Jordan Algebras and the Exceptional Lie Algebra F
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Lecture Notes on Compact Lie Groups and Their Representations
Lecture Notes on Compact Lie Groups and Their Representations CLAUDIO GORODSKI Prelimimary version: use with extreme caution! September , ii Contents Contents iii 1 Compact topological groups 1 1.1 Topological groups and continuous actions . 1 1.2 Representations .......................... 4 1.3 Adjointaction ........................... 7 1.4 Averaging method and Haar integral on compact groups . 9 1.5 The character theory of Frobenius-Schur . 13 1.6 Problems.............................. 20 2 Review of Lie groups 23 2.1 Basicdefinition .......................... 23 2.2 Liealgebras ............................ 25 2.3 Theexponentialmap ....................... 28 2.4 LiehomomorphismsandLiesubgroups . 29 2.5 Theadjointrepresentation . 32 2.6 QuotientsandcoveringsofLiegroups . 34 2.7 Problems.............................. 36 3 StructureofcompactLiegroups 39 3.1 InvariantinnerproductontheLiealgebra . 39 3.2 CompactLiealgebras. 40 3.3 ComplexsemisimpleLiealgebras. 48 3.4 Problems.............................. 51 3.A Existenceofcompactrealforms . 53 4 Roottheory 55 4.1 Maximaltori............................ 55 4.2 Cartansubalgebras ........................ 57 4.3 Case study: representations of SU(2) .............. 58 4.4 Rootspacedecomposition . 61 4.5 Rootsystems............................ 63 4.6 Classificationofrootsystems . 69 iii iv CONTENTS 4.7 Problems.............................. 73 CHAPTER 1 Compact topological groups In this introductory chapter, we essentially introduce our very basic objects of study, as well as some fundamental examples. We also establish some preliminary results that do not depend on the smooth structure, using as little as possible machinery. The idea is to paint a picture and plant the seeds for the later development of the heavier theory. 1.1 Topological groups and continuous actions A topological group is a group G endowed with a topology such that the group operations are continuous; namely, we require that the multiplica- tion map and the inversion map µ : G G G, ι : G G × → → be continuous maps. -
A Taxonomy of Irreducible Harish-Chandra Modules of Regular Integral Infinitesimal Character
A taxonomy of irreducible Harish-Chandra modules of regular integral infinitesimal character B. Binegar OSU Representation Theory Seminar September 5, 2007 1. Introduction Let G be a reductive Lie group, K a maximal compact subgroup of G. In fact, we shall assume that G is a set of real points of a linear algebraic group G defined over C. Let g be the complexified Lie algebra of G, and g = k + p a corresponding Cartan decomposition of g Definition 1.1. A (g,K)-module is a complex vector space V carrying both a Lie algebra representation of g and a group representation of K such that • The representation of K on V is locally finite and smooth. • The differential of the group representation of K coincides with the restriction of the Lie algebra representation to k. • The group representation and the Lie algebra representations are compatible in the sense that πK (k) πk (X) = πk (Ad (k) X) πK (k) . Definition 1.2. A (Hilbert space) representation π of a reductive group G with maximal compact subgroup K is called admissible if π|K is a direct sum of finite-dimensional representations of K such that each K-type (i.e each distinct equivalence class of irreducible representations of K) occurs with finite multiplicity. Definition 1.3 (Theorem). Suppose (π, V ) is a smooth representation of a reductive Lie group G, and K is a compact subgroup of G. Then the space of K-finite vectors can be endowed with the structure of a (g,K)-module. We call this (g,K)-module the Harish-Chandra module of (π, V ). -
Orbits of Real Forms in Complex Flag Manifolds 71
Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) Vol. IX (2010), 69-109 Orbits of real forms in complex flag manifolds ANDREA ALTOMANI,COSTANTINO MEDORI AND MAURO NACINOVICH Abstract. We investigate the CR geometry of the orbits M of a real form G0 of a complex semisimple Lie group G in a complex flag manifold X = G/Q.Weare mainly concerned with finite type and holomorphic nondegeneracy conditions, canonical G0-equivariant and Mostow fibrations, and topological properties of the orbits. Mathematics Subject Classification (2010): 53C30 (primary); 14M15, 17B20, 32V05, 32V35, 32V40, 57T20 (secondary). Introduction In this paper we study a large class of homogeneous CR manifolds, that come up as orbits of real forms in complex flag manifolds. These objects, that we call here parabolic CR manifolds, were first considered by J. A. Wolf (see [28] and also [12] for a comprehensive introduction to this topic). He studied the action of a real form G0 of a complex semisimple Lie group G on a flag manifold X of G.Heshowed that G0 has finitely many orbits in X,sothat the union of the open orbits is dense in X, and that there is just one that is closed. He systematically investigated their properties, especially for the open orbits and for the holomorphic arc components of general orbits, outlining a framework that includes, as special cases, the bounded symmetric domains. We aim here to give a contribution to the study of the general G0-orbits in X, by considering and utilizing their natural CR structure. In [2] we began this program by investigating the closed orbits. -
Inner Derivations of Exceptional Lie Algebras in Prime Characteristic 3
INNER DERIVATIONS OF EXCEPTIONAL LIE ALGEBRAS IN PRIME CHARACTERISTIC PABLO ALBERCA BJERREGAARD, DOLORES MART´IN BARQUERO, AND CANDIDO´ MART´IN GONZALEZ´ Abstract. It is well-known that every derivation of a semisimple Lie algebra L over an algebraically closed field F with characteristic zero is inner. The aim of this paper is to show what happens if the characteristic of F is prime with L an exceptional Lie algebra. We prove that if L is a Chevalley Lie algebra of type {g2, f4, e6, e7, e8} over a field of characteristic p then the derivations of L are inner except in the cases g2 with p = 2, e6 with p = 3 and e7 with p = 2. 1. Introduction This paper deals with the derivation algebra of the Chevalley Lie algebras of exceptional type {g2, f4, e6, e7, e8}. As it is well-known Chevalley constructed Lie algebras over arbitrary fields F starting from any semisimple (finite-dimensional) Lie algebra over an algebraically closed field of characteristic zero. The key point was to realize that on such algebras one can find a suitable basis whose structure constants are integers. Then, by an scalar extension process he constructed those Lie algebras which nowadays are called Chevalley algebras. By gathering results of Zassenhaus (1939), Seligman (1979), Springer and Stein- berg (we will give full details in the next section) one can get convinced that any derivation of a Chevalley F -algebra of any of the types {g2, f4, e6, e7} is in- ner for char(F ) ≥ 5. Also the derivations of a Chevalley algebra of type e8 with char(F ) ≥ 7, are inner. -
Math 222 Notes for Apr. 1
Math 222 notes for Apr. 1 Alison Miller 1 Real lie algebras and compact Lie groups, continued Last time we showed that if G is a compact Lie group with Z(G) finite and g = Lie(G), then the Killing form of g is negative definite, hence g is semisimple. Now we’ll do a converse. Proposition 1.1. Let g be a real lie algebra such that Bg is negative definite. Then there is a compact Lie group G with Lie algebra Lie(G) =∼ g. Proof. Pick any connected Lie group G0 with Lie algebra g. Since g is semisimple, the center Z(g) = 0, and so the center Z(G0) of G0 must be discrete in G0. Let G = G0/Z(G)0; then Lie(G) =∼ Lie(G0) =∼ g. We’ll show that G is compact. Now G0 has an adjoint action Ad : G0 GL(g), with kernel ker Ad = Z(G). Hence 0 ∼ G = Im(Ad). However, the Killing form Bg on g is Ad-invariant, so Im(Ad) is contained in the subgroup of GL(g) preserving the Killing! form. Since Bg is negative definite, this subgroup is isomorphic to O(n) (where n = dim(g). Since O(n) is compact, its closed subgroup Im(Ad) is also compact, and so G is compact. Note: I found a gap in this argument while writing it up; it assumes that Im(Ad) is closed in GL(g). This is in fact true, because Im(Ad) = Aut(g) is the group of automorphisms of the Lie algebra g, which is closed in GL(g); but proving this fact takes a bit of work. -
