Download 1. Topology Here are some basic definitions concerning topological

Lie Groups (GraMPs)
Spring 2009
Hans Plesner Jakobsen
1. Topology
Here are some basic definitions concerning topological spaces. You
may think of the topology as coming from a metric, but other topologies
may occur.
Definition 1.1. A set X is called a topological space with topology T
provided there is a family T of subsets of X for which the following
holds:
• X ∈ T and ∅ ∈ T .
• O1 , . . . , Or ∈ T ⇒ O1 ∩ · · · ∩ Or ∈ T .
• If {Oα }α∈A ⊆ T , then ∪α∈A Oα ∈ T (A is any index set).
The sets in T are called the open subsets of X.
If x ∈ X, a neighborhood of x is by definition any open subset
O with x ∈ O.
The topological space X is Hausdorff if
(1)
∀x1 6= x2 ∈ X : ∃O1 , O2 ∈ T : x1 ∈ O1 , x2 ∈ O2 and O1 ∩ O2 = ∅.
A set X can always be made into a topological space by choosing
T = {X, ∅}. This is the trivial topology. At the other extreme,
T = P(X) (all subsets of X) is also a topology. This is also of no
practical use. We will write (X, T ) when we are in the situation of
Definition 1.1.
Definition 1.2. Let (X1 , T1 ) and (X2 , T2 ) be topological spaces. A
map f : X1 → X2 is continuous (with respect to the given topologies) if
(2)
∀U ∈ T2 : f −1 (U ) ∈ T1 .
Definition 1.3. A subfamily B ⊆ T is called a basis for the topology or a basis for the open sets, if any O ∈ T can be written
as a union of some sets from B. The space X is said to be second
countable if there is a countable basis for the open sets.
Definition 1.4. A topological space (X, T ) is disconnected if there
are two open non-empty sets O1 , O2 of X such that
(3)
X = O1 ∪ O2 and O1 ∩ O2 = ∅.
If X is not disconnected we say that it is connected.
2
Definition 1.5. If (X, T ) is a topological space and Y ⊂ X,
(4)
TY = {O ∩ Y | O ∈ T }
defines a topology on Y called the relative topology or the subset
topology.
2. Differentiable manifolds
Definition 2.1. Let M be a second countable topological space. A
differentiable structure on M (of dimension n) is a family
A = {(χi , Oi )}i∈I , where I is an index set, such that
M1 ∀i ∈ I : Oi is an open subset of M and χ is a homeomorphism
of Oi onto the open set χ(Oi ) ⊆ Rn
M2 M = ∪i∈I Oi .
∞
M3 ∀i, j ∈ I χi ◦ χ−1
map.
j : χj (Oi ∩ Oj ) → χi (Oi ∩ Oj ) is a C
Remark 2.2. Actually, it is a “differentiable structure of class C ∞ ”
we have defined. In a similar fashion one may define structures of
class C k for all k = 0, 1, . . . , as well as class C ω (analytic), but here
we are only concerned with C ∞ .
Definition 2.3. A set M is a differentiable manifold (an ndimensional differentiable manifold) if M is a second countable Hausdorff topological space with differentiable structure A = {(χ, Oi )}i∈I
(of dimension n). Furthermore, A is an atlas, and the individual
elements (χi , Oi ) are called charts. One also refers to (χi , Oi ) as
“local coordinates” on M , or as a local parametrization. The n in the
definition is called the dimension of M . Occasionally we will write
M n.
3. Lie Groups
Definition 3.1. A Lie Group G is is an abstract group which is also
a differentiable manifold and where the two structures are compatible
in the sense that the maps
G × G 3 (g, h) →
7
g·h∈G
G3g →
7
g −1 ∈ G
are both smooth.
3.1. Matrix Lie Groups.
Definition 3.2. A Matrix Lie Group is a closed subgroup of Gl(n, C)
for some n.
3
