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ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Matrix Lie Groups: An Introduction

Lawson J*

Boyd Professor, Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA

Corresponding Author:
Lawson J
Boyd Professor, Department of Mathematics
Louisiana State University, Baton Rouge, LA 70803, United States
Tel:12255783202
E-mail: [email protected]

Received date: August 27, 2015; Accepted date: September 25, 2015; Published date: September 27, 2015

Citation: Lawson J (2015) Matrix Lie Groups: An Introduction. J Generalized Lie Theory Appl 9:229. doi:10.4172/1736-4337.1000229

Copyright: © 2015 Lawson J. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

This article presents basic notions of Lie theory in the context of matrix groups with goals of minimizing the required mathematical background and maximizing accessibility. It is structured with exercises that enhance the text and make the notes suitable for (part of) an introductory course at the upper level undergraduate or early graduate level. Indeed the notes were originally written as part of an introductory course to geometric control theory.

Keywords

Lie groups; Homomorphism; Baker-Campbell-Hausdor formula

Introduction

Lie theory, the theory of Lie groups, Lie algebras, and their applications is a fundamental part of mathematics that touches on a broad spectrum of mathematics, including geometry (classical, differential, and algebraic), ordinary and partial differential equations, group, ring, and algebra theory, complex and harmonic analysis, number theory, and physics (classical, quantum, and relativistic). It typically relies upon an array of substantial tools such as topology, differentiable manifolds and differential geometry, covering spaces, advanced linear algebra, measure theory, and group theory to name a few. However, we will considerably simplify the approach to Lie theory by restricting our attention to the most important class of examples, namely those Lie groups that can be concretely realized as (multiplicative) groups of matrices.

Lie theory began in the late nineteenth century, primarily through the work of the Norwegian mathematician Sophus Lie, who called them “continuous groups,” in contrast to the usually finite permutation groups that had been principally studied up to that point. An early major success of the theory was to provide a viewpoint for a systematic understanding of the newer geometries such as hyperbolic, elliptic, and projective, that had arisen earlier in the century. This led Felix Klein in his Erlanger Programme to propose that geometry should be understood as the study of quantities or properties left invariant under an appropriate group of geometric transformations. In the early twentieth century Lie theory was widely incorporated into modern physics, beginning with Einstein's introduction of the Lorentz transformations as a basic feature of special relativity. Since these early beginnings research in Lie theory has burgeoned and now spans a vast literature.

The essential feature of Lie theory is that one may associate with any Lie group G a Lie algebra image. The Lie algebra image is a vector space equipped with a bilinear non-associative anti-commutative product, called the Lie bracket or commutator and usually denoted [?,?]. The crucial and rather surprising fact is that a Lie group is almost completely determined by its Lie algebra image. There is also a basic bridge between the two structures given by the exponential map exp : image→G. For many purposes structure questions or problems concerning the highly complicated nonlinear structure G can be translated and reformulated via the exponential map in the Lie algebra image, where they often lend themselves to study via the tools of linear algebra (in short, nonlinear problems can often be linearized). This procedure is a major source of the power of Lie theory.

The General Linear Group

Let V be a finite dimensional vector space equipped with a complete norm || ? || over the field image, where imageor image. (Actually since the space V is finite dimensional, the norm must be equivalent to the usual euclidean norm, and hence complete.) Let End(V) denote the algebra of linear self-maps on V , and let GL(V) denote the general linear group, the group (under composition) of invertible self-maps. If V = image, then End(V) may be identified with Mn(image), the n × n matrices, and GL(V) = GLn( image), the matrices of nonvanishing determinant.

We endow End(V ) with the usual operator norm, a complete norm defined by

image

which gives rise to the metric d(A,B) = || B – A || on End(V) and, by restriction, on GL(V).

Exercise 2.1: image and image

Exercise 2.2: Show that GL(V) is a dense open subset of End(V). (Hint: The determinant function is polynomial, hence continuous, and A - (1/n)I converges to A and is singular for at most finitely many values, since the spectrum of A is finite.)

Exercise 2.3: The multiplication and inversion on GL(V) are analytic, i.e., expressible locally by power series. (Hint: the multiplication is actually polynomial and the cofactor expansion shows that inversion is rational.)

A group G endowed with a Hausdorff topology is called a topological group if the multiplication map m : G × G → G and the inversion map on G are continuous. By the preceding exercise GL(V) is a topological group.

The Exponential Map

We define the exponential map on End V by

image

Lemma 1: The exponential map is absolutely convergent, hence convergent on all of End(V). Hence it defines an analytic self-map on End(V).