Lie Algebras from Lie Groups
Preprint typeset in JHEP style - HYPER VERSION Lecture 8: Lie Algebras from Lie Groups Gregory W. Moore Abstract: Not updated since November 2009. March 27, 2018 -TOC- Contents 1. Introduction 2 2. Geometrical approach to the Lie algebra associated to a Lie group 2 2.1 Lie's approach 2 2.2 Left-invariant vector fields and the Lie algebra 4 2.2.1 Review of some definitions from differential geometry 4 2.2.2 The geometrical definition of a Lie algebra 5 3. The exponential map 8 4. Baker-Campbell-Hausdorff formula 11 4.1 Statement and derivation 11 4.2 Two Important Special Cases 17 4.2.1 The Heisenberg algebra 17 4.2.2 All orders in B, first order in A 18 4.3 Region of convergence 19 5. Abstract Lie Algebras 19 5.1 Basic Definitions 19 5.2 Examples: Lie algebras of dimensions 1; 2; 3 23 5.3 Structure constants 25 5.4 Representations of Lie algebras and Ado's Theorem 26 6. Lie's theorem 28 7. Lie Algebras for the Classical Groups 34 7.1 A useful identity 35 7.2 GL(n; k) and SL(n; k) 35 7.3 O(n; k) 38 7.4 More general orthogonal groups 38 7.4.1 Lie algebra of SO∗(2n) 39 7.5 U(n) 39 7.5.1 U(p; q) 42 7.5.2 Lie algebra of SU ∗(2n) 42 7.6 Sp(2n) 42 8. Central extensions of Lie algebras and Lie algebra cohomology 46 8.1 Example: The Heisenberg Lie algebra and the Lie group associated to a symplectic vector space 47 8.2 Lie algebra cohomology 48 { 1 { 9. -
REPRESENTATION THEORY of REAL GROUPS Contents 1
REPRESENTATION THEORY OF REAL GROUPS ZIJIAN YAO Contents 1. Introduction1 2. Discrete series representations6 3. Geometric realizations9 4. Limits of discrete series 14 Appendix A. Various cohomology theories 15 References 18 1. Introduction 1.1. Introduction. The goal of these notes is to survey some basic concepts and constructions in representation theory of real groups, in other words, \at the infinite places". In particular, some of the definitions reviewed here should be helpful towards the formulation of the Langlands program. To be more precise, we first fix some notation for the introduction and the rest of the survey. Let G be a connected reductive group over R and G = G(R). Let K ⊂ G be an R-subgroup such that K := K(R) ⊂ G is a maximal compact subgroup. Let G0 ⊂ GC be the 1 unique compact real form containing K. By a representation of G we mean a complete locally convex topological C-vector space V equipped with a continuous action G × V ! V . We denote a representation by (π; V ) with π : G ! GL(V ): 2 Following foundations laid out by Harish-Chandra, we focus our attention on irreducible admissible representations of G (Definition 1.2), which are more general than (irreducible) unitary representations and much easier to classify, essentially because they are \algebraic". More precisely, by restricting to the K-finite subspaces V fin we arrive at (g;K)- modules. By a theorem of Harish-Chandra (recalled in Subsection 1.2), there is a natural bijection between Irreducible admissible G- Irreducible admissible representations (g;K)-modules ∼! infinitesimal equivalence isomorphism In particular, this allows us to associate an infinitesimal character χπ to each irreducible ad- missible representation π (Subsection 1.3), which carries useful information for us later in the survey. -
Background on Chevalley Groups Constructed from a Root System
