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Lie Algebras Lie algebras Course Notes Alberto Elduque Departamento de Matem´aticas Universidad de Zaragoza 50009 Zaragoza, Spain ©2005-2021 Alberto Elduque These notes are intended to provide an introduction to the basic theory of finite dimensional Lie algebras over an algebraically closed field of characteristic 0 and their representations. They are aimed at beginning graduate students in either Mathematics or Physics. The basic references that have been used in preparing the notes are the books in the following list. By no means these notes should be considered as an alternative to the reading of these books. N. Jacobson: Lie algebras, Dover, New York 1979. Republication of the 1962 original (Interscience, New York). J.E. Humphreys: Introduction to Lie algebras and Representation Theory, GTM 9, Springer-Verlag, New York 1972. W. Fulton and J. Harris: Representation Theory. A First Course, GTM 129, Springer-Verlag, New York 1991. W.A. De Graaf: Lie algebras: Theory and Algorithms, North Holland Mathemat- ical Library, Elsevier, Amsterdan 2000. Contents 1 A short introduction to Lie groups and Lie algebras 1 § 1. One-parameter groups and the exponential map . 2 § 2. Matrix groups . 5 § 3. The Lie algebra of a matrix group . 7 2 Lie algebras 17 § 1. Theorems of Engel and Lie . 17 § 2. Semisimple Lie algebras . 22 § 3. Representations of sl2(k)........................... 29 § 4. Cartan subalgebras . 31 § 5. Root space decomposition . 34 § 6. Classification of root systems . 38 § 7. Classification of the semisimple Lie algebras . 51 § 8. Exceptional Lie algebras . 55 3 Representations of semisimple Lie algebras 61 § 1. Preliminaries . 61 § 2. Properties of weights and the Weyl group . 64 § 3. Universal enveloping algebra . 68 § 4. Irreducible representations . 71 § 5. Freudenthal's multiplicity formula . 74 § 6. Characters. Weyl's formulae . 79 § 7. Tensor products decompositions . 86 A Simple real Lie algebras 91 § 1. Real forms . 91 § 2. Involutive automorphisms . 98 § 3. Simple real Lie algebras . 106 v Chapter 1 A short introduction to Lie groups and Lie algebras This chapter is devoted to give a brief introduction to the relationship between Lie groups and Lie algebras. This will be done in a concrete way, avoiding the general theory of Lie groups. It is based on the very nice article by R. Howe: \Very basic Lie Theory", Amer. Math. Monthly 90 (1983), 600{623. Lie groups are important since they are the basic objects to describe the symmetry. This makes them an unavoidable tool in Geometry (think of Klein's Erlangen Program) and in Theoretical Physics. A Lie group is a group endowed with a structure of smooth manifold, in such a way that both the algebraic group structure and the smooth structure are compatible, in the sense that both the multiplication ((g; h) 7! gh) and the inverse map (g 7! g−1) are smooth maps. To each Lie group a simpler object may be attached: its Lie algebra, which almost determines the group. Definition. A Lie algebra over a field k is a vector space g, endowed with a bilinear multiplication [:; :]: g × g −! g (x; y) 7! [x; y]; satisfying the following properties: [x; x] = 0 (anticommutativity) [[x; y]; z] + [[y; z]; x] + [[z; x]; y] = 0 (Jacobi identity) for any x; y; z 2 g. Example. Let A be any associative algebra, with multiplication denoted by juxtaposi- tion. Consider the new multiplication on A given by [x; y] = xy − yx for any x; y 2 A. It is an easy exercise to check that A, with this multiplication, is a Lie algebra, which will be denoted by A−. 