LECTURE 7: LINEAR LIE GROUPS

1. The general linear group

2 Recall that M(n, R), the set of all n × n real matrices, is diﬀeomorphic to Rn . Deﬁnition 1.1. A linear Lie group, or matrix Lie group, is a submanifold of M(n, R) which is also a Lie group, with group structure the matrix multiplication.

Let’s begin with the “largest” linear Lie group, the general linear group GL(n, R) = {X ∈ M(n, R) | det X 6= 0}. Since the determinant map is continuous, GL(n, R) is open in M(n, R) and thus a sub- manifold. Moreover, GL(n, R) is closed under the group multiplication and inversion operations, so it is a Lie group. Obviously GL(n, R) is an n2-dimensional noncom- pact Lie group, and it is not connected. In fact, it consists of exactly two connected components, GL+(n, R) = {X ∈ M(n, R) | det X > 0} and GL−(n, R) = {X ∈ M(n, R) | det X < 0}. 2 The fact that GL(n, R) is an open subset of M(n, R) ' Rn also implies that the Lie algebra of GL(n, R), as the tangent space at e = In, is the set M(n, R) itself, i.e. gl(n, R) = {A | A is an n × n real matrix}.

To ﬁgure out the Lie bracket operation, we take a matrix A = (Aij)n×n ∈ g, and take the global coordinate system by (Xij). Then the corresponding tangent vector P ∂ at TIn GL(n, R) is Aij ∂Xij , and the corresponding left-invariant vector on G at the ij P ik ∂ matrix X = (X ) is X Akj ∂Xij . It follows that the Lie bracket [A, B] between matrices A, B ∈ g is the matrix corresponding to X ∂ X ∂ X ∂ X ∂ XikA , XpqB = XikA B − XpqB A kj ∂Xij qr ∂Xpr kj jr ∂Xir qr rj ∂Xpj X ∂ = Xik (A B − B A ) . kr rj kr rj ∂Xij In other words, the Lie bracket operation on g is the matrix commutator [A, B] = AB − BA.

Given any A ∈ gl(n, R), we can deﬁne the matrix exponential A2 A3 An eA = I + A + + + ··· + + ··· . n 2! 3! n!

1 2 LECTURE 7: LINEAR LIE GROUPS

It is easy to check that the series converges, and esAetA = e(s+t)A. 0A tA −1 −tA tA tA Notice that e = In, we have (e ) = e . In particular, e ∈ GL(n, R). So e d tA is a one-parameter subgroup of GL(n, R). Since dt |t=0e = A, we conclude that the exponential map exp : gl(n, R) → GL(n, R) is A2 A3 exp(A) = eA = I + A + + + ··· . n 2! 3! Note that exp is not surjective, not even surjective to GL+(n, R). Remark. Alternatively, one can ﬁrst show that etA is the one-parameter subgroup of GL(n, R) generated by A ∈ gl(n, R), and thus the Lie bracket structure on gl(n, R) is

d d tA sB −tA d tA −tA [A, B] = (e e e ) = (e Be ) = AB − BA. dt t=0 ds s=0 dt t=0

2. Lie subgroups of GL(n, R) Before we continue to study other linear Lie groups, we need some general results on Lie subgroups. More details and proofs of these results will be discussed later. Deﬁnition 2.1. A subgroup H of a Lie group G is called a Lie subgroup if it is a Lie group (with respect to the induced group operation), and the inclusion map ιH : H,→ G is an immersion (and therefore a Lie group homomorphism). Suppose H is a Lie subgroup of G, and h be the Lie algebra of H. Since ι : H,→ G is an immersion and is a Lie group homomorphism, dιH : h → g is injective and is a Lie algebra homomorphism. So we can think of h as a Lie subalgebra (= a linear subspace that is closed under the Lie bracket) of g. Note that a one-parameter subgroup of H is automatically a one-parameter of G (with initial vector in TeH), so the exponential map expH : h → H is exactly the restriction of expG : g → G onto h. The following theorem, which we will prove later, is very useful in determine the Lie algebra of a Lie subgroup. Theorem 2.2. Suppose H is a Lie subgroup of G. Then as a Lie subalgebra of g,

h = {X ∈ g | expG(tX) ∈ H for all t ∈ R}. Now we are ready to study other linear Lie groups. By deﬁnition they are Lie subgroups of GL(n, R). Example. (The special linear group) The special linear group is deﬁned as SL(n, R) = {X ∈ GL(n, R) : det X = 1}. It is easy see that SL(n, R) is a subgroup of GL(n, R). In PSET 1 we have seen that SL(n, R) is an n2 − 1 dimensional submanifold of GL(n, R). It follows that SL(n, R) is a (connected non-compact) Lie subgroup of GL(n, R). LECTURE 7: LINEAR LIE GROUPS 3

