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}.
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About mlbaker

just another guy trying to make the diagrams commute.
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