Proof: image

Absolute convergence allows us to rearrange terms and to carry out various algebraic operations and the process of differentiation term wise. We henceforth allow ourselves the freedom to carry out such manipulations without the tedium of a rather standard detailed verification.

Exercise 3.1: (i) Show that the exponential image of a block diagonal matrix with diagonal blocks A1,…,Am is a block diagonal matrix with diagonal blocks exp(A1),…,exp(An). In particular, to compute the exponential image of a diagonal matrix, simply apply the usual exponential map to the diagonal elements.

(ii) Suppose that A is similar to a diagonal matrix, A = PDP-1. Show that exp(A) = P exp(D)P-1.

Proposition 2: If A, B ∈ End V and AB = BA, then exp(A + B) = exp A exp B = exp B exp A.

Proof: Computing term wise and rearranging we have

image

Since A and B commute, the familiar binomial theorem yields

image

and substituting into the previous expression yields the proposition.

Let V,W be finite dimensional normed vector spaces and let f: U → W, where U is a nonempty open subset of V. A linear map L: V → W is called the (Frechet) derivative of f at x ∈ U if in some neighbourhood of x

image where image

If it exists, the derivative is unique and denoted by df(x) or f '(x).

Lemma 3: The identity map on End(V) is the derivative at 0 of exp : End(V ) → End(V ), i.e., d exp(0) = Id.

Proof: For h ∈ End(V), we have

image

where

image

Applying the Inverse Function Theorem, we have immediately from the preceding lemma

Proposition 4: There exist neighbourhoods U of 0 and V of I in End V such that exp |U is a diffeomorphism onto V.

For A∈ End V and r > 0, let Br (A) = {C ∈V :|| C − A||< r}.

Exercise 3.2: Show that exp(Br (0)) ⊆ Bs (1V ) where s = er-1. In particular for r = ln 2, exp(Br (0)) ⊆ B1(1V ).

One-parameter Groups

A one-parameter subgroup of a topological group G is a continuous homomorphism α: image → G from the additive group of real numbers into G.

Proposition 5: For V a finite dimensional normed vector space and A ∈ End V, the mapimage is a one-parameter subgroup of GL(V). In particular, (exp(A))-1 = exp(-A).

Proof: Since sA and tA commute for any s, t ∈ image, we have from Proposition 2 that image is a homomorphism from the additive reals to the End V under multiplication. It is continuous, indeed analytic, since scalar multiplication and exp are. The last assertion follows from the homomorphism property and assures the the image lies in GL(V).

Proposition 6: Choose an r < ln 2. Let A∈ Br(0) and let Q = exp A. Then P = exp(A/2) is the unique square root of Q contained in B1(1V ).

Proof: Since exp(tA) defines a one-parameter subgroup.

P2 = (exp(A / 2))2 = exp(A / 2) exp(A / 2) = exp(A / 2 + A / 2) = exp(A) = Q.

Also A∈ Br(0) implies A/2 ∈ Br(0), which implies exp(A/2) ∈ B1(1V) (Exercise 3.2).

Suppose two elements in B1(1V), say 1+B and 1+C where || B ||, || C || < 1 satisfy (1 + B)2= (1 + C)2. Then expanding the squares, cancelling the 1's, and rearranging gives

2(B - C) = C2 - B2 = C(C - B)) + (C - B)B.

Taking norms yields

2 || B −C || ≤||C || ||C − B || + ||C − B || || B ||= (||C || + || B ||) ||C − B || .

This implies either ||C|| + ||B|| ≥ 2, which is also false since each summand is less than 1, or ||B - C|| = 0, i.e., B = C. We conclude there at most one square root in B1(1V).

Lemma 7: Consider the additive group (image, +) of real numbers.

(i) If a subgroup contains a sequence of nonzero numbers {an} converging to 0, then the subgroup is dense.

(ii) For one-parameter sub groups α, β : image → G, the set {t ∈ image : α(t) = β(t)} is a closed subgroup.

Proof: (i) Let t ∈ image and let ε > 0. Pick an such that |an| < ε. Pick an integer k such that |t/an – k| < 1 (for example, pick k to be the floor of t/ an). Then multiplying by |an| yields |t - kan| < |an| < ε. Since kan must be in the subgroup, its density follows.

(ii) Exercise.

Exercise 4.1: Show that any nonzero subgroup of (image, +) is either dense or cyclic. (Hint: Let H be a subgroup and let r = inf{t ∈ H : t > 0}. Consider the two cases r = 0 and r > 0.)

The next theorem is a converse of Proposition 5.

Theorem 8: Every one parameter subgroup α : image → End(V ) is of the form α(t) = exp(tA) form some A ∈ End V.

