REPRESENTATIONS OF LIE GROUPS AND PHYSICS Maxim Jeffs December 12, 2016 1 Introduction This report re-presents the representation theory of general Lie groups from the perspective of Lie algebras, followed by a mathematical exposition of the representations of significance in Physics. We begin with a presentation of the basic theory of Lie groups; after stating the fundamental theorems relating Lie groups and Lie algebras, we use these to study projective representations. To finish, two important examples in Quantum " Physics are given: SO(3) and the Poincaré group ISO (3; 1). The author (modestly) hopes that this summary might prove useful to future students with a similar state of knowledge; he also wishes to thank Dr. David Ridout and Dr. Simon Wood for suffering gladly through numerous ‘questionings’. 2 Lie Groups A Lie group G is a smooth manifold that is also a group, in such a way that multiplication m : G × G ! G and inversion i : G ! G are smooth maps (for definitions and notation, see [9] [10]). A Lie group homomorphism is a group homomorphism that is also a smooth map; the correct notion of a ‘Lie subgroup’ is more subtle: see Appendix 1. We also have the notion of a universal covering group (for details on smooth fundamental groups, see [11] ch. 6; for a proof of the existence of a universal covering manifold, see [10] p. 91): Proposition 1. If G is a connected Lie group, then its universal covering manifold G~ has a canonical structure of a group in such a way that the covering map θ : G~ ! G is a Lie group homomorphism and the kernel is a discrete normal subgroup isomorphic to π1(G). This is unique up to Lie group isomorphism. Proof. For a detailed proof of Proposition 1, see [10] p.154; we sketch the main idea, following [8]. From standard covering g space theory, we know that, given any map f : G ! G, a point g 2 G, and liftings g;~ f(g) 2 G~, there exists a unique g map f~ : G~ ! G~ such that f~(~g) = f(g). Hence, choosing a lifting of e, we can uniquely lift the multiplication map m in such a way that m~ (~e; e~) =e ~. It is then straightforward to show that the operation m~ actually defines a group on G~, and is essentially the unique compatible way of doing so. Finally, one needs to note that the fundamental group of a smooth manifold is countable [10]. The interaction between algebraic and topological properties plays an important role in Lie theory: Proposition 2. Any discrete normal subgroup of a Lie group is central [8]. Proof. Suppose H is a discrete normal subgroup of G and fix h 2 H. Consider the smooth map f : G0 ! H given by f(g) = ghg−1, where G0 is the identity component of G. Since the image of f must be connected and contain h, it must be exactly h. Proposition 3. If G is a connected Lie group, it is generated by any neighbourhood of the identity [10]. Proof. Suppose U is a neighbourhood of the identity in G and let W be the subgroup it generates. If g 2 W , then gU ⊂ W ; since gU is an open neighbourhood of g, W is open. But W is also closed, since [ W c = gW g2 =W is open as the union of open sets. Thus W = G. 1 The preceding proposition explains why the behaviour of a Lie group homomorphism near the identity largely determines its global properties: every Lie group homomorphism has constant rank [10] and Corollary 1. Suppose f : G1 ! G2 is a Lie group homomorphism such that f∗ : TeG1 ! TeG2 is surjective; then if G2 is connected, f is surjective [8]. Corollary 2. If f : G1 ! G2 is a Lie group homomorphism of connected Lie groups G1;G2 such that f∗ : TeG1 ! TeG2 is an isomorphism, then f is a covering map and G2 is the quotient of G1 by some discrete normal subgroup [10] p. 557. 