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Home , Algebra representation, Covering group, Cover (algebra), Representation theory

REPRESENTATIONS OF LIE GROUPS AND

Maxim Jeffs

December 12, 2016

1 Introduction

This report re-presents the representation of general Lie groups from the perspective of Lie , 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 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 G is a smooth 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 is a 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 is a discrete isomorphic to π1(G). This is unique up to Lie group . 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 ∈ G, and liftings g,˜ f(g) ∈ 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 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 ∈ H. Consider the smooth map f : G0 → H given by f(g) = ghg−1, where G0 is the identity component of G. Since the 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 ∈ W , then gU ⊂ W ; since gU is an open neighbourhood of g, W is open. But W is also closed, since ∪ W c = gW g∈ /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 → 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 ∈ 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 of left-invariant vector fields must be closed under the Lie bracket; we call it g, the Lie of G. As a vector space, g is naturally isomorphic to TeG under the map that takes X ∈ 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 , 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 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: ∈ R X → 1. for each t , ϕt is a diffeomorphism M M; ∈ 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 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 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 [2], that is, given a Lie group homomorphism Φ: G1 → G2, its differential Φ∗ : TeG1 → TeG2 gives a Lie (called the induced homomorphism), related to Φ by Φ(exp(X)) = exp(Φ∗(X)) for all X ∈ 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 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 of finite-dimensional Lie algebras is equivalent to the category of connected simply- connected Lie groups" [8]. We also have the important corollary:

Corollary 4. (‘The Fundamental Corollary’) If g is a Lie algebra, then there exists a unique connected, simply-connected Lie group G with Lie algebra g. If G′ is connected and also has Lie algebra g, then G′ = G/Z for Z a discrete central subgroup of ′ ∼ G and π1(G ) = Z [8].

Proof. Apply Theorem 3 to find a Lie group with Lie algebra g and then take the universal cover G of the identity component. If G′ is another Lie group with Lie algebra g, apply Theorem 2 along with Corollary 2 to find a covering ′ ′ ∼ map of G by G. Proposition 1 then implies that G = G/Z for Z discrete and normal and that π1(G/Z) = Z.

If G is a connected Lie group with Lie algebra g, we say it belongs to the isogeny class for g [4]. If Z(G) is discrete, then G/Z(G) has trivial centre and this Corollary tells us in particular that every isogeny class contains a ‘centreless’ ∼ group G and a simply-connected group G˜ such that π1(G) = Z(G˜). Furthermore, the fundamental corollary along with the quotient theorem for Lie groups shows that the representations of a connected Lie group G are precisely those that descend through the quotient of the universal covering group.

We shall primarily use these theorems to study representations of Lie groups: now we know that if we have a Lie group G that is connected and simply-connected and a Lie group homomorphism Φ: G → GL(V ), then we have a unique Lie algebra homomorphism ϕ : g → gl(V ), and vice-versa. In other words, the representations of G and g are in correspondence. In fact, more is true: we also know that two representations of G are isomorphic if and only if they are isomorphic as representations of g, and

Proposition 7. Suppose that G is connected and simply-connected and that V is a finite-dimensional vector space. Then a subspace of V is invariant under G if and only if it is invariant under the action of g. In particular, the representation Φ of G is irreducible if and only if ϕ is. [5]

Proof. The author has not been able to find a proof of the above result for arbitrary Lie groups; however, he makes no great claims for it since the proof in the general case is a particularly simple generalisation. If a subspace of V is invariant under the action of every X ∈ g, it is certainly invariant under ϕ(X)n for each n and, since V is finite-dimensional (so that every subspace is closed), it hence must be invariant under exp(ϕ(X)). Using Corollary 3 and naturality then gives

3 one implication. Conversely, if a subspace is invariant under G, it must be invariant under each Φ(exp hX) and thus also under ( ) Φ(exp hX) − Φ(I) lim = ϕ(X) h→0 h again because V is finite-dimensional.