Background on Chevalley Groups Constructed from a Root System Paul Tokorcheck Department of Mathematics University of California, Santa Cruz 10 October 2011 Abstract In 1955, Claude Chevalley described a construction which begins with the root sys- tem of a complex, semisimple Lie algebra g, and yields a corresponding Lie group G which has g as its Lie algebra. His method was improved by Robert Steinberg in 1959. In the cases An, Bn, Cn, and Dn, this method gives the simply connected groups SLn+1, Spin2n+1, Sp2n, and Spin2n, respectively. However, this method can also be used to more easily define the exceptional Lie groups, the analogous groups over arbitrary fields, or even over commutative rings. Such groups, like SLn(Fq) or G2(Qp), fall under the blanket term ’groups of Lie type’. 1 Background Historically, the identification of many abstract simple groups was done via a careful consideration of simple complex Lie groups. For example, if one defines the group SL2(C) to be the group of two-by-two matrices of determinant one, then the field C may be easily replaced by any field or commutative ring (with identity) without alteration of the definition. But many other groups, which were known (at least formally) to exist via an association to a Lie algebra or root system, failed to admit such a simple definition. Theoretically, all such algebraic groups can be considered as subgroups of GLn satisfying some polynomial conditions on the entries, though the size of n and the nature of the polynomial conditions 1 proved to be elusive. -
Integration on Manifolds
CHAPTER 4 Integration on Manifolds §1 Introduction Until now we have studied manifolds from the point of view of differential calculus. The last chapter introduces to our study the methods of integraI calculus. As the new tools are developed in the next sections, the reader may be somewhat puzzied about their reievancy to the earlierd material. Hopefully, the puzziement will be resolved by the chapter's en , but a pre liminary ex ampIe may be heIpful. Let p(z) = zm + a1zm-1 + + a ... m n, be a complex polynomial on p.a smooth compact region in the pIane whose boundary contains no zero of In Sectionp 3 of the previous chapter we showed that the number of zeros of inside n, counting multiplicities, equals the degree of the map argument principle, A famous theorem in complex variabie theory, called the asserts that this may also be calculated as an integraI, f d(arg p). òO 151 152 CHAPTER 4 TNTEGRATION ON MANIFOLDS You needn't understand the precise meaning of the expression in order to appreciate our point. What is important is that the number of zeros can be computed both by an intersection number and by an integraI formula. r Theo ems like the argument principal abound in differential topology. We shall show that their occurrence is not arbitrary orStokes fo rt uitoustheorem. but is a geometrie consequence of a generaI theorem known as Low dimensionaI versions of Stokes theorem are probably familiar to you from calculus : l. Second Fundamental Theorem of Calculus: f: f'(x) dx = I(b) -I(a). -
The Symplectic Group Is Obviously Not Compact. for Orthogonal Groups It
The symplectic group is obviously not compact. For orthogonal groups it was easy to find a compact form: just take the orthogonal group of a positive definite quadratic form. For symplectic groups we do not have this option as there is essentially only one symplectic form. Instead we can use the following method of constructing compact forms: take the intersection of the corresponding complex group with the compact subgroup of unitary matrices. For example, if we do this with the complex orthogonal group of matrices with AAt = I and intersect t it with the unitary matrices AA = I we get the usual real orthogonal group. In this case the intersection with the unitary group just happens to consist of real matrices, but this does not happen in general. For the symplectic group we get the compact group Sp2n(C) ∩ U2n(C). We should check that this is a real form of the symplectic group, which means roughly