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Lie Groups, Lie Algebras, and Their Representations

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Lie Groups, Lie Algebras, and Their Representations Lie groups, Lie algebras, and their representations Gwyn Bellamy [email protected] www.maths.gla.ac.uk/∼gbellamy September 16, 2016 Abstract These are the lecture notes for the 5M reading course "Lie groups, Lie algebras, and their representations" at the University of Glasgow, autumn 2015. Contents 1 Introduction i 2 Manifolds - a refresher 2 3 Lie groups and Lie algebras 11 4 The exponential map 20 5 The classical Lie groups and their Lie algebras 30 6 Representation theory 35 7 The structure of Lie algebras 40 8 Complete reducibility 48 9 Cartan subalgebras and Dynkin diagrams 54 10 The classification of simple, complex Lie algebras 65 11 Weyl's character formula 69 12 Appendix: Quotient vector spaces 76 1 Introduction Lie groups and Lie algebras, together called Lie theory, originated in the study of natural symme- tries of solutions of differential equations. However, unlike say the finite collection of symmetries of the hexagon, these symmetries occurred in continuous families, just as the rotational symmetries of the plane form a continuous family isomorphic to the unit circle. 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 classification and representation theory. Lie's ideas played a central role in Felix Klein's grand "Erlangen program" to classify all possible geometries using group theory. Today Lie theory plays an important role in almost every branch of pure and applied mathematics, is used to describe much of modern physics, in particular classical and quantum mechanics, and is an active area of research. You might be familiar with the idea that abstract group theory really began with Galois' work on algebraic solutions of polynomial equations; in particular, the generic quintic. But, in a sense, the idea of groups of transformations had been around for a long time already. As mentioned above, it was already present in the study of solutions of differential equations coming from physics (for instance in Newton's work). The key point is that these spaces of solutions are often stable under the action of a large group of symmetries, or transformations. This means that applying a given transformation to one solution of the differential equation gives us another solution. Hence one can quickly and easily generate new solutions from old one, "buy one, get one free". As a toy example, consider a "black hole" centred at (0; 0) in R2. Then, any particle on R2 r f(0; 0)g, whose position at time t is given by (x(t); y(t)), will be attracted to the origin, the strength of the attraction increasing as it nears (0; 0). This attraction can be encoded by the system of differential equations dx −x dy −y = ; = : (1) dt (x2 + y2)3=2 dt (x2 + y2)3=2 In polar coordinates (r(t); θ(t)), this becomes dr −1 = : (2) dt r2 i 1 2 The circle S = f0 ≤ # < 2πg (this is a group, #1 ?#2 = (#1 +#2) mod 2π) acts on R by rotations: ! ! ! cos # − sin # x x cos # − y sin # # · (x; y) = = : sin # cos # y x sin # + y cos # Then it is clear from (2) that if (x(t); y(t)) is a solution to (1), then so too is # · (x(t); y(t)) = (x(t) cos # − y(t) sin #; x(t) sin # + y(t) cos #): Thus, the group S1 acts on the space of solutions to this system of differential equations. The content of these lecture notes is based to a large extent on the material in the books [5] and [8]. Other sources that treat the material in these notes are [1], [2], [4], [9] and [7]. 1 2 Manifolds - a refresher In this course we will consider smooth real manifolds and complex analytic manifolds. 