5 Projective Representations

In non-relativistic , one considers a linear equation of the form Hψˆ = Eψ. If Hˆ is invariant under the action of a Lie group, or, equivalently, commutes with a of the corresponding Lie algebra, we would expect the space of solutions to this equation to carry a representation Φ of the group: even though each solution need not be invariant, the should carry any solution to another solution. Since the vectors ψ typically live in a complex , we require this representation to be unitary. Since none of the observable quantities are changed, wavefunctions related by a phase factor are physically equivalent. Thus we may also consider projective iθ representations, where Φ(g1g2) = e 12 Φ(g1)Φ(g2) for g1, g2 ∈ G. If we define the projective by the quotient PU(V ) = U(V )/U(1), where U(1) denotes the subgroup consisting of complex scalar multiples of the identity, then a projective is a Lie group homomorphism Φ: G → PU(V ). The group PU(V ) has Lie algebra given by pu(V ) = u(V )/{iaI : a ∈ R}, which is isomorphic to su(V ).

Given a unitary representation Φ of G, composing it with the quotient map π : U(V ) → PU(V ) yields a projective representation. However, given a projective representation of G, can we ‘de-projectivise’: find a ‘genuine’ unitary repre- sentation of G that descends to give the projective representation? The answer is no in general, but it can be carried for Lie algebras

Theorem 4. Suppose Φ: G → PU(V ) is a finite-dimensional projective representation of G such that ϕ : g → pu(v) is the induced Lie . Then there exists a unique Lie algebra homomorphism σ : g → su(V ) such that ϕ = π ◦ σ, where π is the projection of u(V ) onto pu(V ) [6].

Proof. For each X ∈ g, ϕ(X) is an equivalence class of skew-Hermitian matrices that differ by iaI for a ∈ R. Choosing a representative Y and setting λ = − tr(Y )/ dim(V ), we have λ ∈ iR and tr(X + λI) = 0. If we take σ(X) = Y + λI, then tr(σ) = 0 and σ is clearly independent of representative. Since ϕ is a projective representation of the Lie algebra, σ[X,Y ] = [σ(X), σ(Y )] + iaI for some a ∈ R. But because σ has zero and the trace of any is zero, we see that a = 0, and hence that σ is a Lie algebra homomorphism.

In turn, this allows us to lift the results back up to the universal covering group

Theorem 5. Suppose Φ: G → PU(V ) is a finite-dimensional projective unitary representation of G. Then there exists a unique special-orthogonal representation of the universal covering group, Σ: G˜ → SU(V ) such that π ◦ Σ = Φ ◦ θ, where θ is the covering map G˜ → G. Furthermore, Σ is irreducible if and only if Φ is [6].

which follows from the previous theorem and the correspondence between Lie group and Lie algebra representations. Indeed, these give all of the irreducible projective unitary representations:

Proposition 8. If Σ is an irreducible unitary representation of G˜, the universal covering group of G, then the kernel of π ◦ Σ contains the kernel of the covering map θ : G˜ → G. Hence any ‘genuine’ unitary irreducible representation of G˜ always descends to a projective representation of G [6].

Proof. Since the fundamental group of a manifold is countable [10], it is not hard to show that the kernel of Σ is a discrete normal subgroup of G and hence must be central by Proposition 2. On an irreducible representation, central elements act as multiples of the identity and hence are sent to the identity under the map π [14].

Note finally that this ‘de-projectivisation’ process cannot be carried out in general for infinite-dimensional represen- tations without further assumptions on G, such as semisimplicity [6].

4 6 Applications to Physics

6.1 Spin As discussed, in non-relativistic quantum mechanics, if rotational symmetry is present then the space of solutions must carry a projective unitary representation of the group SO(3); finding the unitary representations of SO(3) provides a perfect example of the theory described above. The group SO(3) is not simply-connected, but its Lie algebra so(3), with ‘physicist’s basis’ given by       0 0 0 0 0 1 0 −1 0       F1 = 0 0 −1 F2 = 0 0 0 F3 = 1 0 0 0 1 0 −1 0 0 0 0 0 is isomorphic to the Lie algebra su(2) under the map Fi 7→ Ei, where the basis Ei is given by ( ) ( ) ( ) 1 i 0 1 0 1 1 0 i E = E = E = 1 2 0 −i 2 2 −1 0 3 2 i 0

