1 LIE GROUPS Lie groups and their algebras
1 Lie groups
A topological group is a group that is also a topological space, in which both the left translations
λg : G → G h 7→ gh
(for g ∈G) and the inversion map
ιG : G → G g 7→ g-1 are continuous. 1 LIE GROUPS
A (real) Lie group is a topological group that is locally Euclidean, i.e. covered by open sets W each homeomorphic to an open subset
U⊆Rn (Rn is the set of n-tuples of real numbers with the usual topology associated to the Euclidean metric); the local homeomorphism α: W →U is called a local chart.
Since U is a subset of Rn, local charts allow to parametrize (locally) group elements; the group element g ∈W will have parameter vector
α~ (g) = (α1(g) ,..., αn(g))
Remark. Overlapping local charts give rise to dierent local parametriza- tions of the same element, related to each other by transition functions. 1 LIE GROUPS
If the positive integer n is the same for all local charts (e.g. if G is connected), then it is called the dimension of G (put dierently, G is an n parameter Lie group).
Example: 3D space translations form a three parameter Lie group, with
(global) coordinates given by the vector components (with respect to a given basis) of the image of the origin ~0.
Remark. While Rn has no distinguished point as a topological space, its linear structure distinguishes the origin ~0=(0,..., 0), hence it is natural (but by no means necessary) to associate this point to the only distin- guished group element, the identity element of G; whenever possible, ~ we shall use the convention α~ (1G)=0. 1 LIE GROUPS
Given a local chart α : W → U in a neighborhood of the identity, the group structure implies the existence of continuous maps µ : U×U → U and ι: U→U (the structure functions) for which
α~ (gh) = µ α~ (g) , α~ (h) α~ g-1 = ι(α~ (g))
µ α~ , β~ is the parameter vector of the product of the group elements with parameter vectors α~ and β~ , while ι(α~ ) is that of the inverse. Associativity of the group product implies the relation
µ α~ , µ β~ , γ~ = µ µ α~ , β~ , γ~ for α~ , β~ , γ~ ∈U. 1 LIE GROUPS
Since ~0 is, by convention, the parameter vector of the identity element
µ α~ ,~0 = µ ~0, α~ = α~ µ α~ , ι α~ = µ ι α~ , α~ = ~0
Remark. Since the parameter vectors are n-tuples of real numbers, the maps µ and ι (that characterize locally the group structure) can be studied by means of calculus in several variables.
Gleason-Montgomery-Zippin: the structure functions µ : U×U → U and ι:U→U are analytic, i.e. their Taylor-series around the origin have a positive radius of convergence. 1 LIE GROUPS
Examples
1. The additive group (R, +) of real numbers is a one dimensional Lie group, with structure functions µ(α, β) = α+β and ι(α) = −α
in case of the trivial parametrization α(z)=z.
2. The circle group U(1) = {z ∈C | |z|=1} of complex phases is again a one dimensional Lie group, with structure functions µ(α, β)=
α +β and ι(α) = −α in case of the exponential parametrization
α(z)=−i log z.
3. The isospin group † SU(2)= U ∈Mat2 (C) | det U =1,U U =1 is a three parameter Lie group. 1 LIE GROUPS
4. The 3D rotation group of rotations around axes having a common
point (the center) is a three parameter Lie group, since rotations
can be parametrized e.g. by the 3 Euler angles; topologically, it is
the interior of a ball, with nontrivial identi cation of the bound-
ary. Since rotations preserve orientation, and the distance from the
center is invariant under them, the rotation group can be identi ed
with the group SO(3) of 3-by-3 orthogonal matrices.
5. The Poincaré group P (the symmetry group of 4D Minkowski space- time) is 10 dimensional: 4 parameters correspond to space-time
translations, while 6 to 4D rotations, out of which 3 describe 3D
rotations, and another 3 the Lorentz boosts. 1 LIE GROUPS
A Lie homomorphism φ: G1 →G2 between the Lie groups G1 and G2 is a continuous (analytic) group homomorphism.A local isomorphism is an analytic map that is a bijective homomorphism when restricted to a suitable neighborhood of the identity (identical structure functions).
Remark. Note that Lie (resp. local) isomorphism is a more (resp. less) restrictive notion than ordinary algebraic isomorphism.
A closed subgroup of a Lie group is itself a Lie (sub)group.
A one-parameter subgroup of the Lie group G is a continuous (Lie) ho- momorphism from the additive group (R, +) of real numbers into G (its image, a Lie subgroup of dimension 1, is a continuous curve in G). 2 LIE ALGEBRAS
2 Lie algebras
A Lie algebra is a linear space L endowed with a binary operation, the
Lie bracket [, ]: L×L→L, that is bilinear and skew symmetric, i.e.