2.1 Topological terminology Every manifold is in particular a topological space, therefore we will occasionally need some basic topological terms. We begin by recalling that a topological space is a pair (X; T ) where X is a set and T is a collection of subsets of X, the open subsets of X. The collection of open subsets T must include X and the emptyset ;, be closed under arbitrary unions and finite intersections. Recall that a topological space M is disconnected if there exist two non-empty open and closed subsets U; V ⊂ M such that U \ V = ;; otherwise M is connected. It is path connected if, for any two points x; y 2 X, there is a path from x to y (every path connected space is connected, but the converse is not true). We say that X is simply connected if it is path connected and the fundamental group π1(X) of X is trivial i.e. every closed loop in X is homotopic to the trivial loop. The space X is compact if every open covering admits a finite subcover. The space X is said to be Hausdorff if for each pair of distinct points x; y 2 X there exists open sets x 2 U, y 2 V with U \ V = ;. n ◦ n For each ` 2 R+ and x 2 R , we denote by B`(x), resp. B`(x) , the set of all points y 2 R such that jjx − yjj ≤ `, resp. jjx − yjj < `. We will always consider Rn as a topological space, equipped with the Euclidean topology i.e. a base for this topology is given by the open balls ◦ B`(x) . Recall that the Heine-Borel Theorem states: Theorem 2.1. Let M be a subset of Rn. Then, M is compact if and only if it is closed and contained inside a closed ball B`(0) for some ` 0. n Remark 2.2. If M is some compact subspace of R and fXigi2N a Cauchy sequence in M, then the limit of this sequence exists in M. This fact is a useful way of showing that certain subspaces of Rn are not compact. 2.2 Manifolds Recall that a real n-dimensional manifold is a Hausdorff topological space M with an atlas A(M) = fφi : Ui ! Vi j i 2 Ig, where fUi j i 2 Ig is an open cover of M, each chart φi : Ui ! Vi is a n homeomorphism onto Vi, an open subset of R , and the composite maps −1 φi ◦ φj : Vi;j ! Vj;i 2 n n are smooth morphisms, where Vi;j = φj(Ui \ Uj) ⊂ R and Vj;i = φi(Ui \ Uj) ⊂ R . 1 2 2 2 Example 2.3. The circle S = f(x1; x2) j x1 +x2 = 1g ⊂ R is a manifold. We can use stereographic projection to construct charts on S1. Let N = (0; 1) and S = (0; −1) be the north and south poles respectively. N P = (x1; x2) Q = (z; 0) S 1 A line through N and P meets the x1-axis in a unique point. This defines a map S r fNg ! R. To get an explicit formula for this, the line through N and P is described by ft(x1; x2) + (1 − x1 1 t)(0; 1) j t 2 g. This implies that z = . Therefore we define U1 = S fNg, V1 = and R 1−x2 r R x1 φ1 : U1 ! V1; φ1(x1; x2) = : 1 − x2 Similarly, if we perform stereographic projection from the south pole, then we get a chart x1 φ2 : U2 ! V2; φ2(x1; x2) = ; x2 + 1 1 where U2 = S r fSg and V2 = R. −1 For this to define a manifold, we need to check that the transition map φ2 ◦φ1 is smooth. Since 1 −1 U1 \ U1 = S r fN; Sg, φ1(U1 \ U2) = φ2(U1 \ U2) = R r f0g. Hence φ2 ◦ φ1 : R r f0g ! R r f0g. −1 1 A direct calculation shows that φ2 ◦ φ1 (z) = z . This is clearly smooth. −1 A continuous function f : M ! R is said to be smooth if f ◦ φi : Vi ! R is a smooth, i.e. infinitely differentiable, function for all charts φi. The space of all smooth functions on M is denoted C1(M). Example 2.4. The real projective space RPn of all lines through 0 in Rn+1 is a manifold. The n n+1 set of points in RP are written [x0 : x1 : ··· : xn], where (x0; x1; : : : ; xn) 2 R r f0g and × [x0 : x1 : ··· : xn] represents the line through 0 and (x0; x1; : : : ; xn). Thus, for each λ 2 R , [x0 : x1 : ··· : xn] = [λx0 : λx1 : ··· : λxn]: 3 n n ∼ n The n + 1 open sets Ui = f[x0 : x1 : ··· : xn] 2 RP j xi 6= 0g cover RP . The maps φi : Ui −! R , x0 xn n [x0 : x1 : ··· : xn] 7! ( ;:::; xi;:::; ) define an atlas on . Thus, it is a manifold. xi b xi RP Exercise 2.5. Check that the maps φi are well-defined i.e. only depends on the line through n −1 (x0; x1; : : : ; xn). Prove that RP is a manifold by explicitly describing the maps φj ◦ φi : φi(Ui \ n n Uj) ⊂ R ! φj(Ui \ Uj) ⊂ R and checking that they are indeed smooth map. The following theorem is extremely useful for producing explicit examples of manifolds. n m Theorem 2.6. Let m < n and f : R ! R , f = (f1(x1; : : : ; xn); : : : ; fm(x1; : : : ; xn)), a smooth map. Assume that u is in the image of f. Then f −1(u) is a smooth submanifold of Rn, of dimension n − m if and only if the differential @f i=1;:::;m d f = i : n ! m v @x R R j j=1;:::;njx=v is a surjective linear map for all v 2 f −1(u).
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