1 2 CHAPTER 1. INTRODUCTION TO LIE GROUPS AND LIE ALGEBRAS As for any algebraic structure, one can immediately define in a natural way the concepts of subalgebra, ideal, homomorphism, isomorphism, ..., for Lie algebras. The most usual Lie groups and Lie algebras are \groups of matrices" and their Lie algebras. These concrete groups and algebras are the ones that will be considered in this chapter, thus avoiding the general theory. § 1. One-parameter groups and the exponential map Let V be a real finite dimensional normed vector space with norm k:k. (So that V is n isomorphic to R .) Then EndR(V ) is a normed space with kAvk kAk = sup : 0 6= v 2 V kvk = sup fkAvk : v 2 V and kvk = 1g The determinant provides a continuous (even polynomial) map det : EndR(V ) ! R. Therefore −1 GL(V ) = det R n f0g is an open set of EndR(V ), and it is a group. Moreover, the maps GL(V ) × GL(V ) ! GL(V ) GL(V ) ! GL(V ) (A; B) 7! AB A 7! A−1 are continuous. (Actually, the first map is polynomial, and the second one rational, so they are smooth and even analytical maps. Thus, GL(V ) is a Lie group.) One-parameter groups A one-parameter group of transformations of V is a continuous group homomorphism φ :(R; +) −! GL(V ): Any such one-parameter group φ satisfies the following properties: 1.1 Properties. (i) φ is differentiable. R t 0 Proof. Let F (t) = 0 φ(u)du. Then F (t) = φ(t) for any t and for any t; s: Z t+s F (t + s) = φ(u)du 0 Z t Z t+s = φ(u)du + φ(u)du 0 t Z t Z t+s = φ(u)du + φ(t)φ(u − t)du 0 t Z s = F (t) + φ(t) φ(u)du 0 = F (t) + φ(t)F (s): § 1. ONE-PARAMETER GROUPS AND THE EXPONENTIAL MAP 3 But F (s) F 0(0) = lim = φ(0) = I s!0 s (the identity map on V ), and the determinant is continuous, so F (s) det F (s) lim det = lim = 1 6= 0; s!0 s s!0 sn and hence a small s0 can be chosen with invertible F (s0). Therefore −1 φ(t) = F (t + s0) − F (t) F (s0) is differentiable, since so is F . (ii) There is a unique A 2 EndR(V ) such that 1 ! X tnAn φ(t) = etA = : n! n=0 P1 An n n (Note that the series exp(A) = n=0 n! converges absolutely, since kA k ≤ kAk , and uniformly on each bounded neighborhood of 0, in particular on Bs(0) = fA 2 EndR(V ): kAk < sg, for any 0 < s 2 R, and hence it defines a smooth, in fact 0 analytic, map from EndR(V ) to itself.) Besides, A = φ (0). Proof. For any 0 6= v 2 V , let v(t) = φ(t)v. In this way, we have defined a map R ! V , t 7! v(t), which is differentiable and satisfies v(t + s) = φ(s)v(t) for any s; t 2 R. Differentiate with respect to s for s = 0 to get ( v0(t) = φ0(0)v(t); v(0) = v; which is a linear system of differential equations with constant coefficients. By elementary linear algebra(!), it follows that 0 v(t) = etφ (0)v for any t. Moreover, 0 φ(t) − etφ (0) v = 0 for any v 2 V , and hence φ(t) = etφ0(0) for any t. tA (iii) Conversely, for any A 2 EndR(V ), the map t 7! e is a one-parameter group. Proof. If A and B are two commuting elements in EndR(V ), then n p n q n r A B X A X B X (A + B) e e = lim = lim + Rn(A; B) ; n!1 p! q! n!1 r! p=0 q=0 r=0 4 CHAPTER 1. INTRODUCTION TO LIE GROUPS AND LIE ALGEBRAS with X Ap Bq R (A; B) = ; n p! q! 1≤p;q≤n p+q>n so 2n r X kAkp kBkq X kAk + kBk kR (A; B)k ≤ ≤ ; n p! q! r! 1≤p;q≤n r=n+1 p+q>n whose limit is 0. Hence, eAeB = eA+B. Now, given any A 2 EndR(V ) and any scalars t; s 2 R, tA commutes with sA, so φ(t + s) = etA+sA = etAesA = φ(t)φ(s), thus proving that φ is a group homomor- phism. The continuity is clear. (iv) There exists a positive real number r and an open set U in GL(V ) contained in r Bs(I), with s = e − 1, such that the \exponential map": exp : Br(0) −! U A 7! exp(A) = eA is a homeomorphism. Proof. exp is differentiable because of its uniform convergence. Moreover, its dif- ferential at 0 satisfies: etA − e0 d exp(0)(A) = lim = A; t!0 t so that d exp(0) = id (the identity map on EndR(V )) and the Inverse Function Theorem applies. A P1 An A P1 kAkn kAk Moreover, e − I = n=1 n! , so ke − Ik ≤ n=1 n! = e − 1. Thus U ⊆ Bs(I). Note that for V = R (dim V = 1), GL(V ) = R n f0g and exp : R ! R n f0g is not onto, since it does not take negative values. 2 0 −1 Also, for V = R , identify EndR(V ) with Mat2(R). Then, with A = 1 0 , it 2 −1 0 3 0 1 4 tA cos t − sin t follows that A = 0 −1 , A = −1 0 and A = I. It follows that e = sin t cos t . In particular, etA = e(t+2π)A and, therefore, exp is not one-to-one. Adjoint maps −1 1. For any g 2 GL(V ), the linear map Ad g : EndR(V ) ! EndR(V ), A 7! gAg , is an inner automorphism of the associative algebra EndR(V ). The continuous group homomorphism Ad : GL(V ) −! GL(EndR(V )) g 7! Ad g; is called the adjoint map of GL(V ).
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