In the case of a matrix Lie Group, the fundamental observation is that
there exists an open ball Uε = B0 (ε) = {X ∈ Mn (C) | |X| < ε} such
that Vε = exp(Uε ) is an open set in GL(n, C) with the property that
G ∩ Vε = exp(Uε ∩ g), where g is the Lie algebra of G.
For any g ∈ G one may define a chart (χg , g · Vε ) where the Vε is as
above, and where
(g · Vε ) 3 h 7→ χg (h) := log(g −1 h).
Since the operations of taking inverse and of multiplication from the
left are smooth in GL(n, C), and since log is smooth, it follows easily
that the “change of coordinates” (χa )◦(χb )−1 are smooth for all a, b ∈ G
where ever they are defined. Thus, a Matrix Lie Group is indeed a Lie
Group.
4. The Matrix Lie Groups, their Lie algebras, and the
exponential map
We give here some fundamental definitions and results - the latter
without proofs - relating to certain families of Lie groups and Lie algebras.
Some of the definitions are particular for this course.
Definition 4.1. By a matrix Lie group we understand a closed subgroup of either GL(n, R) or GL(n, C).
Definition 4.2. A Lie algebra is a vector space g with a bilinear map
[·, ·] : g × g 7→ g satisfying
• (Skew symmetry)
∀X, Y ∈ g : [X, Y ] = −[Y, X].
• (The Jacobi identity)
∀X, Y, Z ∈ g : [[X, Y ], Z] + [[Y, Z], X] + [[Z, X], Y ] = 0.
The map [·, ·] is called the Lie bracket. The linear structures may
in fact be over any field F. If F = R we say that the Lie algebra is
real, if F = C we say that it is complex.
Definition 4.3. The Lie algebras gl(n, R) and gl(n, C) are defined
to be the sets of all n × n real and complex,respectively, matrices
equipped with the Lie bracket
[X, Y ] = XY − Y X.
Here XY is given by matrix multiplication, etc.
4
Remark 4.4. Any real subspace of gl(n, R) – or of gl(n, C)(!) –
which is invariant under the Lie bracket is a real Lie algebra. Any
complex subspace of gl(n, R) which is invariant under the Lie bracket
is a complex Lie algebra.
Example 4.5. u(n) = {H ∈ gl(n, C) | H ∗ = −H} is a real Lie
algebra. su(n) = {H ∈ gl(n, C) | H ∗ = −H and Tr(H) = 0} is a
real Lie algebra.
Remark 4.6. By the famous Ado’s Theorem, any finite dimensional
real Lie algebra is equivalent to one obtained from a subspace of
gl(n, R) for some n. In contrast, there are very many finite-dimensional abstract Lie groups that do not have isomorphic images inside
some GL(n, R) or GL(n, C).
Definition 4.7. The exponential map gl(n, C) 7→ Gl(n, C) is given
as
∞
X
Ai
A
.
∀A ∈ gl(n, C) : exp(A) = e =
i!
i=0
P Ai
P∞ kAki
kAk
Observe that k ∞
k
≤
, where kAk denotes the
i=0 i!
i=0 i! = e
operator norm of A. A similar estimate holds for the Hilbert-Schmidt
norm k · kHS ; c.f. Exercise 2.3.2.
Proposition 4.8. The following useful facts hold:
i) ∀s, t ∈ R : exp((s + t)A) = exp(sA) exp(tA).
ii) More generally, if [A, B] = 0 then exp(A+B) = exp(A) exp(B).
iii) det(eA ) = eT r(A) .
iv) exp(A) is invertible; (exp(A))−1 = exp(−A).
v) (exp(A))∗ = (exp(A∗ )).
Moreover, we mention without proof
Lemma 4.9. If R 3 t 7→ a(t) ∈ GL(n, C) and R 3 t 7→ b(t) ∈
GL(n, C) are differentiable functions, then
da(t)
db(t)
d
(a(t) · b(t)) =
· b(t) + a(t) ·
(= a0 (t)b(t) + a(t)b0 (t)).
dt
dt
dt
Moreover,
d exp(t · A)
= A exp(t · A) = exp(t · A)A.
dt
5
The first major theorem relating Lie groups and Lie algebras is the
following:
Theorem 4.10. Let G be a matrix Lie group. Then
g := {H ∈ gl(n, R) | ∀t ∈ R : exp(tH) ∈ G}