Proof: Pick r < ln 2 such that exp restricted to Br(0) is a diffeomorphism onto an open subset containing 1 = 1V. This is possible by Proposition 4. Note that 1 image By continuity of α, pick 0 < ε such that α(t) ∈ exp(Br(0)) for all - ε < t < ε. Then α(1/2k)∈exp(B1(0)) ⊆ B1(1) for all 1/2k< ε.

Pick 1/2n < ε. Then :=α (1/ 2n )∈exp(Br (0)) Q Br implies Q =α (1 / 2n ) = exp(B) for some B ∈ Br(0). Set A = 2n B. Then Q = exp((1 / 2n )A).

Then α(1/2n+1) and exp(B/2) are both square roots of Q contained in B1(1), and hence by Proposition 6 are equal. Thus α(1/2n+1) = exp((1/2n+1)A). By induction α(1/2n+k) = exp((1/2n+k)A) for all positive integers k. By Lemma 7(ii) the two one-parameter subgroups agree on a closed subgroup, and by Lemma 7 this subgroup is also dense. Hence α(t) and exp(tA) agree everywhere.

The preceding theorem establishes that a merely continuous one-parameter subgroup must be analytic. This is a very special case of Hilbert's fifth problem, which asked whether a locally euclidean topological group was actually an analytic manifold with an analytic multiplication. This problem was solved positively some fifty years later in the 1950's by Gleason, Montgomery, and Zippin.

Exercise 4.2: Show that if exp(tA) = exp(tB) for all t ∈ image, then A = B. (Hint: Use Proposition 4)

Remark 9: The element A ∈ EndV is called the infinitesimal generator of the one-parameter group image.We conclude from the preceding theorem and remark that there is a one-toone correspondence between one-parameter subgroups and their infinitesimal generators.

Curves in End V

In this section we consider basic properties of differentiable curves in End V. Let I be an open interval and let A(?) : I → End V be a curve. We say that A is Cr if each of the coordinate functions Aij(t) is Cr on image. We define the derivative image. The derivative exists iff the derivative image of each coordinate function exists, and in this case image is the linear operator with coordinate functions image

Items (1) and (2) in the following list of basic properties for operator valued functions are immediate consequences of the preceding characterization, and item (5) is a special case of the general chain rule.

(1) image

(2) image

(3) image

(Note: Order is important since multiplication is noncommutative.)

(4) image

(5) If image

Exercise 5.1: Establish properties (3) and (4). (Hints: (3) Mimic the proof of the product rule in the real case. (4) Note A-1(t) is differentiable if A(t) is, since it is the composition with the inversion function, which is analytic, hence Cr for all r. Differentiate the equation A(t).A-1(t) = I and solve for Dt(A-1(t)).)

We can also define the integral imageby taking the coordinate integrals image The following are basic properties of the integral that follow from the real case by working coordinate wise.

(6) If B image, then image

(7) If image, the image

We consider curves given by power series: image Define the nth-partial sum to be image The power series converges for some value of t if the partial sums Sn(t) coverage in each coordinate to some S(t). This happens iff the coordinatewise real power series all converge to the coordinates of S(t).

Since for an operator A, |aij| ≤ || A || for each entry aij (exercise), we have that absolute convergence, the convergence of image imples the absolute convergence of each of the coordinate series, and their uniform convergence over any closed interval in the open interval convergence of the real power series image These observations justify term wise differentiation and integration in the interval of convergence of image

Exercise 5.2: (i) Show that the power series

image

is absolutely convergent for all t (note that An = (1/n!)An in this series).

(ii) Use termwise differentiation to show Dt(exp(tA)) = A exp(tA).

(iii) Show that X(t) = exp(tA)X0 satisfies the differential equation on End V given by

image

(iv) Show that the equation image on V has solution x(t) = exp(tA)x0.

The Baker-Campbell-Hausdorff Formalism

It is a useful fact that the derivative of the multiplication map at the identity I of End V is the addition map.

Proposition 10: Let m : End(V) × End(V) → End(V) be the multiplication map, m(A,B) = AB. Then the derivative at (I,I), d(I,I)m : End(V) × End(V) → End(V) is given by dm(I,I)(U, V ) = U + V.

Proof: Since the multiplication map is polynomial, continuous partials of all orders exist, and in particular the multiplication map is differentiable. By definition the value of the derivative at (I,I) evaluated at some (U,0) ∈' End(V) × End(V) is given by

image

We have seen previously that the exponential function is a diffeomorphism from some open neighbourhood B of 0 to some open neighbourhood U of I. Thus there exists an analytic inverse to the exponential map, which we denote by log : U → B. Indeed if one defines

image

then just as for real numbers this series converges absolutely for || A || < 1. Further since exp(log A) = A holds in the case of real numbers, it holds in the algebra of formal power series, and hence in the linear operator or matrix case. Indeed one can conclude that exp is 1 - 1 on Bln 2(0), carries it into B1(I), and has inverse given by the preceding logarithmic series, all this without appeal to the Inverse Function Theorm.

