Introduction to Lie Groups and Lie Algebras
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Introduction to Lie Groups and Lie Algebras Alexander Kirillov, Jr. Department of Mathematics, SUNY at Stony Brook, Stony Brook, NY 11794, USA E-mail address: [email protected] URL: http://www.math.sunysb.edu/~kirillov Contents Chapter 1. Introduction 7 Chapter 2. Lie Groups: Basic Deﬁnitions 9 §2.1. Lie groups, subgroups, and cosets 9 §2.2. Action of Lie groups on manifolds and representations 12 §2.3. Orbits and homogeneous spaces 13 §2.4. Left, right, and adjoint action 14 §2.5. Classical groups 15 Exercises 18 Chapter 3. Lie Groups and Lie algebras 21 §3.1. Exponential map 21 §3.2. The commutator 23 §3.3. Adjoint action and Jacobi identity 24 §3.4. Subalgebras, ideals, and center 25 §3.5. Lie algebra of vector ﬁelds 26 §3.6. Stabilizers and the center 28 §3.7. Campbell–Hausdorﬀ formula 29 §3.8. Fundamental theorems of Lie theory 30 §3.9. Complex and real forms 34 §3.10. Example: so(3, R), su(2), and sl(2, C). 35 Exercises 36 Chapter 4. Representations of Lie Groups and Lie Algebras 39 §4.1. Basic deﬁnitions 39 §4.2. Operations on representations 41 §4.3. Irreducible representations 42 §4.4. Intertwining operators and Schur lemma 43 §4.5. Complete reducibility of unitary representations. Representations of ﬁnite groups 45 §4.6. Haar measure on compact Lie groups 46 3 4 Contents §4.7. Orthogonality of characters and Peter-Weyl theorem 48 §4.8. Universal enveloping algebra 51 §4.9. Poincare-Birkhoﬀ-Witt theorem 53 Exercises 55 Chapter 5. Representations of sl(2, C) and Spherical Laplace Operator 59 §5.1. Representations of sl(2, C) 59 §5.2. Spherical Laplace operator and hydrogen atom 62 Exercises 65 Chapter 6. Structure Theory of Lie Algebras 67 §6.1. Ideals and commutant 67 §6.2. Solvable and nilpotent Lie algebras 68 §6.3. Lie and Engel theorems 70 §6.4. The radical. Semisimple and reductive algebras 71 §6.5. Invariant bilinear forms and semisimplicity of classical Lie algebras 74 §6.6. Killing form and Cartan criterion 75 §6.7. Properties of semisimple Lie algebras 77 §6.8. Relation with compact groups 78 §6.9. Complete reducibility of representations 79 Exercises 82 Chapter 7. Complex Semisimple Lie Algebras 83 §7.1. Semisimple elements and toroidal subalgebras 83 §7.2. Cartan subalgebra 85 §7.3. Root decomposition and root systems 86 Exercises 90 Chapter 8. Root Systems 91 §8.1. Abstract root systems 91 §8.2. Automorphisms and Weyl group 92 §8.3. Pairs of roots and rank two root systems 93 §8.4. Positive roots and simple roots 95 §8.5. Weight and root lattices 97 §8.6. Weyl chambers 98 §8.7. Simple reﬂections 101 §8.8. Dynkin diagrams and classiﬁcation of root systems 103 §8.9. Serre relations and classiﬁcation of semisimple Lie algebras 106 Exercises 108 Chapter 9. Representations of Semisimple Lie Algebras 111 §9.1. Weight decomposition 111 §9.2. Highest-weight representations and Verma modules 112 Contents 5 §9.3. Classiﬁcation of irreducible ﬁnite-dimensional representations 114 §9.4. Bernstein–Gelfand–Gelfand resolution 116 §9.5. Characters and Weyl character formula 117 §9.6. Representations of sl(n, C) 117 §9.7. Proof of Theorem 9.19 117 Exercises 117 Appendix A. Manifolds and immersions 119 Appendix B. Jordan Decomposition 121 Exercises 122 Appendix C. Root Systems and Simple Lie Algebras 123 §C.1. An = sl(n + 1, C) 124 §C.2. Bn = so(2n + 1, C) 125 §C.3. Cn = sp(n, C) 127 §C.4. Dn = so(2n, C) 128 List of Notation 131 Linear algebra 131 Diﬀerential geometry 131 Lie groups and Lie algebras 131 Representations 131 Semisimple Lie algebras and root systems 131 Index 133 Bibliography 135 Chapter 1 Introduction What are Lie groups and why do we want to study them? To illustrate this, let us start by considering this baby example. Example 1.1. Suppose you have n numbers a1, . , an arranged on a circle. You have a transfor- an+a2 a1+a3 mation A which replaces a1 with 2 , a2 with 2 , and so on. If you do this suﬃciently many times, will the numbers be roughly equal? To answer this we need to look at the eigenvalues of A. However, explicitly computing the characteristic polynomial and ﬁnding the roots is a rather diﬃcult problem. But we can note that the problem has rotational symmetry: if we denote by B the operator of rotating the circle by 2π/n, −1 i.e. sending (a1, . , an) to (a2, a3, . , an, a1), then BAB = A. Since the operator B generates the group of cyclic permutation Zn, we will refer to