a real form of a complex lie algebra cf. Complexification of a lie algebra encyclopedia of mathematics. Berkeley lectures on lie groups and quantum groups richard borcherds, mark haiman, theo johnsonfreyd, nicolai reshetikhin, and vera serganova last updated january 31, 2020. Part of the springer series in soviet mathematics book series sssov. The existence of a compact group whose lie algebra has as complexi fication a given semisimple complex lie algebra makes it possible to give another.
Our study of real semisimple lie groups and algebras is based on the theory of complex semisimple lie groups developed in ch. General properties of lie algebras, jordanchevalley decomposition, semisimple lie algebras, classification of complex semisimple lie algebras, cartan subalgebras, classification of connected coxeter graphs and complex semisimple lie. Lectures on real semisimple lie algebras and their representations. Sl 3 3, which is isomorphic to a maximal subgroup of the orthogonal group. Any two cartan involutions are conjugate via inn g0. For example, when g is a semisimple lie algebra, the representations of g form a tan. This page contains list of freely available ebooks, online textbooks and tutorials in lie algebra. Ifg 0 is the compact real form, then it is transitive on. For example, the compact real form of a 1 is just the wellknown 3dimensional vector cross product given. The cartan matrix gives you the structure of the lie algebra using the serre relations. Introduction lie groups were initially introduced as a tool to solve or simplify ordinary and partial di erential equations.
On the compact real forms of the lie algebras of type e 6. Use features like bookmarks, note taking and highlighting while reading lie groups, lie algebras, and representations. A symmetric construction of the compact real form of the. Here we consider three values for i, namely, 1, 2, and j, where j is the number of isomorphism types of real forms. The categories of nitedimensional real lie algebras, and of connected simplyconnected lie groups, are equivalent. In preparing this book we have completely rearranged these notes and added two new chapters.
In particular we show that the algebra is uniquely determined by this subgroup. An ideal i of g is an invariant subalgebra of g, namely, g,i. Chapter 2, the structure of locally compact groups, deals with the solution of hilberts fifth problem given. We can also start with a complex lie algebra g and define a real lie algebra by forgetting. Invariant bilinear forms and semisimplicity of classical lie algebras. Download it once and read it on your kindle device, pc, phones or tablets. Check our section of free ebooks and guides on lie algebra now. Clifford algebras and spin groups math g4344, spring 2012 well now turn from the general theory to examine a speci c class class of groups. In these lectures we will start from the beginning the theory of lie algebras and their representations. Lax equations associated with a least squares problem and compact lie algebras nakamura, yoshimasa, 1993.
Jacobsons book contains a fairly extensive list of references and. Taking k 1 and k 2 to be real division algebras, we obtain a 4. A compact lie algebra can be seen as the smallest real form of a corresponding complex lie algebra, namely the complexification. While the advice and information in this book are believed to be true and accurate at the date of publica tion, neither the. The lie algebra g is called reductive with compact real form g0 if there exists a. This result underpins the theory of real lie algebras, their maximal compact subgroups and the classi. There is also a discus sion of the compact form and other real forms of a complex semisimple lie algebra, and a section on automorphisms.
Since a compact connected group is semisimple or trivial modulo its center and the latter lies in a torus, the semisimple case is crucial for maximal subgroup problems. This volume presents lecture notes based on the authors courses on lie algebras and the solution of hilberts fifth problem. Firstly, the computation of the kazhdanlusztigvogan polynomials 35 involved the split real form of e 8. Eventually these notes will consist of three chapters, each about 100 pages long, and a short appendix. For classical lie algebras there is a more explicit construction. Conversely, we prove from first principles that the algebra satisfies the jacobi identity, and thus give an elementary proof of existence of a lie algebra of type e 6. Chapter 2, the structure of locally compact groups, deals with the solution of hilberts fifth problem given by gleason, montgomery, and zipplin in 1952. The symplectic group sp 2nr fx 2gl 2nr j xtjx jgrepresents automorphisms of r2npreserving the nondegenerate skewsymmetric innerproduct represented by the matrix j. Lie groups, lie algebras, and their representations. A unitary representation of a real lie algebra g0 on a hilbert space h is a representation of the. Before leaving the reals behind us, note the following theorems we will not prove. If l is a real semisimple lie algebra, then the use of a compact real form of lc leads to the construction of a cartan involution of l. In table 2 we list the runtimes for realformbyidt,n,i,f.