3 The Exponential Map A vector field X on G is called left-invariant if (Lg)∗X = X for all g 2 G, where Lg : G ! G is the left-multiplication map defined by Lg(h) = gh. As the Lie bracket is natural with respect to pushforwards, that is, ϕ∗[X; Y ] = [ϕ∗X; ϕ∗Y ] for any smooth map ϕ (where we interpret ϕ∗X as ϕ-relatedness if ϕ is not a diffeomorphism [9]), the vector space of left-invariant vector fields must be closed under the Lie bracket; we call it g, the Lie algebra of G. As a vector space, g is naturally isomorphic to TeG under the map that takes X 2 TeG and ‘pushes it around’ to give a vector field: X(g) := (Lg)∗(X). Therefore we have the following: ∼ Proposition 4. If H is a Lie subgroup of G, then h = TeH is a subalgebra of g. Proof. Since TeH ⊂ TeG, we can consider h ⊂ g. Naturality of the Lie bracket with respect to the inclusion map ι : H ! G then shows that h is a subalgebra. We might ask whether the converses to the above statements are true: for every Lie algebra, is there a corresponding Lie group, and for every Lie subalgebra, is there a corresponding Lie subgroup? These will be answered in the next section, but first, to study the relationship between Lie groups and Lie algebras generally, we need an analogue of the matrix exponential. The matrix exponential is the solution of a certain ODE, and we now use this to define it. X R × ! Definition 1. A global flow for a vector field X on a manifold M is a map ϕt : M M such that: 2 R X ! 1. for each t , ϕt is a diffeomorphism M M; 2 R X X ◦ X 2. for all t; s , ϕt+s = ϕt ϕs ; X X X 3. the tangent vector to ϕt (p) is X(ϕt (p)), that is, each ϕt (p) is an integral curve for X. Existence theory for ODE can be used to show that any vector field has a so-called ‘local’ flow, but these cannot generally be extended to global flows [9]. However, a left-invariant vector field on a Lie group always induces a global X X X flow; since X is left-invariant, we can extend any (local) integral curve ϕt (e) to all of G by taking ϕt (g) := gϕt (e). Uniqueness for ODE then implies that this gives a global flow. Therefore we have a Lie group homomorphism R ! G 7! X given by t ϕt (e), by property 2, and we call this the 1-parameter subgroup associated with X (the homomorphism, ! 7! X not the subgroup!) We also get a map g G given by X ϕ1 (e), called the exponential map and denoted exp(X); the 1-parameter subgroup generated by X is then given by exp(tX). This map is smooth (this follows from ODE theory; see [10] p.520 for a detailed proof) and has the following key properties: Proposition 5. The exponential map is a local diffeomorphism from a neighbourhood of 0 in g to a neighbourhood of e in G [8]. The local smooth inverse is sometimes denoted by log. ∼ ∼ Proof. The differential of exp, from T0(g) = g to TeG = g is simply the identity map. The proposition then follows from the inverse function theorem. Corollary 3. If G is a connected Lie group, it is generated by exponentials of elements of g. Proposition 6. The exponential map is natural with respect to Lie group homomorphisms [2], that is, given a Lie group homomorphism Φ: G1 ! G2, its differential Φ∗ : TeG1 ! TeG2 gives a Lie algebra homomorphism (called the induced homomorphism), related to Φ by Φ(exp(X)) = exp(Φ∗(X)) for all X 2 g. 2 Proof. The fact that Φ∗ is a Lie algebra homomorphism follows from the naturality of the Lie bracket as stated above. Secondly, since Φ(exp(tX)) is a 1-parameter subgroup of G2 with tangent vector at e given by Φ∗(X), uniqueness for ODE implies that Φ(exp(tX)) = exp(tΦ∗(X)) for all t, from which the result follows. Combining Proposition 6 with Corollary 3 shows that if G1 is connected, then any homomorphism Φ is completely determined by Φ∗. We might now ask whether, given a Lie algebra homomorphism ϕ : g1 ! g2, there is a Lie group homomorphism Φ: G1 ! G2 that induces it. The answer to this question, as well as the ones posed earlier, is provided by the following three Fundamental Theorems. 4 The Fundamental Theorems These theorems are highly difficult to prove, requiring deep results such as the Frobenius distribution theorem; proofs are sketched in Appendix 2. Theorem 1. (Subgroup Theorem) If G is a Lie group, then there is a bijective correspondence between connected Lie subgroups H of G and Lie subalgebras of g, given by taking the tangent space to the identity TeH [8]. Furthermore, H is a normal subgroup of G if and only if h is an ideal in g [9]. Theorem 2. (Homomorphism Theorem) If G1;G2 are Lie groups, G1 is connected, simply-connected, and ϕ : g1 ! g2 is a Lie algebra homomorphism, then there exists a Lie group homomorphism Φ: G1 ! G2 that induces ϕ [8]. Theorem 3. (Lie’s Theorem) If g is a finite-dimensional Lie algebra, then it is the Lie algebra of some connected Lie group G [8]. Or, if you must: “the category of finite-dimensional Lie algebras is equivalent to the category of connected simply- connected Lie groups" [8].