The local diffeomorphic property of the exponential function allows one to pull back the multiplication in GL(V) locally to a neighbourhood of 0 in End V . One chooses two points A, B in a sufficiently small neighbourhood of 0, forms the product exp(A) ? exp(B) and takes the log of this product:

image

This Baker-Campbell-Hausdorff multiplication is defined on any Br(0) small enough so that exp(Br(0)) ? exp(Br(0)) is contained in the domain of the log function; such exist by the local diffeomorphism property and the continuity of multiplication. Now there is a beautiful formula called the Baker-Campbell-Hausdorff formula that gives A * B as a power series in A and B with the higher powers being given by higher order Lie brackets or commutators, where the ( firstorder) commutator or Lie bracket is given by [A,B]:= AB − BA. The Baker- Campbell- Hausdorff power series is obtained by manipulating the power series for log(exp(x) ? exp(y)) in two noncommuting variables x, y in such a way that it is rewritten so that all powers are commutators of some order. To develop this whole formula would take us too far afield from our goals, but we do derive the first and second order terms, which suffice for many purposes.

Definition 11: An open ball Br(0) is called a Baker-Campbell- Hausdorff neighbourhood, or BCH-neighborhood for short, if r < 1/2 and exp(Br (0) ⋅ exp(Br (0) ⊆ Bs (0) for some s,r such that exp restricted to Bs(0) is a diffeomorphism onto some open neighbourhood of I. By the local diffeomorphism property of the exponential map and the continuity of multiplication at I, BCH-neighbourhoods always exist. We define the Baker-Campbell-Hausdorff multiplication on any BCHneighbourhood Br(0) by

image

Note that A * B exists for all A, B ∈ Br(0), but we can only say that A * B ∈ End V, not necessarily in Br(0).

Proposition 12: Let Br(0) be a BCH-neighbourhood. Define μ : Br(0) × Br(0) → End V by μ(A,B) = A * B. Then

(i) A* B = A + B + R(A,B) where image

(ii)There exists 0 < s ≤ r such that || A* B ||≤ 2(|| A|| + || B ||) for A,B ∈ Bs(0).

Proof: (i) We have that μ = log°m°(exp× exp), so by the chain rule, the fact that the derivatives of exp at 0 and log at I are both the identity map Id : End V → End V (Lemma 3 and the Inverse Function Theorem) and Proposition 10, we conclude that

(0,0) (U,V ) = Id°dm(I ,I)°(Id × Id)(U,V ) =U +V.

By definition of the derivative, we have

image (1)

(Note that the second equality is just the definition of the derivative, where the norm on End V × End V is the sum norm.) This gives (i).

(ii) Using (i), we obtain the following string:

image

Now || R(A,B) ||→0 as A, B → 0, so the right-hand sum is less than or equal 2(|| A|| + || B ||) on some Bs (0) ⊆ Br (0)

Exercise 6.1: Use the fact that 0 * 0 = 0 and imitate the proof of Proposition 10 to show directly that dm(0,0) (U,V ) =U +V.

We now derive the linear and second order terms of the Baker- Campbell-Hausdorff series.

Theorem 13: Let Br(0) be a BCH-neighbourhood. Then

image

Proof: Pick (0) (0) s r B ⊆ B so that condition (ii) of Proposition 12 is satisfied. Setting C = A * B, we have directly from the definition of A* B that exp C = exp A ? exp B. By definition

(2) image

For A, B ∈ Bs(0), we have from Proposition 12 that ||C || = || A* B || ≤ 2(|| A|| + || B ||) <1 since r < 1/2. Thus we have the estimate

(3) image

Recalling the calculations in the proof of Proposition 2, we have

(4) image

where image

We have the estimate

(5) image

If in the equation exp C = exp A ? exp B, we replace exp C by the right side of equation (2), exp A ? exp B by the right side equation (4), and solve for C, we obtain

(6) image

Since

image

we obtain alternatively

(7) image

where image

To complete the proof, it suffices to show that the limit as A, B → 0 of each of the terms of S(A,B) divided by (|| A|| + || B ||)2 is 0. First we have in Bs(0)

image

where the second inequality and last equality follow by applying appropriate parts of Proposition 12. Proposition 12 also insures that

image

That image follows directly from equation (5). Finally by equation (3) and Proposition 12(ii)

image

which goes to 0 as A, B → 0.