A symmetric construction of the compact real form of the lie. Free lie algebra books download ebooks online textbooks. Contents introduction the lie algebra of r and in general. If time permits we may study lie algebras over the field of real numbers or look at jordan algebras. The theory as we know it today began with the ground breaking work of the norwegian mathematician sophus lie, who introduced the notion of continuous transformation groups and showed the crucial role that lie algebras play in their classi cation and representation theory. As above, this group is compact because it is closed and bounded with respect to the hilbertschmidt norm. A real lie algebra g0 is called a real form of a complex lie algebra g if g is. Clifford algebras and spin groups math g4344, spring 2012. This is possible because the complexification of a real semisimple lie algebra is also semisimple see 1. Lie algebras and locally compact groups irving kaplansky. Sl33, which is isomorphic to a maximal subgroup of the orthogonal group. Compact groups as algebraic groups 4 ill deduce this theorem from the more precise result that chevalley proved.
The largest nitedimensional exceptional lie algebra, namely e 8, appears in supergravity 33 in its maximally noncompact split real form, whereas the compact real form appears in heterotic string theory 34. Several traditional topics of lie algebra theory, however, are left entirely disregarded, e. One writes t a,t b f ab c t c, 1 where the structure constants f ab c are real if the lie algebra is a real vector space, or complex if the lie algebra is a complex vector space. An introduction to matrix groups and their applications andrew baker. The references are all in knapps book mentioned above. To apply dynkins results to compact lie groups one has to use weyls approach. Let g 0 be a real lie algebra of matrices over r that is closed under the transpose map. In the case of the complex special linear group sln,c, the compact real form is the. Any nitedimensional real lie algebra is isomorphic to the lie algebra of a lie group. Real form maximal compact subgroup cartan decomposition compact real form real algebraic group. Is every connected complex semisimple lie group the complexification of a compact lie group.
Intrinsically and algebraically, a compact lie algebra is a real lie algebra whose killing form is negative definite. Cartan and iwasawa decompositions in lie theory 5 theorem 3. If time permits we may study lie algebras over the field of real numbers or look at jordan. An introduction to matrix groups and their applications andrew baker 1472000 department of mathematics, university of glasgow, glasgow g12 8qw, scotland. Conversely, we prove from first principles that the algebra satisfies the jacobi identity, and thus.
The most important example of a lie group and it turns out, one. Exceptional lie algebras, su3 and jordan pairs part 2. The lie algebra l is the compact real form of the complex lie algebra e 8. It is known that every simple complex lie algebra has a unique compact real form. Universal envelopping algebras, levis theorem, serres theorem, kacmoody lie algebra, the kostants form of the envelopping algebra and a beginning of a proof of the chevalley. This group is a compact form of the group spn, c in the sense we will describe. The problem let z gqbe a complex ag manifold where gis a complex connected semisimple lie group and qis a parabolic subgroup. On the compact real forms of the lie algebras of type. Zhelobenko, compact lie groups and their representations, amer. Prove that o2 and n2 are abelian 1dimensional lie algebras, hence they are isomorphic to kwith zero bracket.
Rather surprisingly, in recent times the popular press has been dealing with e 8 more than once. Lie algebra has a compact real form that is to say, the complex lie algebra is the complexi. Check our section of free e books and guides on lie algebra now. Notes on lie algebras cornell department of mathematics. Prove that the lie algebra from example 2 is isomorphic to o3 by comparing the structure. We give an elementary construction of the compact real form of the lie algebra g 2.
Sl33, which is isomorphic to a maximal subgroup of the. 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. There is also a discussion of the compact form and other real forms of a complex semisimple lie algebra, and a section on automorphisms. Jun 25, 2008 compact real forms of a complex semisimple lie algebra. Contents preface pagexi 1 introduction 1 2 liegroups.