is a Lie subalgebra. A similar result is true for gl(n, C).1 We say
that g is the algebra of G.
Example 4.11. To find the Lie algebra su(2) of SU (2) we observe
that for X to be in su(2) we must have ∀t : (exp(tX))(exp(tX))∗ = 1
and ∀t : det(exp(tX)) = 1. If we use v) in Proposition 4.8 and
differentiate the first equation at t = 0 we obtain: X + X ∗ = 0, i.e.
X is skew adjoint (iX is self adjoint). The second equation becomes
etT r(X) = 1 for all t which implies that T r(X) = 0. Thus,
ix λ
| x ∈ R, λ ∈ C .
su(2) =
−λ −ix
A basis of su(2) is given by e.g.
1 i 0 1 0 −i 1 0 1 (5) X =
, Y =
, Z=
.
2 0 −i
2 −i 0
2 −1 0
Notice that
(6)
[X, Y ] = Z, [Y, Z] = X, [Z, X] = Y.
Notice that “the same” computations give that u(n) = {X ∈ gl(n, C) |
X+X ∗ = 0} and su(n) = {X ∈ gl(n, C) | X+X ∗ = 0, and T r(X) =
0}.
Example 4.12. The Lie algebra so(3) of SO(3) is given as
(
!
)
0 a b
−a 0 c | a, b, c ∈ R .
so(3) =
−b −c 0
A basis of so(3) is given by e.g.
!
!
0 1 0
0 0 1
0 0 0 , C=
A = −1 0 0 , B =
0 0 0
−1 0 0
1The condition is still that ∀t ∈ R : exp(tH) ∈ G
0 0 0
0 0 −1
0 1 0
!
.
6
Definition 4.13. Let g, h be real Lie algebras. A real linear map
dπ : g 7→ h that satisfies
∀X, Y ∈ g : dπ([X, Y ]) = [dπ(X); dπ(Y )],
where the Lie brackets are computed in the relevant Lie algebras, is
called a homomorphism. The definition for the complex case is
analogous.
In case h = gl(n, C) (viewed as a real Lie algebra) we say that dπ is
a Lie algebra representation. In the latter case, if dπ(g) ⊆ u(n)
we say that dπ is infinitesimally unitary.
The second major theorem relating Lie groups and Lie algebras is:
Theorem 4.14. Let φ : G 7→ H be a homomorphism between two
matrix Lie groups G and H. Let the Lie algebras be denoted g, h,
respectively. Then there exists a Lie algebra homomorphism dφ :
g 7→ h such that
∀t ∈ R, ∀X ∈ g : φ(expG (tX)) = expH (tdφ(X)).
(7)
GO
φ
/
expG
HO
expH
g
dφ
/
h
In the special case where φ = π is an n-dimensional complex (unitary)
representation of a matrix Lie group G we obtain a Lie algebra homomorphism dπ : g 7→ gl(n, C) (u(n)). In this case, dπ(X) is called the
infinitesimal generator corresponding to X.
Remark 4.15. It is of course not at all clear a priori why a Lie
group homomorphism φ, which to begin with only is assumed to be
continuous, is differentiable and hence can give rise to the map dφ not even in the special case of a continuous representation.
For later use we mention that any homomorphism dφ : g 7→ h between
real Lie algebras can be extended to the complexifications, gC = g ⊗R C
and hC = h ⊗R C simply by setting
dπ C (X1 + iX2 ) = dπ(X1 ) + idπ(X2 ).
This is extremely useful. Since any matrix A ∈ gl(n, C) can be written
uniquely as A = K1 + iK2 with K1 , K2 skew adjoint, it follows that any
representation dπ of u(n) can be extended (complexified) to gl(n, C).
7
But even more so, any representation of gl(n, R) can be complexifed
to arepresentation of gl(n, C). Hence, to any finite dimensional irreducible (contemplate this!) representation of gl(n, R) there corresponds
an irreducible representation of u(n) (in the same space) and vice versa.
5. The classical groups
5.1. Bilinear forms and pseudo-orthogonal groups. We let B ∈
B(V1 , V2 , . . . Vn ) denote the set of multi linear maps V1 × V2 × · · · ×
Vn 7→ F in analogy to the case with the bilinear maps. Further, we set