The Trotter and Commutator Formulas

In the following sections we show that one can associate with each closed subgroup of GL(V) a Lie subalgebra of End V , that is, a subspace closed under Lie bracket. The exponential map carries this Lie algebra into the matrix group and using properties of the exponential map, one can frequently transfer structural questions about the Lie group to the Lie algebra, where they often can be treated using methods of linear algebra. In this section we look at some of the basic properties of the exponential map that give rise to these strong connections between a matrix group and its Lie algebra.

Theorem 14: (Trotter Product Formula) Let A,B ∈ End V and let limn nAn = A, limn nBn = B. Then

(i) image

(ii) image

Proof: (i) Let ε > 0 For large n, n|| An || ≤ || A || + || nAn – A || < || A || + ε, and thus || An || ≤ (1/n)(|| A || + ε). It follows that limn An = 0 and similarly limn Bn = 0. By Proposition 12(i) we have

image

provided that limnnR(An, Bn) = 0. But we have

image

(ii) The first equality follows directly by applying the exponential function to (i):

image

where the last equality follows from the fact that exp is a local isomorphism from the BCH-multiplication to operator multiplication, and penultimate equality from the fact that exp(nA) = exp(A)n, since exp restricted to imageA is a one-parameter group. The second equality in part (i) of the theorem follows from the first by setting An = A/n, Bn = B/n.

The exponential image of the Lie bracket of the commutator can be calculated from products of group commutators.

Theorem 15: (Commutator Formula) Let A, B ∈ End V and let limn nAn = A, limn nBn = B. Then

(i) [A, B] = limn n2 (An ∗ Bn − Bn ∗ An) = limn n2(An ∗ Bn ∗ (− An)∗ (− Bn)) ;

(ii) image

image

Proof. (i) From Theorem 13 we have for A, B in a BCHneighbourhood:

image

image

since [A, B] = − [B, A]. Therefore

image

provided provided limn n2S (An, Bn) = limn n2S (Bn, An)) = 0. To see this, we note

image

and similarly limn n2 || S (Bn, An) || = 0

To see second equality in item (i), observe first that on a BCHneighbourhood where the exponential map is injective,

image

which implies (−A) ∗ (−B) = − ( B ∗ A). Hence we have by Theorem 13 that

image

Applying this equality to the given sequences, we obtain

image

Now if we show that the two terms in the second expression approach 0 as n→∞, then the first expression approaches 0, and thus the two limits in (i) will be equal. We observe first that by the Trotter Product Formula

image

since [C,-C] = =[C,C] = 0 for any C. Thus the first right-hand term approaches 0. For the second as n → ∞.

image

(ii) The proof follows from an application of the exponential function to part (i), along the lines of the Trotter Product Formula.

Exercise 7.1: Give the proof of part (ii) in the preceding theorem.

The Lie Algebra of a Matrix Group

In this section we set up the fundamental machinery of Lie theory, namely we show how to assign to each matrix group a (uniquely determined) Lie algebra and an exponential map from the Lie algebra to the matrix group that connects the two together. We begin be defining the notions and giving some examples.

By a matrix group we mean a closed subgroup of GL(V ), where V is a finite dimensional vector space.

Examples 8.1: The following are standard and basic examples.

1. The general linear group GL(V ). If V = imagen, then we write the group of n × n invertible matrices as GLn(image).

2. The special linear group {g∈GL(V ) : det(g) = 1}.

3. Let V be a real (resp. complex) Hilbert space equipped with an inner product ?⋅ , ⋅?. The orthogonal group (resp. unitary group) consists of all transformations preserving the inner product, i.e.,

O(V )( resp. U(V )) = {g ∈ GL(V ) : ∀x, y ∈ V, ?gx, gy? = ?x, y?}

If V = imagen (resp. imagen) equipped with the usual inner product, then the orthogonal group On (resp. unitary group Un) consists of all g ∈ GL(V ) such that gt = g-1 (resp. g* = g-1 ).

(4) Let V = imagenimagen equipped with the sympletic form

image

The real sympletic group is the subgroup of GL(V ) preserving Q:

image

(5) Let 0 < m, n and consider the group of block upper triangular real matrices

image

This is the subgroup of image that carries the subspace image of image n into itself.

Exercise 8.1. (i) Verify that the subgroups in (2)-(5) are closed.

(ii) Verify the alternative characterizations of elements of the subgroup in items (3) and (5).

Exercise 8.2. Establish the following equivalence:

image

image

If M has block matrix form image(where all submatrices are n × n), then

A* C, B* D are symmetric, and A* D − C* B=I

Definition 16. A real Lie algebra g is a real vector space equipped with a binary operation

[⋅ , ⋅ ] : g × g → g

satisfying the identities

(i) (Bilinearity) For all image and image

image

image

image

(iii) (Jacobi identity) For all image

[X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0.

