Lie Groups I
5.1 Definitions and Examples
One approach to geometry is to view geometry as the study of invariance and symmetry. In our case we are interested in studying symmetries of differentiable manifolds, Riemannian manifolds, symplectic manifolds etc. Now the usual way to talk about symmetry in mathematics is by the use of the notion of a transformation group. The wonderful thing for us is that the groups that arise in the study of manifold symmetries are themselves differentiable (even analytic) manifolds. Perhaps the reader should keep in mind the prototypical example of the Euclidean motion group and the subgroup of rotations about a point in Euclidean space Rn. The rotation group is usually represented as the group of orthogonal matricesO(n) and then the action on Rn is given in the usual way as
x7→Qx
forQ∈O(n) andxa column vector from Rn. It is pretty clear that this group must play a big role in physical theories. Of course, translation is important too and the combination of rotations and translations generate the group of Euclidean motions E(n). We may represent a Euclidean motion as a matrix using the following trick: If we want to represent the transformationx→Qx+b where x, b∈Rn we can achieve this by letting column vectors
1 x
∈Rn+1 represent elementsx. The we form
1 0 b Q
and notice that 1 0 b Q
1 x
=
1 b+Qx
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Thus we may identifyE(n) with the group of (n+ 1)×(n+ 1) matrices of the above form where Q ∈ O(n). Because of the above observation we say that E(n) is the semidirect product ofO(n) andRn and writeO(n)iRn.
Exercise 5.1 Make sense of the statementE(n)/O(n)∼=Rn.
Lie groups place a big role in classical mechanics (symmetry and conserva-tion laws), fluid mechanic where the Lie group may be infinite dimensional and especially in particle physics (Gauge theory). In mathematics, Lie groups play a prominent role Harmonic analysis (generalized Fourier theory) and group repre-sentations as well as in many types of geometry including Riemannian geometry, algebraic geometry, K¨ahler geometry, symplectic geometry and also topology.
Definition 5.1 AC∞ differentiable manifoldGis called a Lie group if it is a group (abstract group) such that the multiplication mapµ:G×G→Gand the inverse mapν :G→Ggiven byµ(g, h) =ghandν(g) =g−1areC∞functions.
If G and H are Lie groups then so is the product group G×H where multiplication is (g1, h1)·(g2, h2) = (g1g2, h1h2).Also, if H is a subgroup of a Lie groupGthat is also a regular closed submanifold thenH is a Lie group itself and we refer toH as a (regular)Lie subgroup. It is a nontrivial fact that an abstract subgroup of a Lie group that is also a closed subset is automatically a Lie subgroup (see 8.4).
Example 5.1 Ris a one-dimensional (abelian Lie group) were the group mul-tiplication is addition. Similarly, any vector space, e.g. Rn, is a Lie group and the vector addition.
Example 5.2 The circle S1={z∈C: |z|2= 1} is a 1-dimensional (abelian) Lie group under complex multiplication. More generally, the torus groups are defined byTn=S1×. . .×S1
n-times .
Now we have already introduced the idea of a discrete group action. We are also interested in more general groups action. Let us take this opportunity to give the relevant definitions.
Definition 5.2 A left action of a Lie groupGon a manifoldM is a smooth map λ:G×M →M such thatλ(g1, λ(g2, m)) =λ(g1g2, m))for allg1, g2∈G.Define the partial map λg : M →M by λg(m) =λ(g, m) and then the requirement is that λe : g 7→ λg is a group homomorphism G → Diff(M). We often write λ(g, m)asg·m.
Example 5.3 (The General Linear Group) The set of all non-singular ma-tricesGL(n,R)under multiplication.
Example 5.4 (The Orthogonal Group) The set of all orthogonal matrices O(n,R) under multiplication. Recall that an orthogonal matrix A is one for
5.1. DEFINITIONS AND EXAMPLES 83 which AAt=I. Equivalently, an orthogonal matrix is one which preserves the standard inner product onRn:
Ax·Ay=x·y.
More generally, let V, b=h., .i be any n- dimensional inner product space (b is just required to be nondegenerate) and let k be the maximum of the dimensions of subspaces on whichb is positive definite. Then define
Ok,n−k(V,b) ={A∈GL(V) :hAv, Awi=hv, wifor allv, w∈V}. Ok,n−k(V,b)turns out to be a Lie group. As an example, take inner product on Rn defined by
hx, yi:=
Xk i=1
xiyi− Xn i=1+1
xiyi
and denote the corresponding group by Ok,n−k(Rn). We can identify this group with the group O(k, n−k)of all matricesAsuch that AΛAt= Λ where
Λ =diag( 1, ...,
k times −1, ...
n−k times
).
O(1,3)is called the Lorentz group. If we further require that the elements of our group have determinant equal to one we get the groups denoted SO(k, n−k).
Exercise 5.2 Show that if 0 < k < n then SO(k, n−k) has two connected components.
Example 5.5 (The Special Orthogonal Group) The set of all orthogonal matrices having determinant equal to one SO(n,R) under multiplication. This is thek=ncase of the previous example. As a special case we have the rotation groupSO(3,R) =SO(3).
Example 5.6 (The Unitary Group) U(2) is the group of all 2×2 complex matricesA such that AA¯t=I.
Example 5.7 (The Special Unitary Group) SU(2) is the group of all 2x2 complex matrices A such that AA¯t = I and det(A) = 1. There is a special relation betweenSO(3)andSU(2)that is important in quantum physics related to the notion of spin. We will explore this in detail later in the book.
Notation 5.1 A∗:= ¯At
Example 5.8 The set of all A ∈ GL(n,R) with determinant equal to one is called the special linear group and is denoted SL(n,R).
Example 5.9 Consider the matrix2n×2n matrix J =
0 1
−1 0
.
J defines a skew-symmetric bilinearω0 form on R2n byω0(v, w) =vtJw. The set of all matrices inM2n×2n that preserve this bilinear form is called the sym-plectic groupSp(n,R)and can be defined also as
Sp(n,R) ={S∈M2n×2n:StJS=J.
This group is important in Hamiltonian mechanics. If e1, ..., en, ..., e2n is a the standard basis ofR2n then ω0=Pn
i=1ei∧ei+n. We will see that in some basis every nondegenerate 2-form on R2n takes this form.
Exercise 5.3 Show thatSU(2)is simply connected.
Exercise 5.4 Let g ∈G(a Lie group). Show that each of the following maps G→Gis a diffeomorphism:
1) Lg:x7→gx(left translation) 2) Rg:x7→xg (right translation) 3) Cg:x7→gxg−1 (conjugation).
4) inv:x7→x−1 (inversion)