that the Lie algebras of both groups have the same complexification sp2n(C), and in particular has the right dimension. We show that if V is any †-invariant complex subspace of Mk(C) then the Hermitian matrices of V are a real form of V . This follows because any matrix v ∈ V can be written as a sum of hermitian and skew hermitian matrices v = (v + v†)/2 + (v − v†)/2, in the same way that a complex number can be written as a sum of real and imaginary parts. (Of course this argument really has nothing to do with matrices: it works for any antilinear involution † of any complex vector space.) Now we need to check that the Lie algebra of the complex symplectic group is †-invariant. -
Chevlie: Constructing Lie Algebras and Chevalley Groups
Journal of Software for Algebra and Geometry ChevLie: Constructing Lie algebras and Chevalley groups MEINOLF GECK vol 10 2020 JSAG 10 (2020), 41–49 The Journal of Software for https://doi.org/10.2140/jsag.2020.10.41 Algebra and Geometry ChevLie: Constructing Lie algebras and Chevalley groups MEINOLF GECK ABSTRACT: We present ChevLie-1.1, a module for Julia and, ultimately, the emerging OSCAR system. It provides functions for constructing simple Lie algebras and the corresponding Chevalley groups (of adjoint or other types), using a recently established approach via Lusztig’s “canonical bases”. These programs, combined with the Julia interface to SINGULAR, supply an efficient, user-friendly way to establish a key part of a new characterisation of Lusztig’s “special” nilpotent orbits in simple Lie algebras. 1. THE -CANONICAL CHEVALLEY BASIS OF A LIE ALGEBRA. Let g be a finite-dimensional, simple Lie algebra over C. By the classical Cartan–Killing theory (see[Humphreys 1978]), one can associate with g a Dynkin diagram 0 (that is, one of the graphs in Figure 1 below) and g is, up to isomorphism, uniquely determined by 0. Conversely, an elegant way to construct a Lie algebra g corresponding to such a diagram 0 is given by taking the quotient of the free Lie algebra on generators fei ; fi j i 2 I g (where I is an index set for the nodes of 0) by the ideal generated by the Serre relations in[Humphreys 1978, (18.1)] (which only depend on 0). The general theory then shows that g has a basis B D fhi j i 2 I g [ feα j α 2 8g; where the elements hi VD Tei ; fi U (i 2 I ) span a Cartan subalgebra h ⊆ g, the set 8 is the root system determined by 0, and the eα are chosen such that Thi ; eαU 2 Ceα for all i 2 I ; the eα are unique up to nonzero scalar multiples. -
Math 261A: Lie Groups, Fall 2008 Problems, Set 1 1. (A) Show That the Orthogonal Groups On(R)
Math 261A: Lie Groups, Fall 2008 Problems, Set 1 1. (a) Show that the orthogonal groups On(R) and On(C) have two connected com- ponents, the identity component being the special orthogonal group SOn, and the other consisting of orthogonal matrices of determinant −1. (b) Show that the center of On is {±Ing. ∼ (c) Show that if n is odd, then SOn has trivial center and On = SOn × (Z=2Z) as a Lie group. (d) Show that if n is even, then the center of SOn has two elements, and On is a semidirect product (Z=2Z) n SOn, where Z=2Z acts on SOn by a non-trivial outer automorphism of order 2. 2. Problems 5-9 in Knapp Intro x6, which lead you through the construction of a smooth group homomorphism Φ: SU(2) ! SO(3) which induces an isomorphism of Lie algebras and identifies SO(3) with the quotient of SU(2) by its center {±Ig. 3. Construct an isomorphism of GL(n; C) (as a Lie group and an algebraic group) with a closed subgroup of SL(n + 1; C). 4. Show that the map C∗ × SL(n; C) ! GL(n; C) given by (z; g) 7! zg is a surjective homomorphism of Lie and algebraic groups, find its kernel, and describe the corresponding homomorphism of Lie algebras. 5. Find the Lie algebra of the group U ⊆ GL(n; C) of upper-triangular matrices with 1 on the diagonal. Show that for this group, the exponential map is a diffeomorphism of the Lie algebra onto the group.