Exercise 8.3. Verify that End V equipped with the Lie bracket or commutator operation [A, B] = AB − BA is a Lie algebra.

It follows directly from the preceding exercise that any subspace of End V that is closed with respect to the Lie bracket operation is a Lie subalgebra.

We define a matrix semigroup S to be a closed multiplicative subsemigroup of GL(V ) that contains the identity element. We define the tangent set of S by

image

We define a wedge in End V to be a closed subset containing {0} that is closed under addition and scalar multiplication by nonnegative scalars.

Proposition 17. If S is a matrix semigroup, then L(S) is a wedge.

Proof. Since I = exp(t.0) for all t ≥ 0 and I ∈ S, we conclude that 0 ∈ L(S). If A ∈ L(S), then exp(tA) ∈ S for all t ≥ 0, and thus exp(rtA) ∈ S for all r, t ≥ 0 It follows that rA ∈ L(S) for r ≥ 0. Finally by the Trotter Product Formula if A, B ∈ L(S), then

image

since S is a closed subsemigroup of GL(V ). Thus A + B ∈ L(S).

Theorem 18. For a matrix group G ⊆ GL(V ), the set

image

is a Lie algebra, called the Lie algebra of G.

Proof. As in the proof of Proposition 17, g is closed under addition and scalar multiplication, i.e., a subspace of End V. By the Commutator Formula for A, B ∈g,

image

since G is a closed subgroup of GL(V ). Replacing A by tA, which again is in g, we have exp(t[A, B]) = exp([tA, B]) ∈ G for all t ∈ image. Thus [A, B] ∈ g.

Exercise 8.4. Show for a matrix group G (which is a matrix semigroup, in particular) that g = L(G).

Lemma 19. Suppose that G is a matrix group, {An} is a sequence in End V such that An → 0 and exp(An) ∈ G for all n. If snAn has a cluster point for some sequence of real numbers sn, then the cluster point belongs to g.

Proof. Let B be a cluster point of snAn. By passing to an appropriate subsequence, we may assume without loss of generality that snAn converges to B. Let t ∈ image and for each n pick an integer mn such that | mn− tsn | < 1. Then

|| mnAn− tB || = || mn− tsn || An +t (snA − B) ||

≤ | mn− tsn | || An || + | t | || snAn − B ||

≤ || An | + | t | || snAn − B || → 0

which implies mnAn → tB. Since exp(mnAn) = (exp An)mn ∈ G for each n and G is closed, we conclude that the limit of this sequence exp(tB) is in G. Since t was arbitrary, we see that B ∈ g.

We come now to a crucial and central result.

Theorem 20. Let G ⊆ GL(V ) be a matrix group. Then all sufficiently small open neighborhoods of 0 in g map homeomorphically onto open neighborhoods of I in G.

Proof. Let Br(0) be a BCH-neighborhood around 0 in End V , which maps homeomorphically under exp to an open neighborhood exp(Br(0)) of I in GL(V ) with inverse log. Assume that exp(Br(0) ∩ g) does not contain a neighborhood of I in G. Then there exists a sequence gn contained in G but missing exp(Br(0) ∩ g) that converges to I. Since exp(Br(0)) is an open neighborhood of I, we may assume without loss of generality that the sequence is contained in this open set. Hence An = log gn is defined for each n, and An → 0. Note that An ∈ Br (0), but An ∉ g, for each n, since otherwise exp(An) = gn ∈ exp(g ∩ Br(0)).

Let W be a complementary subspace to g in End V and consider the restriction of the BCH-multiplication μ (A, B) = A ∗ B to (g ∩ Br(0)) × (W ∩ Br(0)). By the proof of Proposition 12, the derivative dμ(0,0) of μ at (0, 0) is addition, and so the derivative of the restriction of μ to (g ∩ Br(0)) × (W ∩ Br(0)) is the addition map + : g × W → End V. Since g and W are complementary subspaces, this map is an isomorphism of vector spaces. Thus by the Inverse Function Theorem there exists an open ball Bs(0), 0 < s ≤ r, such that μ restricted to (g ∩ Bs(0)) × (W ∩ Bs(0)) is a diffeomorphism onto an open neighborhood Q of 0 in End V . Since An ∈ Q for large n, we have An = Bn ∗ Cn (uniquely) for Bn ∈ (g ∩ Bs(0)) and Cn ∈ (W ∩ Bs(0)). Since the restriction of μ is a homeomorphism and 0 ∗ 0 = 0, we have (Bn, Cn) → (0, 0), i.e., Bn → 0 and Cn → 0.