[λa+µb, c]=λ[a, c]+µ[b, c] [a, λb+µc]=λ[a, b]+µ[a, c] [b, a]=− [a, b] and satis es the Jacobi identity
[a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0 for all a, b, c∈L and arbitrary scalars λ, µ. 2 LIE ALGEBRAS
Examples:
1. R3 with the cross product as Lie bracket; 2. with the commutator ; Matn (R) [A,B]=AB−BA
3. general linear algebra gl(V ) of all linear operators A: V → V with
the commutator [A,B]=AB−BA;
4. observable quantities (i.e. functions on phase space) of a Hamilto-
nian system with the Poisson bracket as Lie bracket;
5. angular momentum operators in quantum mechanics. 2 LIE ALGEBRAS
A Lie algebra morphism is a linear map φ:L1 →L2 such that
φ [a, b]=[φ(a), φ(b)]
Given a basis B={b1, . . . , bn} of L, the Lie brackets
n X ij [bi, bj] = ck bk k=1 of the basis vectors completely determine the algebra.
The coe cients ij, the structure constants of , satisfy ck L
jk kj skew symmetry ci + ci = 0 X jm kl km lj lm jk Jacobi identity ci cm + ci cm + ci cm = 0 m and characterize the Lie algebra up to isomorphism. 3 THE LIE ALGEBRA OF A LIE GROUP
3 The Lie algebra of a Lie group
The analytic functions on a Lie group G form a ring A(G) with the pointwise operations of addition and multiplication.
A derivation of A(G) is a map ∇: A(G)→A(G) that is additive
∇(a + b) = ∇a + ∇b vanishes on constant functions, and satis es the Leibnitz rule
∇(ab) = (∇a) b + a∇b
Derivations of A(G) form a Lie algebra Der(A(G)) with the commutator
[∇1, ∇2] = ∇1 ◦∇2 − ∇2 ◦∇1 as Lie bracket. 3 THE LIE ALGEBRA OF A LIE GROUP
For a∈A(G) and g ∈G, the function
g a : G → R h 7→ a g-1h is analytic, ag ∈A(G), and gives rise to an additive map
λg : A(G) → A(G) a 7→ ag for which
λgh = λg ◦λh
A (left) invariant derivation ∇∈Der(A(G)) is one that satis es
∇◦λg = λg ◦∇ for all g ∈G. 3 THE LIE ALGEBRA OF A LIE GROUP
The sum and commutator of invariant derivations is itself invariant
invariant derivations form a Lie algebra Lie(G)
Every Lie group has a canonical parametrization such that n 1 X µ α~ , β~ = α + β + cjkα β + higher order terms i i i 2 i j k j,k=1 and ι α~ =−α~ .
Lie's theorem: the coe cients jk are the structure constants of ci Lie(G) (with respect to a suitable basis), and the higher order terms are deter- mined by the quadratic ones.
Local structure of G re ected by the linear object Lie(G)! 3 THE LIE ALGEBRA OF A LIE GROUP
Q: how can we compute the Lie algebra?
Lie transformation group: continuous group of dierentiable coordinate transformations 0 of m, where n. xi 7→xi(x1, . . . , xm|α~ ) R α~ ∈U⊆R
In nitesimal generators( i=1, . . . , n)
m 0 X ∂xj ∂ Ti = ∂α α~ =~0 ∂x j=1 i j are rst order partial dierential operators, with commutators
n X ij [Ti,Tj] = Ti ◦Tj − Tj ◦Ti = ck Tk k=1 with ij the structure constants of the Lie algebra. ck 3 THE LIE ALGEBRA OF A LIE GROUP
Examples 1. The group of 3D translations
3 3 tα~ : R → R x~ 7→ x~ + α~ with α~ ∈R3. The in nitesimal generators read
X ∂(xj +αj) ∂ ∂ T = = i ∂α ∂x ∂x j i j i
The Lie algebra is commutative, since all Lie brackets vanish: [Ti,Tj]=0. 2. The one-parameter group of 2D rotations
x cosα x−sinα y R(α): 7→ y sinα x+cosα y 3 THE LIE ALGEBRA OF A LIE GROUP for α∈[0, 2π]. The only generator reads
∂(cosα x−sinα y) ∂ ∂(sinα x+cosα y) ∂ ∂ ∂ T = + =−y + x ∂α ∂x ∂α ∂y ∂x ∂y
3. The a ne group Affn consisting of the transformations
a(λ, ~a):x~ 7→λx~ +~a for ~a ∈Rn and λ∈R. The generators and their commutators read n n X ∂(λxi +ai) ∂ X ∂ D = = x ∂λ ∂x i ∂x i=1 λ=1,~a=~0 i i=1 i n X ∂(λxi +ai) ∂ ∂ T = = j ∂a ∂x ∂x i=1 j λ=1,~a=~0 i j
n n ! X ∂ ∂ ∂ X ∂ ∂ [D,T ]= x − x =− =−T j i ∂x ∂x ∂x i ∂x ∂x j i=1 i j j i=1 i j 3 THE LIE ALGEBRA OF A LIE GROUP
If G is a linear Lie group, i.e. a closed subgroup of GL(n, R), then its Lie algebra Lie(G) is isomorphic to the Lie algebra
2 g= A∈Matn (R) | 1n +εA∈G, ε =0 of matrices in nitesimally close to the identity, with the usual commu- tator as Lie bracket.