B(V, V ) = B2 (V ), and, in general, B(V, V, . . . , V ) = Bn (V ).
| {z }
n
Definition 5.1. Let B ∈ B(V, W ). We say that B is non degenerate
if
∀v ∈ V : B(v, w) = 0 ⇒ w = 0
∀w ∈ W : B(v, w) = 0 ⇒ v = 0.
Let, as usual, Sn denote the symmetric group in n letters and let
Sgn(σ) = (−1)σ denote the sign of the permutation σ ∈ Sn .
Definition 5.2. Let B ∈ Bn (V ). We say that B is symmetric if
∀v1 , v2 , . . . , vn ∈ V og ∀σ = (σ1 , . . . , σn ) ∈ Sn :
B(vσ1 , . . . , vσn ) = B(v1 , . . . , vn ).
Definition 5.3. Let B ∈ Bn (V ). We say that B is alternating if
∀v1 , v2 , . . . , vn ∈ V og ∀σ = (σ1 , . . . , σn ) ∈ Sn :
B(vσ1 , . . . , vσn ) = Sgn(σ)B(v1 , . . . , vn ).
Assume that B ∈ B2 (V ) is non degenerate and symmetric. Then we
set
(8)
O(B, F) = {g ∈ AutF (V ) | ∀v1 , v2 ∈ V : B(gv1 , gv2 ) = B(v1 , v2 )} .
One can say that O(B, F) is the invariance group of B. We also refer to
O(B, F) as a pseudo-orthogonal group though this terminology is used
mostly for the case with F = R.
A real vector space V with a positive definite inner product (dot product) (·, ·) (here after just called an inner product) is, as is well known,
called a (pre) Euclidean space. If V ' Rn we denote by (·, ·)E the usual
inner product
(9)
(v, w)E = x1 y1 + x2 y2 · · · xn yn ,
8
where v = (x1 , x2 , · · · , xn ) and w = (y1 , y2 , · · · , yn ) and we will also
use this terminology if V ' Cn .
As is well known, if (V, (·, ·)) is a real Euclidean vector space, a linear
operator o : V 7→ V is said to be orthogonal if
(10)
∀x, y ∈ V : (o(x), o(y)) = (x, y).
We set O(V ) = {o | o orthogonal}, but if V = Rn we denote it
as O(V ) = O(n) and we refer to it as the “orthogonal group in n
dimensions”. In all cases, O(V ) is a group.
Now let B be a symmetric non degenerate form on Rn .
It is straightforward to show (see the problem sessions) that there exists
a symmetric linear operator JB so that
(11)
∀x, y ∈ Rn : B(x, y) = (x, JB (y))E .
One must have det(JB ) 6= 0. Conversely, any such operator J (matrix)
defines a symmetric non degenerate form BJ by
(12)
∀x, y ∈ Rn : BJ (x, y) = (x, J(y))E .
Definition 5.4. If J is symmetric and has p strictly positive eigenvalues and q strictly negative eigenvalues with p + q = n we define
OJ (p, q) = {o ∈ Gl(n, R) | ∀x, y ∈ Rn : BJ (o(x), o(y)) = BJ (x, y)}
= {o ∈ Gl(n, R) | ∀x, y ∈ Rn : (Jo(x), o(y))E = (Jx, y)E }
= {o ∈ Gl(n, R) | o∗ Jo = J}2.
The groups thus defined depend explicitly on J but are all isomorphic.
For this reason, the J is usually dropped and one just writes O(p, q).
Likewise, if Ba ∈ B2 (V ) is non degenerate and alternating (skew symmetric)
(13)
Sp(Ba , F) = {g ∈ AutF (V ) | ∀v1 , v2 ∈ V : Ba (gv1 , gv2 ) = Ba (v1 , v2 )}.
Again one could say that Sp(Ba , F) is the invariance group of Ba . These
are the (real or complex, depending on whether F = R or F = C.)
symplectic groups.
There is a skew symmetric linear operator JBa such that
(14)
∀x, y ∈ V : Ba (x, y) = (x, JBa (y))E .
2Notice that one has: [∀x, y ∈ Rn : (A(x), y) = 0] ⇔ A = 0 - one can merely look at x =
ei , y = ej for all the vectors ei in the canonical basis.
9
To be completely specific, if V = R2n (or V = C2n )3 we set
0 In
JBa = −I 0
(15)
n
- where In is the identity on Rn (or Cn ) - and we set
(16)
Sp(n, R) = {g ∈ GL(2n, R) | ∀v1 , v2 ∈ V : (JBa gv1 , gv2 ) = (JBa v1 , v2 )},
with an analogous definition in the complex case.