Fortunately, SU(2) is diffeomorphic to S3 and hence it provides the universal covering group of SO(3). Since Z(SU(2)) = ∼ {I} and Z(SO(3)) = {I}, we know that SO(3) = SU(2)/Z2 and hence that the genuine representations of SO(3) are precisely those of SU(2) that are trivial on −I. We already know the unitary representations of SU(2), since ∼ ∼ so(3)C = su(2)C = sl(2, C), and we know the representations of sl(2, C) to be skew-Hermitian (see Appendix 3). These ∈ { 1 3 } representations are classified by a single number ℓ 0, 2 , 1, 2 , 2,... , half of the highest weight, called the spin. The terminology arises as follows: the wavefunction of a single particle has multiple components and so lives in V ⊗ L2(R3), where V is some finite-dimensional vector space. The group SO(3) acts on L2(R3) by R · ψ(x) = ψ(R−1x), and the space V must also carry a projective unitary representation of SO(3), that is, a representation of so(3). Hence we can classify the possible types of spin for a particle using . Not all of these representations correspond to genuine representations of SO(3), only the ones with integer spin: we can see this by supposing that Φ is a representation of SO(3) that induces a half- ϕ, so that ( ) Φ(I) = Φ e2πiF3 = e2πiϕ(F3) = −I (1) since ϕ(F3) has only half-integer eigenvalues, giving a contradiction. On the other hand, if the spin is an integer, then any representation of SU(2) inducing ϕ will satisfy ( ) Φ(−I) = Φ e2πE3 = e2πϕ(E3) = I and thus will descend to a genuine representation of SO(3). The fact that these half-integer representations do not descend to genuine representations of SO(3) is of great physical significance; particles with half-integer spin are called fermions and those with integer spin are called bosons. Equation (1) is the source of mysterious statements often made by physicists along the lines of ‘fermions are particles that are flipped under a of 360◦’, and is also the origin of many of the properties of fermions. For more on spin from a mathematical perspective, see [7] [6] or [16]; the classic [15] was also useful.

6.2 The Poincaré Group In the relativistic setting, we may proceed in the opposite direction; begin with the complete of space- time and use this to deduce the possible relativistic wave equations by studying the representations of the symmetry group! In turn, this allows us to classify the possible particles in a relativistic quantum theory. This is the Wigner Classification, and was carried out in special cases by E. Wigner himself in the 1930s [1]. The mathematics is significantly ↑ more subtle in this case, since the full isometry group of space-time is the non-compact Poincaré group ISO (3, 1) = R4 ⋊ SO(3, 1)↑, where SO(3, 1)↑ is the proper orthochronous , the collection of orientation-preserving linear transformations of R4 that preserve the indefinite Minkowski metric and the ‘direction of time’. We sketch the ideas behind the classification, known as the method of induced representations. Since the Lie ↑ 4 algebra of ISO (3, 1) has an abelian subalgebra consisting of vectors P0,P1,P2,P3 coming from the Lie algebra of R , if we assume that these operators can all be diagonalised, we can find a simultaneous eigenbasis of vectors ψq labelled

5 ↑ by their eigenvalues: Pµψq = qµψq. If L ∈ SO(3, 1) , then the action of L on ψq produces an eigenvector of eigenvalue L(q):

µ −1 µ µ µ P Lψq = L(L P L)ψq = LL(P )ψq = L(q) Lψq

↑ and thus Lψq must be a linear combination of the states ψL(q). If we want the representation of ISO (3, 1) to be irreducible, then every such L(q) must be in the orbit of some fixed q0. It is known that the only functions of q preserved by the action of SO(3, 1)↑ are the sign of q0 and the norm q2 of q [1]; these divide the vectors q into orbits. Geometrically, R4 they are hyperboloids in . Therefore, fixing some q0 in a given orbit, we can write, for any q in this orbit, ψq = L(q)ψq0 ↑ ↑ for some properly chosen L of SO(3, 1) with normalised action. Now, if we act on ψq with T ∈ SO(3, 1) , we have