Example: the Lie algebra of contains those matrices SU(2) A∈Mat2 (R) for which 1=det (12 +εA)=1+εTr(A) and
† † 12 =(12 +εA) 12 +εA =12 +ε A+A
(recall that ε is in nitesimal), i.e. Tr(A)=0 and A† =−A. But any such matrix is a combination of Pauli matrices with imaginary coe cients, hence the algebra is isomorphic to that of angular momentum operators. 4 GLOBAL PROPERTIES
4 Global properties
The Lie algebra Lie(G) (equivalently, the structure functions µ and ι) re ects only the local structure of the Lie group G (in a suitable neigh- borhood of the identity), the global structure is captured by topology.
The group is
compact, if every open covering contains a nite subcovering;
connected, if any two group elements may be connected by a continuous curve;
simply connected, if any closed curve can be deformed continu- ously to a point. 4 GLOBAL PROPERTIES
Every Lie group has a connected Lie subgroup G0 cosets of in connected components of G0 G ! G Every connected Lie group G is locally isomorphic with a unique simply connected Lie group Gˆ, its universal cover, and there is a discrete central subgroup Z G =∼ G/Zˆ Remark. Z is nite if G is compact. 4 GLOBAL PROPERTIES Examples 1. (R, +) and U(1) are locally isomorphic, but while (R, +) is simply connected and non-compact, U(1) is compact and connected, but not simply connected (R, +) is the universal cover of U(1). 2. SU(2) and SO(3) are locally isomorphic, but while the former is simply connected and compact, hence SU(2) is the universal cover of SO(3), the latter is compact and connected, but not simply connected. 3. the Poincaré group is neither connected (re ections!) nor compact (translations!). 5 THE HAAR MEASURE 5 The Haar measure An invariant measure µ on a topological group G is a Lebesgue-measure such that for every g ∈ G and every measurable set U ⊆ G the translate gU ={gx | x∈U} is also measurable, and µ(gU) = µ(U) Haar's theorem: every compact group admits an invariant measure, unique up to normalization. Normalized Haar measure: µ(G)=1. Allows the integration (averaging) of real valued functions on G. 6 THE 3D ROTATION GROUP 6 The 3D rotation group Group of rotations around axes having a common point (the origin) in common. Since Cartesian coordinates transform linearly, rotations are orientation preserving and the distance from the center is invariant rotation group may be identi ed with the matrix group SO(3) (with SO(n) in dimension n). Any rotation can be decomposed into a product of three consecutive rotations around perpendicular axes: O(α~ ) = Ox(αx) Oy(αy) Oz(αz) 6 THE 3D ROTATION GROUP where x cosα x−sinα y Oz(α):y 7→ sinα x+cosα y z z The in nitesimal generators read ∂ ∂ L = x − y z ∂y ∂x and ∂ ∂ L = y − z x ∂z ∂y ∂ ∂ L = z − x y ∂x ∂z The Lie-algebra is spanned by linear combinations of Lx,Ly,Lz. 6 THE 3D ROTATION GROUP ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ [L ,L ] = y −z z −x − z −x y −z x y ∂z ∂y ∂x ∂z ∂x ∂z ∂z ∂y ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = y z −x − z z −x − z y −z ∂z ∂x ∂z ∂y ∂x ∂z ∂x ∂z ∂y ∂ ∂ ∂ ∂ ∂ + x y −z = y − x = −L ∂z ∂z ∂y ∂x ∂y z and similarly [Lx,Lz] =Ly [Ly,Lz] = − Lx The generator of rotations around an axis parallel to n~ is Ln~ = nxLx + nyLy + nzLz and their commutator reads [Ln~ ,Lm~ ] = Ln~ ×m~ 6 THE 3D ROTATION GROUP the Lie algebra of the rotation group is isomorphic with that of 3D vectors (with the cross product as Lie bracket). 2 3 Remark. In nitesimal generators are anti-hermitian operators on L R hf, L gi = f(x, y, z)L g(x, y, z) dxdydz = − hL f, gi i ˆ i i The self-adjoint operators do not belong to the Lie algebra Ji = −i ~Li properly, but to its complexi cation, and have Lie brackets (ijk is the Levi-Civita tensor) [Ji,Jj] = i ijk~Jk the commutation rules of the angular momentum operators! 6 THE 3D ROTATION GROUP The operator 2 2 2 2 J = Jx + Jy + Jz that does not belong to the Lie algebra, but only to its universal en- veloping algebra (consisting of non-commutative polynomial expressions in the generators), commutes with all of the generators 2 Ji,J =0 Such an element of the enveloping algebra is called a Casimir operator. SU(2) is simply connected, its center Z has order 2, and SU(2) and SO(3) have the same Lie algebras, hence SU(2) is the universal cover of SO(3) SO(3) =∼ SU(2)/Z