By compactness of the unit sphere in End V, we have that Cn/||Cn|| clusters to some C ∈ W with ||C || = 1. Furthermore,

gn = exp(An) = exp(Bn ∗ Cn) = exp(Bn) exp(Cn)

so that exp(Cn) = (exp Bn)-1 gn ∈ G. It follows from Lemma 19 that C ∈ g. But this is impossible since g ∩ W = {0}and C ≠ 0. We conclude that exp(Br(0) ∩ g) does contain some neighborhood N of I in G.

Pick any open neighborhood U (Br(0) ∩ g) of 0 in g such that exp(U) ⊆ N. Then exp U is open in exp(Br(0) ∩ g) (since exp restricted to Br(0) is a homeomorphism), hence is open in N, and thus is open in G, being an open subset of an open set.

Although we treat matrix groups from the viewpoint of elementary differential geometry in Chapter 5, we sketch here how that theory of matrix groups develops from what we have already done in that direction. Recall that a manifold is a topological space M, which we will assume to be metrizable, that has a covering of open sets each of which is homeomorphic to an open subsets of euclidean space. Any family of such homeomorphisms from any open cover of M is called an atlas, and the members of the atlas are called charts. The preceding theorem allows us to introduce charts on a matrix group G in a very natural way. Let U be an open set around 0 in g contained in a BCH- neighborhood such that W = exp U is an open neighborhood of I in G. Let λg: G → G be the left translation map, i.e., λg (h) = gh. We define an atlas of charts on G by taking all open sets g-1N, where N is an open subset of G such that I N ⊆ W and defining the chart to be log ολg: g-1N → g (to view these as euclidean charts, we identify g with some imagen via identifying some basis of g with the standard basis of imagen). One can check directly using the fact that multiplication of matrices is polynomial that for two such charts and φ and ψ, the composition φ ο ψ -1, where defined, is smooth, indeed analytic. This gives rise to a differentiable structure on G, making it a smooth (analytic) manifold. The multiplication and inversion on G, when appropriately composed with charts are analytic functions, and thus one obtains an analytic group, a group on an analytic manifold with analytic group operations. This is the unique analytic structure on the group making it a smooth manifold so that the exponential map is also smooth.

The Lie Algebra Functor

We consider the category of matrix groups to be the category with objects matrix groups and morphisms continuous (group) homomorphisms and the category of Lie algebras with objects subalgebras of some End V and morphisms linear maps that preserve the Lie bracket., The next result shows that the assignment to a matrix group of its Lie algebra is functorial.

Proposition 21. Let α : G → H be a continuous homomorphism between matrix groups. Then there exists a unique Lie algebra homomorphism

dα : g → h such that the following diagram commutes:

image

exp↑ ↑ exp

image

Proof. Let A ∈ g. Then the map β (t) := α(exp(tA)) is a oneparameter subgroup of H. Hence it is a unique infinitesimal generator à ∈ h

Define dα (A) =Ã. We show that dα is a Lie algebra homomorphism.

For r ∈ image,

α (exp(trA)) = exp(trÃ),

so the infinitesimal generator for the left-hand side is rÃ. This shows that dα (rA) = rà = rdα (A), so dα is homogeneous.

Let A, B ∈ G. Then

image

This shows that dα (A + B) = Ã + image = dα (A) + dα (B), and thus dα is linear. In an analogous way using the commutator, one shows that dα preserves the commutator.

If dα (A) = Ã, then by definition for all t, α (exp(tA)) = exp(tÃ).

For t = 1, α (exp A) = exp(Ã) = exp(dα (A)). Thus α ο exp = dα οexp. This shows the square commutes. If γ : g → h is another Lie algebra homomorphism that also makes the square commute, then for A ∈ g and all t ∈ image,

exp(tdα (A)) = exp(dα (tA)) = α (exp(tA)) = exp(γ(tA)) = exp(tγ(A)).

The uniqueness of the infinitesimal generator implies dα (A) = γ (A), and hence dα = γ.

Exercise 9.1. Show that dα preserves the commutator.

Exercise 9.2. Let α : G → H be a continuous homomorphism of matrix groups. Then the kernel K of α is a matrix group with Lie algebra the kernel of dα .

10. Computing Lie Algebras

In this section we consider some tools for computing the Lie algebra of a matrix group, or more generally a closed subsemigroup of a matrix group. We begin with a general technique.

Proposition 22. Let β (⋅ , ⋅) be a continuous bilinear form on V and set

G = {g ∈ GL(V ) : ∀x, y ∈ V, β (gx, gy) = β (x, y)}.

Then

g = {A ∈ End V : ∀x, y ∈V, β (Ax, y) + β (x, Ay) = 0}.