−1 T ψq = TL(q)ψq0 = L(T q)(L (T q)TL(q))ψq0

−1 Since we know the action of L(T q) on ψq0 and since the action of L (T q)TL(q) always preserves q0, we only need ↑ to understand the action of the subgroup of ISO(3, 1) that leaves q0 fixed! This is called the little group associated 2 − 2 ̸ to q0; the nature of this subgroup depends on the choice of orbit for q0. If q0 = m = 0, the resulting representation is said to be massive. Since q0 is time-like, we can choose a frame in which q0 = (m, 0, 0, 0) and it follows that the 2 2 ̸ little group is simply SO(3). Similarly, if q0 = m = 0, the representation is tachyonic (corresponding to a particle that ↑ 2 travels ‘faster than light’) and the little group must be SO (2, 1). Finally, if q0 = 0, the representation is massless and 4 one can show that the little group is ISO(2) = R ⋊ SO(2). The trivial orbit q0 = 0 corresponds to a zero-momentum representation. Fortunately, we have already studied the representation theory of these little groups, excepting the case of ISO(2), to which we may apply the method of induced representations again to reduce down to SO(2), the short little group. The problem is now, in principle, reduced to one previously solved. However, the explicit description of these representations is rather technical: the intrepid reader is referred to [1], or [17] for more...

References

[1] X. Baekert and N. Boulanger. Unitary Representations of the Poincare Group in any . arXiv: hep-th/0611263v1, 2006.

[2] T. Broecker and T. tom Dieck. Representations of Compact Lie Groups. Springer GTM, 1985.

[3] J. F. Cornwell. in Physics, an Introduction, volume II. Academic Press, 1986.

[4] W. Fulton and J. Harris. Representation Theory, A First Course. Springer, 2004.

[5] B. C. Hall. Lie Groups, Lie Algebras, and Representations. Springer GTM, 2010.

[6] B. C. Hall. Quantum Theory for Mathematicians. Springer GTM, 2013.

[7] K. Hannabuss. An Introduction to Quantum Theory. Oxford GTM, 1997.

[8] A. Kirillov. An Introduction to Lie Groups and Lie Algebras. Cambridge, 2008.

[9] S. Kobayashi and K. Nomizu. Foundations of Differential , volume I. Wiley Interscience, 1963.

[10] J. M. Lee. Introduction to Smooth . Springer GTM, 2nd edition, 2003.

[11] L. Nicolaescu. Lectures on the Geometry of Manifolds. World Scientific, 1996.

[12] Y. Ohnuki. Unitary Representations of the Poincaré Group and Relativistic Wave Equations. World Scientific, 1988.

[13] D. Ridout. Lie Groups, Lie Algebras and Applications. ANU, 2011.

[14] D. Ridout. Representations of Semisimple Lie Algebras. ANU, 2014.

[15] L. I. Schiff. Quantum Mechanics, 3rd Ed. McGraw Hill, 1968.

6 [16] L. A. Takhtajan. Quantum Mechanics for Mathematicians. AMS GSM, 2008. [17] S. Weinberg. The Quantum Theory of Fields, volume 1. Cambridge, 1995.

[Note: The author has managed to find the book [12], but has not had time to peruse it; it may perhaps be helpful.]

7 Appendix 1: Lie Subgroups

Formulating the correct notion of a ‘Lie subgroup’ is a subtle question, due to the multiple notions of ‘submanifold’ in geometry. A subgroup that is also an immersed submanifold is a Lie subgroup, and a subgroup that is an embedded submanifold is a closed Lie subgroup (this convention is by no means universal, but follows [8]). The following theorem clarifies this terminology Theorem 6. A subgroup of a Lie group G is closed if and only if it is embedded [8]. This theorem is both highly surprising and highly difficult to prove (see [10], p.526). Here-Be-Dragons 1. If H is a Lie subgroup of G, then the smooth structure on H is part of the data; there can certainly be multiple smooth structures on H with respect to which it is an immersed submanifold of G! This smooth structure is however unique up to diffeomorphism if we require H to be connected (see [9] p.40 for details). These two notions of subgroup have different properties: Proposition 9. If H is a normal closed Lie subgroup of G, then G/H has a canonical Lie group structure such that the projection map is a Lie group homomorphism [8], p.8. In particular, we note that discrete subgroups are closed Lie subgroups of dimension 0. Proposition 10. If Φ: G → G′ is a Lie group homomorphism, then im Φ is a Lie subgroup of G′ and ker Φ is a normal closed Lie subgroup of G [8], p.36. These allow us to formulate the standard quotient homomorphism theorem in terms of Lie groups.