5.2. Hermitean forms. We remind you that a sesquilinear form
Hs in a complex vector space V is a map
V × V 3 v, w 7→ Hs (v, w) ∈ C,
(17)
satisfying:
∀λ ∈ C : ∀v1 , v2 , w, ∈ V :
Hs (v1 + λv2 , w) = Hs (v1 , w) + λHs (v2 , w) (linearity in the 1. var.)
and
∀λ ∈ C : ∀v, w1 , w2 , ∈ V :
Hs (v, w1 + λw2 ) = Hs (v, w1 ) + λHs (v, w2 ) (anti-linearity in the 2. var.)
We say that the sesquilinear form Hs (henceforth denoted H) is hermitian provided
(18)
∀v, w ∈ V : H(w, v) = H(v, w).
(and call it skew hermitian if ∀v, w ∈ V : H(w, v) = −H(v, w) - but
we will not get involved with those here...)
Finally, a hermitian form H is positive definite if
(19)
∀v ∈ V : v 6= 0 ⇒ H(v, v) > 0.
As is well known, a complex vector space V with a positive definite
hermitian form h·, ·i (hereafter just an inner product) is called a (pre)
Hilbert space. If V ' Cn , we denote by h·, ·i0 the usual inner product;
(20)
hv, wi0 = x1 y1 + x2 y2 · · · xn yn ,
if v = (x1 , x2 , · · · , xn ) and w = (y1 , y2 , · · · , yn ).
We set U(H) = {U | U unitary}, but if H = Cn we usually write
U(H) = U (n) and call it the “unitary group in n dimensions”. Quite
generally, U(H) is a group.
3Notice that it says 2n!
10
Now, let H be a Hermitean form on Cn . If we do not maintain the
assumption about it being positive definite, it is natural to assume in
stead that H is non degenerate.4 So, this is what we do.
It is straightforward to prove /see the problem sessions) that there is
a Hermitean (or self adjoint) operator JH 5
(21)
∀x, y ∈ Cn : H(x, y) = hx, JH (y)i0 .
One must have that det(JH ) 6= 0. Conversely, any such operator operator (matrix) J defines a Hermitean form HJ by
(22)
∀x, y ∈ Cn : HJ (x, y) = hx, J(y)i0 .
Definition 5.5. If J is Hermitean and has p strictly positive eigenvalues and q strictly negative eigenvalues with p + q = n we define
UJ (p, q) = {u ∈ Gl(n, C) | ∀x, y ∈ Cn : HJ (u(x), u(y)) = HJ (x, y)}
= {u ∈ Gl(n, C) | ∀x, y ∈ Cn : hJu(x), u(y)i0 = hJx, yi0 }
= {u ∈ Gl(n, C) | u∗ Ju = J}.
As in Definition 5.4, one usually drops the J and just writes U (p, q).
6. The Canonical Commutation Relations
Let Ba be a non-degenerate skew-symmetric form on a finite-dimensional
real vector space V . We know that V then is even dimensional; dim V =
2n
Definition 6.1. The Heisenberg Algebra hV , or just h(n), based on
V is the vector space V × R equipped with the Lie bracket
(23)
∀(v, c), (v1 , c1 ) ∈ V × R : [(v, c), (v1 , c1 )] = (0, Ba (v, v1 )).
The Heisenberg Group H(V ), or just H(n), based on V is the vector
space V × R equipped with the product
(24)
1
∀(v, c), (v1 , c1 ) ∈ V ×R : (v, c)?(v1 , c1 ) = (v +v1 , c+c1 + Ba (v, v1 )).
2
One can easily bring the skew-symmetric form into ‘canonical form’:
Specifically, there is a basis where one may write the elements v of V as
v = (q, p) with q, p ∈ Rn and such that
4This defined completely analogous to the case with bilinear forms.
5So. J satisfies: ∀x, y ∈ Cn : hJ (x), yi = hx, J (y)i. More generally, the adjoint T ∗ to
H
H
H
the linear operator T on the Hilbert space (H, h·, ·i) is defined by the equation ∀x, y ∈ V :
hT ∗ (x), yi = hx, T (y)i. T is then Hermitian if T ∗ = T .
11
q
q1
(25)
Ba ( p , p ) = q · p1 − q1 · p
1
where we use the usual ·-product in Rn . We shall use this form in the
sequel and both on the algebra and on the group level.
Consider


