Lie Groups

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A Lie group is a group \(G\) which is a differential manifold (locally \(\mathbb{R}^n\) and functions are differentiable). Additionally the map

\[G \times G \ni (a, b) \mapsto a^{-1} b \in G\]

must be smooth.

For any \(a \in G\), the left and right translations \(\gamma_a, \delta_a\) are diffeomorphisms (maps and inverses both smooth)

\[\gamma_a:G \to G, \quad \gamma_a(b) := ab\] \[\delta_a:G \to G, \quad \delta_a(b) := ba\]

as is the internal automorphism of \(G\)

\[ad_{a}: G \to G, \quad ad_{a}(b) := aba^{-1}\]

since \(ad_{a} = \delta_{a^{-1}} \circ \gamma_a\).

A vector field \(A\) in the space \(V(G)\) of vector fields over \(G\) is left-invariant, if for any \(a \in G\)

\[\gamma_{a}^{*}A = A\]

where \(\gamma_{a}^{*}\) denotes the induced map on vector fields.

Since

\[h^{*}[A, B] = [h^{*}A, h^{*}B]\]

for any diffeomorphism \(h\), the set \(\mathcal{G} \subset V(G)\) of all left-invariant vector fields on \(G\) is closed under the bracket. \(G'\) is the Lie algebra of \(G\). Any \(A \in \mathcal{G}\) is uniquely identified by its value \(A_{e}\) at the unit element of \(G\) as

\[T_{e} \gamma_{a}(A_{e})\]

So \(\mathcal{G}\) may be identified with \(T_{e}G\), the tangent space at the identity, and \(\textrm{dim} \mathcal{G} = \textrm{dim} G\). If \((e_i)\) is a frame in \(\mathcal{G}\), then

\[[e_i, e_j] = c^{k}_{ij} e_{k}\]

where \(c^{k}_{ij}\) are the structure constants of \(G\). From the properties of the bracket they are skew in (i,j) and satisfy the Jacobi identity.

Irreducible representations of Lie algebras in practice

MRI and NMR machines at the hospital: SU(2)
standard model of particle physics: SU(3) × SU(2) × U(1)

Examples

\(\mathbb{R}^n\) as a differentiable manifold, with vector addition as group multiplication. Translation group.
\(U(1)\), set of all \(e^{i\phi}\) with \(\phi \in [0, 2\pi)\). Corresponding inverse elements are \(e^{i(2\pi -\phi)}\). The domain being a half-open interval, the manifold is not trivial. Define the charts as identity maps about appropriate open intervals. Define a simple ‘wrapping’ chart \(\psi_0\) about the special boundary point 0: \(\psi_0(p) = p\) for p greater than or equal to 0, \(\psi_0(p) = p - 2\pi\) for p less than \(2\pi\). U(1) is clearly a differentiable manifold, and \((\phi_1, \phi_2) \mapsto e^{i(\phi_2 - \phi_1)}\) is smooth, so this gives a Lie group.
\(SU(2)\), the set of all pairs of complex numbers (z1, z2) such that \(z_1^2 + z_2^2 = 1\). Isomorphic to the 3-sphere embedded in \(\mathbb{R}^4\). Equivalent to set of all unitary 2 by 2 matrices of determinant 1.