8 Appendix 2: Proofs of the Fundamental Theorems

We begun by summarising the theory of distributions, following [9]:

Definition 2. A smooth distribution S on a manifold M assigns a subspace Sp (of fixed dimension) of TpM to each p ∈ M such that there is a basis of Sp described locally by a collection of smooth vector fields. A vector field X belongs to S if X(p) ∈ Sp for all p ∈ M, and S is involutive if for every X,Y belonging to S, [X,Y ] belongs to S. If N is a connected embedded submanifold of M, it is an integral submanifold for S if TpN = Sp for all p ∈ N. Theorem 7. (Frobenius Distribution) If S is an involute distribution on M, then for all p ∈ M, there is a unique maximal integral submanifold for S passing through p such that any other integral manifold through p is an open submanifold (see [9], or [10] for a proof).

Proof. (of Theorem 1) Suppose h is a subalgebra of g, and consider the smooth distribution S defined by Sp = {Xp : X ∈ h}. Then this distribution is involutive, since h is a subalgebra and hence by the Frobenius Theorem above, there is a maximal integral submanifold H for S through e that has the desired properties; one only needs to show now that the maximality properties imply that H is indeed a subgroup. This argument is long but fairly straightforward: see [10] p. 506 for all the details.

Now we can play clever games to deduce Theorem 2 from Theorem 1:

Proof. (of Theorem 2) G = G1 × G2, so that its Lie algebra is g = g1 ⊕ g2, and define a subalgebra of g by h = {(X, ϕ(X)),X ∈ g1}. Therefore, by Theorem 1, there exists H, an embedded submanifold of G with inclusion map ι and Lie algebra h. Composing with the projection π1 : G → G1, we get a map f : H → G1 such that f∗ : h → g1 is an isomorphism. Hence, by Corollary 2, f is a covering map, and since G1 is simply connected and H is connected, f must be a Lie group isomorphism, by Proposition 1. Now we can take Φ: G1 → G2 to be the Lie group homomorphism −1 π2 ◦ ι ◦ f , and verify that Φ∗ = ϕ [8].

7 Finally, Theorem 3 follows from combining Theorem 1 with the classical

Theorem 8. (Ado) Every finite-dimensional Lie algebra is a subalgebra of some gl(n, K) [4].

9 Appendix 3: C vs. R

We usually study unitary representations of Lie groups because of the following

Theorem 9. (Weyl) If G is a compact Lie group, then every finite-dimensional representation is unitary [2].

The proof of this claim uses the ‘averaging trick’ for integration on manifolds: see [2] or [11]. However, note that

Proposition 11. If G is simple and non-compact, then G has no non-trivial finite-dimensional unitary representations [1].

Proof. If G is simple, then every representation of G must be either trivial or faithful. In the latter case, the image of G would be a non-compact subgroup of U(n), impossible in finite dimensions.

Also, we usually study the representations of the complexification of g, rather than those of g itself. This is resolved by

Proposition 12. If g is a real Lie algebra, then its complex representations are in correspondence with the representations of its complexification gC [8]. If G is a connected Lie group with Lie algebra g, a representation of G is unitary if and only if the of g is skew-Hermitian [5].

Proof. If ϕ is a representation of g, set ϕ(x + iy) = ϕ(x) + iϕ(y) to get a representation of gC. The second claim is a consequence of naturality: if the representation ϕ of g is skew-Hermitian, we have

(Φ(exp(X))v, Φ(exp(X))v) = (exp(ϕ(X))v, exp(ϕ(X))v) = (exp(ϕ(X))† exp(ϕ(X))v, v) = (exp(ϕ(X)†) exp(ϕ(X))v, v) = (exp(−ϕ(X)) exp(ϕ(X)), v) = (Iv, v) using easily-proven properties of the matrix exponential. Since G is connected, applying Corollary 3 implies the result. The proof of the converse is similar.

Given a complex Lie algebra g, there may exist multiple real Lie algebras that have g as their complexifications, called the real forms of g. If G is a complex Lie group and K a closed real Lie subgroup such that kC = g, we say that K is a of G. The relationship between these two concepts is

Proposition 13. Every real form of the Lie algebra of a connected, simply-connected complex Lie group arises from a real form of the Lie group. [8]

which follows from the fundamental theorems. The reverse process is however far more complicated: passing from real Lie groups to complex ones (in a sensible way) is not possible in general [8].

8