0 √12 qr √12 pr
c
0 0
√1 ps 
0
,

2
(26)
0 0
1
0 − √2 q s 
0 0
0
0
where the vectors in the first row are written in row form and the vectors
in the last column are written in column form. This formula defines a
faithful representation of h(n) as can be easily seen.
Likewise,


1 √12 qr √12 pr
c
0 1
√1 ps 
0

,
2
(27)
0 0
1
1 − √2 q s 
0 0
0
1
is a faithful 2 + 2n-dimensional representation of H(n).
Other matrix versions of h(n) and H(n) are also in use, for instance



0 q1 q2 · · · qn z






0
0
0
.
.
.
0
p




1





.
.
.
.
.


0
.
.
.
0
.
g
h(n) =  .
 | qi , pi , z ∈ R .


 ..

0
p
n−1




 0 0 0 ... 0 p 




n


0 0 0 ... 0
0
The analogous formula with 1’s in the diagonal is then an isomorphic
version of H(n). In the sequel we will use directly the definitions (23)
and (24) together with (25).
The following formula6 gives the Stone-von Neumann representation
of the canonical commutation relations.
Theorem 6.2. The following formula defines a strongly continuous
unitary representation of H(n) in L2 (Rn ):
(28)
1
(U (q, p, c)f ) (x) = ei~(x·q+c− 2 q·p) f (x − p).
6Notice two small changes from the lectures which is introduced to make the formula more
natural.
12
Proof: The group property follows easily. To prove continuity, it suffices
to prove that the map
(q, p, c) 7→ U (q, p, c)f
is continuous for f in a dense subspace. Here one may choose the
Schwartz Space S(Rn ), or the space Cc∞ (Rn ) of smooth functions with
compact support. For such functions the result follows easily by Lebesgue’s
Theorem on Dominated Convergence.
We will later indicate a proof of the Stone-von Neumann Uniqueness
Theorem which states that this representation is irreducible and essentially unique7. However, we wish to use this result immediately and for
this reason we now turn to the Symplectic Group:
Theorem 6.3. Let, for g ∈ Sp(n, R), g • (q1 , · · · , qn , p1 , · · · , pn )
denote the natural linear action of g on R2n . The function
Sp(n, R) × H(n) 7→ H(n) :
(g, (q1 , · · · , qn , p1 , · · · , yp ), c ) 7→ g ? (q1 , · · · , qn , p1 , · · · , pn ), c
= g • (q1 , · · · , qn , p1 , · · · , pn ), c
defines an action of Sp(n, R) on H(n) by automorphisms.
7Two different choice of the positive constant ~ will give inequivalent representations, but
besides this, the representation is unique.