Lie groups Flashcards
Lie groups and their parametrization
What’s a topological group?
A group G that is at the same time a topological space such that for g ∈ G both
- the left translations: λg : G → G, h → gh
- and the inversion map: ιG : G → G, g → g^(-1)
are continuous.
Lie groups and their parametrization
What do the following concepts mean: manifold, local chart, transition functions?
Manifold: a topological space that is locally Euclidean; it is covered by open sets W, each homeomorphic to an open subset U ⊆ Rn, where Rn denotes n-dimensional Euclidean space
- in other words, it’s a generalization of curves, smooth surfaces in higher dimenions
Local chart: a local homeomorphism αW : W → U
- it allows to parametrize each point x ∈ W by real-valued curvilinear coordinates, the components of αW(x)
Transition functions: the relations between the different local parametrization of the same point allowed by the overlapping of local charts
- if the positive integer n is the same for all local charts, then n is called the dimension of the space.
Lie groups and their parametrization
What are Lie groups and how can they be parametrized?
An n-parameter topological group that is locally Euclidean of dimension n.
The local charts allow to parametrize (locally) the group elements: to a group element g ∈ W is associated its parameter vector:
α(g) = (α1(g) , . . . , αn(g))
- convention: α(1G) = 0 (not necessary tho)
- example: 3D space translations τ : R3 → R3 form a 3-parameter Lie group
Lie groups and their parametrization
What are structure functions?
Given a local chart αW : W → U in a neighborhood W ⊆ G of the identity element 1G, the group structure implies the existence of continuous maps μ : U×U → U and ι : U → U (the structure functions) such that:
α(gh) = μ(α(g)), α(h)) and 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: μ(α, μ(β, γ)) = μ(μ(α, β), γ)
- identity: μ(α, 0) = μ(0, α) = α
- inverses: μ(α, ι(α)) = μ(ι(α), α) = 0
Lie groups and their parametrization
What’s the Gleason-Montgomery-Zippin theorem?
The structure functions μ : U×U → U and ι : U → U are analytic, meaning their Taylor series around the origin have a positive radius of convergence, if they are twice diferentiable.
Lie groups and their parametrization
What are some examples of Lie groups?
- additive group (R, +) of real numbers: one parameter, μ(α, β) = α+β, ι(α) = −α, in case of the trivial parametrization α(z) = z
- 3D rotation group of rotations around axes having a common point: three parameters, parametriazation with Euler angles for example (can be identified with the group SO3(R))
- Poincaré group: 10 dimensional, 4 (space time translations) + 3 (3D rotations) + 3 (Lorentz boosts) parameters
3D rotations and Lorentz boosts together make the 4D rotations.
Lie groups and their parametrization
What’s the canonical parametrization of Lie groups? Lie’s theorems?
Every Lie group G has a (infinitely many) canonical parametrization such that
μ(α, β)(i) = α(i) + β(i) + 1/2 sum(j,k = 1, n) c^(jk)(i) α(j)β(k) + higher order terms and ι(α) = -α.
Lie’s theorems: the coeffiecients c^(jk)(i) satisfy skew-symmetry and the Jacobi identity and determine all the higher order terms.
- any system of real coeffcients c^(ij)(k) that satisfy the above requirements corresponds to some Lie group
- the coefficients are the structure constants, they characterize the algebraic structure locally
Lie groups and their parametrization
What’s a Lie homomorphism and what’s a local isomorphism?
Lie homomorphism: a mapping ϕ : G1 → G2 between the Lie groups G1 and G2 that is a continuous (analytic) group homomorphism
- a natural notion for comparing Lie groups
Local isomorphism: an analytic map that is a bijective homomorphism when restricted to a suitable neighborhood of the identity
- such groups have locally indentical structure functions with suitable parametrizations, hence they look the same locally
- much less restrictive than a Lie homomorphism
Lie groups and their parametrization
What’s a defining property of one-parameter subgroups of a Lie group?
For a Lie group G, they are homomorphic images (inside G) of the additive group (R, +) of real numbers.
Lie algebras
What is a Lie algebra and what’s a Lie algebra homomorphism?
A Lie algebra is a linear space L endowed with a binary operation, the Lie bracket [,] : L×L → L, that:
- is bilinear
- is skew-symmetric
- satisfies the Jacobi identity
Lie algebra homomorphism: a linear map ϕ : L1 → L2 such that ϕ([a, b]) = [ϕ(a), ϕ(b)]
Lie algebras
What are two important examples of Lie algebras?
General linear algebra of all linear operators: V → V with the commutator [A,B] = AB −BA as Lie bracket
Observable quantities (continuous functions on the phase space): Hamiltonian systems with the Poisson bracket as Lie bracket
Lie algebras
What determine and characterize the Lie algebras?