this as the Zn symmetry. So we might try to make use of this symmetry to ﬁnd eigenvectors and eigenvalues of A. Naive idea would be to just consider B-invariant eigenvectors. It is easy to see that there is only one such vector, and while it is indeed an eigenvector for A, this is not enough to diagonalize A. Instead, we can use the following well-known result from linear algebra: if linear operators A, B in a vector space V commute, and V is an eigenspace for B, then AV ⊂ V . Thus, if the operator L λ λ λ B is diagonalizable so that V = Vλ, then A preserves this decomposition and thus the problem reduces to diagonalizing A on each of Vλ separately, which is a much easier problem. In this case, since Bn = id, its eigenvalues must be n-th roots of unity. Denoting by ε = e2πi/n the primitive n-th root of unity, it is easy to check that indeed, each n-th root of unity λ = εk, k = k 2k (n−1)k 0 . , n − 1, is an eigenvalue of B, with eigenvector vk = (1, ² , ² , . , ² ). Thus, in this case each eigenspace for B is one-dimensional, so each vk must also be an eigenvector for A. Indeed, one εk+ε−k immediately sees that Avk = 2 vk. This is the baby version of a real life problem. Consider S2 ⊂ R3. Deﬁne the Laplace operator ∞ 2 ∞ 2 ˜ ˜ 3 ∆sph : C (S ) → C (S ) by ∆sphf = (∆f)|S2 , where f is the result of extending f to R − {0} 3 (constant along each ray), and ∆ is the usual Laplace operator in R . It is easy to see that ∆sph is a second order diﬀerential operator on the sphere; one can write explicit formulas for it in the spherical coordinates, but they are not particularly nice. For many applications, it is important to know the eigenvalues and eigenfunctions of ∆sph. In particular, this problem arises in qunatum mechanics: the eigenvalues are related to the energy levels of a hydrogen atom in quantum mechanical description. Unfortunately, trying to ﬁnd the eigenfunctions by brute force gives a second-order diﬀerential equation which is very diﬃcult to solve. 7 8 1. Introduction But much as in the baby example, it is easy to notice that this problem also has some symmetry — namely, the group SO(3, R) acting on the sphere by rotations. However, trying to repeat the approach used in the baby example (which had Zn-symmetry) immediately runs into the following two problems: • SO(3, R) is not a ﬁnitely generated group, so we can not use just one (or ﬁnitely many) operators Bi and consider their common eigenspaces. • SO(3, R) is not commutative, so diﬀerent operators from SO(3, R) can not be diagonalized simultaneously. The goal of the theory of Lie groups is to give tools to deal with these (and similar) problems. In short, the answer to the ﬁrst problem is that SO(3, R) is in a certain sense ﬁnitely generated — namely, it is generated by three generators, “inﬁnitesimal rotations” around x, y, z axes (see details in Example 3.10). The answer to the second problem is that instead of decomposing the C∞(S2) into a direct sum of common eigenspaces for operators B ∈ SO(3, R), we need to decompose it into “irreducible representations” of SO(3, R). In order to do this, we need to develop the theory of representations of SO(3, R). We will do this and complete the analysis of this example in Chapter 5. Chapter 2 Lie Groups: Basic Deﬁnitions 2.1. Lie groups, subgroups, and cosets Deﬁnition 2.1. A Lie group is a set G with two structures: G is a group and G is a (smooth, real) manifold. These structures agree in the following sense: multiplication and inversion are smooth maps. A morphism of Lie groups is a smooth map which also preserves the group operation: f(gh) = f(g)f(h), f(1) = 1. In a similar way, one deﬁnes complex Lie groups. However, unless speciﬁed otherwise, “Lie group” means a real Lie group. Remark 2.2. The word “smooth” in the deﬁnition above can be understood in diﬀerent ways: C1,C∞, analytic. It turns out that all of them are equivalent: every C0 Lie group has a unique analytic structure. This is a highly non-trivial result (it was one of Hilbert’s 20 problems), and we are not going to prove it (proof of a weaker result, that C2 implies analyticity, is much easier and can be found in [10, Section 1.6]). In this book, “smooth” will be always understood as C∞. Example 2.3. The following are examples of Lie groups (1) Rn, with the group operation given by addition (2) R∗, × R+, × (3) S1 = {z ∈ C : |z| = 1}, × 2 (4) GL(n, R) ⊂ Rn . Many of the groups we will consider will be subgroups of GL(n, R) or GL(n, C).