PMATH 763

This is just a really quick bunch of chapter summaries for quick reference. Hopefully I can find the time to make more insightful posts soon.

Summary of Chapter 5:

• You can exponentiate matrices by using the usual power series formula of $\exp$. Everything converges and works out nicely:
• If $XY = YX$ then $\exp (X+Y) = \exp X \exp Y$. Consequently $\exp(-X) = (\exp X)^{-1}$ so that $\exp X \in \mathrm{GL}_n(\mathbb{F})$.
• Conjugation passes through: $g(\exp X)g^{-1} = \exp(gXg^{-1})$.
• $\det \exp X = e^{\mathrm{Tr} \; X}$ (use Jordan canonical form).
• $\exp : \mathrm{M}_n(\mathbb{F}) \to \mathrm{GL}_n(\mathbb{F})$ isn’t injective except in trivial cases.
• You can rearrange absolutely summable infinite series (with terms in $\mathbb{F}$ or even $\mathrm{M}_n(\mathbb{F})$) without altering the sum’s value; this relies on completeness.
• For $g \in \mathrm{M}_n(\mathbb{F})$ with $\| g - I \| < 1$ you can define the matrix logarithm by the usual power series for $\log(x) = \log(1+(x-1))$. For such matrices $\exp(\log g) = g$. Similar things about $\log(\exp X)$.
• There are neighbourhoods $U$ of $0$ and $V$ of $I$ such that $\exp : U \to V$ is a $\mathcal{C}^\infty$ diffeomorphism.

Summary of Chapter 6:

• One-parameter subgroup of $\mathrm{GL}_n(\mathbb{F})$ is a continuous group homomorphism $\gamma : (\mathbb{R}, +) \to \mathrm{GL}_n(\mathbb{F})$ (we conflate $\gamma$ with its image as usual).
• Every one-parameter subgroup is of the form $\gamma(t) = \exp(tA)$ for some $A \in \mathrm{M}_n(\mathbb{F})$, and $A$ is called the infinitesimal generator of $\gamma$. Show $\gamma$ is differentiable, then let $A = \gamma'(0)$ and apply matrix product rule.

Summary of Chapter 7:

• matrix Lie group is a closed subgroup $G \leq \mathrm{GL}_n(\mathbb{F})$. The associated Lie algebra is $\mathfrak{g} = \mathrm{Lie}(G) = \{ X \in \mathrm{M}_n(\mathbb{F}) : \exp(tX) \in G, \; \forall t \in \mathbb{R} \}$.
• Define the Lie bracket by $[X,Y] = XY - YX$ for $X, Y \in \mathfrak{g}$. Then $\mathfrak{g}$ is a $\mathbb{R}$-vector space closed under the Lie bracket.
• This motivates us to define a matrix Lie algebra as a $\mathbb{R}$-vector subspace $\mathfrak{g}$ of $\mathrm{M}_n(\mathbb{F})$ which is closed under taking Lie brackets.
• $\frac{d}{dt} \bigg|_{t=0} \exp(tX) = X$.
• $\mathfrak{gl}_n(\mathbb{F}) = \mathrm{M}_n(\mathbb{F})$.
• $\mathfrak{sl}_n(\mathbb{F})$ consists of the “traceless” (trace 0) matrices.
• Other examples of the Lie algebras associated to certain matrix Lie groups.
• $\mathrm{Lie}(G \cap H) = \mathfrak{g} \cap \mathfrak{h}$.

Summary of Chapter 8:

• $\mathcal{C}^1$ coordinate system: $\{ (\varphi_i, U_i) \}_{i \in I}$, typical definition with open covers/charts, such that the transition maps $\varphi_i \circ \varphi_j^{-1}$ are $\mathcal{C}^1$.
• Two coordinate systems are $\mathcal{C}^1$ equivalent when the transition maps $\varphi_i \circ \psi_j^{-1}$ are $\mathcal{C}^1$.
• $\mathcal{C}^1$ manifold consists of a topological space $M$ together with an equivalence class of $\mathcal{C}^1$ coordinate systems on $M$.
• Coordinates at identity: given by the exponential map (maps a neighbourhood of $0$ in $\mathfrak{g}$ homeomorphically to a neighbourhood of $I$ in $G$).
• Matrix Lie groups are analytic manifolds.
• The connected component of the identity of a matrix Lie group $G$ is an open normal subgroup generated by $\exp \mathfrak{g}$.