Given a basis B = {b1, . . . , bn} of L, the Lie brackets
[b(i), b(j) ] = sum(k=1,n) c^(ij)(k)b(k)
of the basis vectors determine the algebra because of bilinearity (insert képlet).
The structure constants c^(ij)(k) of L satisfy skew-symmetry and the Jacobi identity and characterize the Lie algebra up to isomorphism.
Lie algebras
What’s the connection between Lie groups and Lie algebras?
The structure constants of a Lie group satisfy the same identities, so there’s a correspondence between Lie groups and Lie algebras. Questions about Lie groups can be reduced to questions about Lie algebras.
- and the question of Lie algebras can be reduced to linear algebra because of the linear structure of Lie algebras
Lie algebras
How can we compute the Lie algebra of a Lie group?
With effective methods for Lie transformation groups, meaning continuous groups of differentiable coordinate transformations: insert képlet
- infinitesimal generators: first order differential operators
- the latter have commutators that contain the structure constants of the Lie algebra of the group
Global properties
What are the three fundamental topological properties?
The Lie algebra alone doesn’t tell us about the global structure, only the local one. Topology is what does the former.
- compactness: if every open covering contains a finite subcovering
- connectedness: if any two group elements may be connected by a continuous curve
- simply connectedness: if any closed curve can be deformed continuously to a point
These are good criteria to decide whether two groups are isomorphic or not for example.
Global properties
What’s the component of the identity and a universal cover?
Component of the identity: in every Lie group, the endpoints of all continuous curves starting at the identity element form a connected subgroup G0 < G
- there is a one-to-one correspondence between the cosets of G0 in G and the connected components of G
Universal cover: every connected Lie group G is locally isomorphic with a unique simply connected Lie group G^, which is the universal cover
Every Lie algebra corresponds to a unique (up to isomorphism) simply connected Lie group, hence the study of Lie algebras parallels that of simply connected groups.
The 3D rotation group
How can we describe the 3D rotation group as a Lie group?
It’s the group of 3D rotations around axes having a point in common (the rotation center).
- because rotations transform Cartesian coordinates linearly, they form a Lie transformation group
- detO > 0 (rotations preserve orientation) + the distance from the rotation center is invariant: O is an orthogonal matrix
There exists a one-to-one correspondence between 3D rotations and 3-by-3 orthogonal matrices whose determinant equals 1. The group of 3D rotations is isomorphic with the matrix group SO(3).
- it’s a 3-parameter Lie group (3 angular coordinates to parametrize the 3D rotations)
The 3D rotation group
How to obtain the Lie algebra of the rotation group?
- Any rotation can be decomposed into a product of three consecutive rotations around perpendicular axes: O(α) = Ox(αx) Oy(αy) Oz(αz).
- We can calculate the infinitesimal generators: Lx, Ly and Lz: the Lie algebra is spanned by the lineracombinations of these
- Lie brackets form commutators of infinitesimal generators
- The generator of rotations around an axis parallel to n is: L(n) = n(x)Lx + n(y)Ly + n(z)Lz and their commutator is [L(n), L(m)] = L(n x m)
The Lie algebra of the rotation group is isomorphic with that of 3D vectors (with the cross product as Lie bracket).
Each O operator rotates a given plane around the given axis.
The 3D rotation group
What’s the relation between the SO(3) and SU(2) groups?
SU(2) and SO(3) have the same Lie algebra, hence they are locally isomorphic, because the Pauli matrices (algebra of SU(2)) have the same Lie brackets as the generators J(i) for SO(3).
While the former is simply connected and compact, the latter is compact and connected, but not simply connected: SU(2) is the universal cover of SO(3).
- SO(3) is a factor group by a central subgroup: SO(3) ∼= SU(2)/Z
- Z is the center of SU(2)
SU(2) is the isospin group consisting of the traceless self-adjoint 2x2 matrices with the commutator of matrices as the Lie bracket.
The 3D rotation group
What’s the relation between angular momentum and infinitesimal generators of the rotation group?
Noether’s theorem implies that there’s some kind of connection.
Guess: the infinitesimal generators are linear operators acting on the Hillbert space L^2(R3) of square-integrable functions (the space of the wave functions), so they could be related to the components of the angular momentum operator
Problem: the eigenvalues of the inf. generators are dimensionless so they have to be rescaled by Planck’s constant, but they will still be anti-hermition op.s on L^2(R3)
- purely imaginary eigenvalues (≠ observable quantities)
Solution: operators J(i) = −iℏL(i)
- self-adjoint
- their Lie brackets: [J(i), J(j)] = iϵ(ijk)ℏJ(k) reproduce the commutation rules of the components of the angular momentum operator