To determine its Lie algebra sl(n, R), we notice that det eA = eTr(A). So for an n × n matrix A, eA ∈ SL(n, R) if and only if Tr(A) = 0. We conclude sl(n, R) = {A ∈ gl(n, R) | Tr(A) = 0}. Example. (The orthogonal group) Next let’s consider the orthogonal group T O(n) = {X ∈ GL(n, R): X X = In}. n(n−1) This is another subgroup of GL(n, R). In PSET 1 we have seen that O(n) is an 2 n(n−1) dimensional submanifold of GL(n, R). So O(n) is an 2 dimensional Lie subgroup of GL(n, R). Note that O(n) is compact. To ﬁgure out its Lie algebra o(n), we note that (eA)T = eAT , so

tA T tA tAT −tA (e ) e = In ⇐⇒ e = e For all t. Since the exponential map is locally bijective, we conclude that A ∈ o(n) if and only if AT = −A. So T o(n) = {A ∈ gl(n, R) | A + A = 0}, which is the space of n × n skew-symmetric matrices. Notice that O(n) is not connected. It consists of two connected components, and the connected component of identity is the called the special orthogonal groups T SO(n) = {X ∈ GL(n, R): X X = In, det X = 1} = O(n) ∩ SL(n, R). Its Lie algebra so(n) is the same as o(n). Example. (The symplectic group) The symplectic group is by deﬁnition T Sp(2n, R) = {X ∈ GL(2n, R): X JX = J}, 0 I where Let J = n . It is a Lie group of dimension n(2n + 1) with Lie algebra −In 0 T sp(2n, R) = {A ∈ gl(2n, R) | JA + A J = 0} A1 A2 T T T = {A = | Ai ∈ M(n, R),A1 = −A4 ,A2 = A2 ,A3 = A3 }. A3 A4 This, as well as the orthogonal group in the previous example, are special cases of the following more general example. Let β : Rn × Rn → R be a bilinear form on Rn. Consider the set of all invertible n × n matrices that preserves β, n GLβ(n, R) = {X ∈ GL(n, R) | β(Xu, Xv) = β(u, v)for all u, v ∈ R }. In matrix form, there is a matrix B such that β(u, v) = uT Bv. Then T GLβ(n, R) = {X ∈ GL(n, R) | X BX = B}. 4 LECTURE 7: LINEAR LIE GROUPS

Lemma 2.3. GLβ(n, R) is a linear Lie group with Lie algebra T glβ(n, R) = {A ∈ gl(n, R) | A B + BA = 0}.

Proof. One can easily check that GLβ(n, R) is a subgroup of GL(n, R), and it is topo- logically a closed subset. According to the Cartan’s closed subgroup theorem that we will prove later, it is a Lie subgroup. tA To describe its Lie algebra, notice that A ∈ glβ(n, R) if and only if e ∈ GLβ(n, R), i.e. etAT BetA = B. By taking t derivative at t = 0, we get AT B + BA = 0. Conversely, if AT B + BA = 0, i.e. tAT B = B(−tA), one can easily derive by deﬁnition that tAT −tA tAT tA e B = Be , i.e. e Be = B. This completes the proof. n Notice that in the case B = In (β = the standard inner product on R ), we get the orthogonal group O(n), and in the case B = J(β = the standard symplectic form on R2n), we get the symplectic group above. If we take β to be the standard inner product of signature (p, n − p), p n X X β(x, y) = xiyi − xiyi, i=1 i=p+1 then we will get the indeﬁnite orthogonal group O(p, n − p), T O(p, n − p) = {X ∈ GL(n, R) | X I(p, n − p)X = I(p, n − p)}, where I(p, n − p) = diag(Ip, −In−p). Its Lie algebra is T o(p, n − p) = {X ∈ gl(n, R) | A I(p, n − p) + I(p, n − p)A = 0.}. Example. (The unitary group) All the previous examples generalize to complex matrices. In particular, the unitary groups T U(n) = {X ∈ GL(n, C): X X = In} is a Lie subgroup of GL(n, C) with Lie algebra T u(n) = {A ∈ gl(n, C) | A + A = 0}, the space of skew-Hermitian matrices. Also the special unitary groups SU(n) = U(n) ∩ SL(n, C) has Lie algebra T su(n) = {A ∈ M(n, C) | A + A = 0, Tr(A) = 0}. Also we have the compact symplectic group Sp(n) = U(2n) ∩ Sp(2n, C). This is a real compact Lie group of dimension n(2n + 1).