Proof. If A ∈ g, then β (exp(tA)x, exp(tA)y) = β (x, y) for all x, y ∈ V . Differentiating the equation with respect to t by the product rule (which always holds for continuous bilinear forms), we obtain

β (A exp(tA)x, exp(tA)y) + β (exp(tA)x, A exp(tA)y) = 0.

Evaluating at t = 0 yields β (Ax, y) + β (x, Ay) = 0.

Conversely suppose for all x, y ∈ V, β (Ax, y) + β (x, Ay) = 0. Then from the computation of the preceding paragraph the derivative of

f(t) := β (exp(tA)x, exp(tA)y)

is f′ (t) = 0. Thus f is a constant function with the value β (x, y) at 0. It follows that exp(tA) ∈ G for all t, i.e., A ∈ g.

Exercise 10.1. Apply the preceding proposition to show that the Lie algebra of the orthogonal group On(image) (resp. the unitary group Un(image)) is the Lie algebra of n × n real (resp. complex) skew symmetric matrices.

Exercise 10.2. (i) Use the Jordan decomposition to show for any A ∈ Mn(image), exp(tr A) = det(exp A).

(ii) Use (i) and Exercise 9.2 to show that the Lie algebra of the group SLn(image) of complex matrices of determinant one is the Lie algebra of matrices of trace 0. (Hint: the determinant mapping is a continuous homomorphism from GLn(image) to the multiplicative group of nonzero complex numbers.)

(iii) Observe that L(G ∩ H) = L(G) ∩ L(H). What is the Lie algebra of SUn(image), the group of unitary matrices of determinant one?

Exercise 10.3. Let V = imagenimagen equipped with the canonical sympletic form

image

The Lie algebra of Sp(V ) is given by

image

(Hint: If (exp tA)* J(exp tA) = J for all t, differentiate and evaluate at t = 0 to obtain A* J + JA = 0. Multiply this out to get the preceding conditions. Conversely any block matrix satisfying the conditions can be written as

image

Show directly that each of the summands is in sp(V) and use the fact that sp(V) is a subspace.)

We introduce another general technique, this time one that applies to semigroups and groups.

Proposition 23. Let W be a closed convex cone in the vector space End V that is also closed under multiplication. Then S := (I + W) ∩ GL(V) is a closed subsemigroup of GL(V ) and L(S) = W.

Proof. Let X, Y ∈ W. Then (I +X) (I +Y) = I +X +Y +XY ∈ I +W since W is closed under multiplication and addition. Thus I + W is a closed subsemigroup, and thus its intersection with GL(V) is a subsemigroup closed in GL(V).

Let A ∈ W. Then for image has all infinite partial sums in I + W since W is closed under multiplication, addition, and scalar multiplication. Since the whole sum is the limit, it follows that exp(tA) is in the closed set I+W, and since the exponential image is invertible, it is in S. Thus A ∈ L(S).

Conversely assume that exp(tA) ∈ S for all t ≥ 0. Then

image

where the last assertion follows from the fact that exp(tA) ∈ I + W for t > 0, and hence exp(tA) ? I and therefore (1 ⁄ t)(exp(tA) ? I) are in W. Since W is closed the limit is also in W.

Exercise 10.4. Use Proposition 23 to show the following in GLn(image) or GLn(image).

i. The group of unipotent (diagonal entries all 1) upper triangular matrices has Lie algebra the set of strictly upper triangular matrices.

ii. The group of invertible upper triangular matrices has Lie algebra the set of all upper triangular matrices.

iii. The group of stochastic matrices (invertible matrices with all row sums 1) has Lie algebra the set of matrices with all row sums 0.

iv. The semigroup S of all invertible matrices with all entries non-negative has as its Lie wedge the set of matrices whose nondiagonal entries are nonnegative.

Suggestions for Further Reading

The author is indebted to two sources in particular that have greatly influenced these notes [1-6]. The first is Roger Howe's “Very Basic Lie Theory" [6]. Comparison with that source will show our indebtedness to that source, particularly in treatment of matters connected with the Baker-Campbell-Hausdorff formula. A second important source is what arose as the class notes on introductory Lie theory by Karl Hofmann, much of which was has been incorporated in Chapter 5 of the book “The structure of Compact Groups” [5]. The idea of introducing Lie theory via matrix Lie theory also has been worked out by Morton Curtis [1]. A novel feature of these notes is the inclusion toward the end of the fairly recent notion of a Lie semigroup. More on this topic can be found in literature of Hilger [2,3]. For a comprehensive treatment of Lie theory, one somewhat in the spirit of these notes, especially in its earlier parts, the author recommends the recent volume of Hilger and Neeb [4].

References

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