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Chapter 19: Problem 3
Show that \(G L(n, C)\) and \(S L(n, \mathbb{C})\) are connected Lie groups. Is \(U(n)\) a connected group?
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Let \(E_{f}^{i}\) be the matrix whose \((i, j)\) th component is 1 and all other components vanish. Show that these matrices form a basis of \(G \mathcal{C}(n, \mathbb{R})\), and have the commutator relations $$ \left[E_{t}^{\prime}, E_{t}^{k}\right]=\delta_{i}^{t} E_{t}^{k}-\delta^{k}, E_{t^{-}}^{\prime} $$ Write out the structure constants with respect to this algebra in this basis.
Asin Problem \(9.2\) every Lorentz transformation \(L=\left[L^{1}\right]\) has det \(L=\pm 1\) and either \(L_{4}^{4} \geq 1\) or \(L_{4}^{4} \leq-1\). Hence show that the Lorentz group \(G=O(3,1)\) has four connected components, $$ \begin{aligned} G_{0}=& G^{++}: \operatorname{det} L=1, L_{4}^{4} \geq 1 & G^{+-}: \operatorname{det} L=1, L_{4}^{4} \leq-1 \\ G^{+}+: \operatorname{det} L=-1, L_{4}^{6} \geq 1 & G^{--}: \operatorname{det} L=-1, L_{4}^{4} \leq-1 \end{aligned} $$ Show that the group of components \(G / G_{0}\) is isomorphic with the discrete abelian group \(Z_{2} \times Z_{2}\).
Show that the special orthogonal group \(S O(n)\), the pseudo-orthogonal groups \(O(p \cdot q)\) and the symplectic group \(S p(n)\) arc all closed subgroups of \(G L(n, \mathbb{R})\). (a) Show that the complex groups \(S L(n, \mathbb{C}), O(n, C), U(n), S U(n)\) are closed subgroups of \(G L(n, C)\) (b) Show that the unitary groups \(U(n)\) and \(S U(n)\) are compact groups.
A function \(f: G \rightarrow \mathbb{R}\) is said to be an aralytic function on \(G\) if it can be expanded as a Taylor serics at any point \(g \in G\). Show that if \(X\) is a left-invariant vector ficld and \(f\) is an analytic function on \(G\) then $$ f(g \exp t X)=\left(c^{t x} f\right)(g) $$ where, for any vector ficld \(Y\), we define $$ \mathrm{e}^{\gamma} f=\boldsymbol{f}+Y \boldsymbol{f}+\frac{1}{21} Y^{2} f+\frac{1}{3 !} Y^{3} f+\cdots=\sum_{i=0}^{\infty} \frac{Y^{n}}{n !} f $$ The operator \(Y^{\pi}\) is defined inductively by \(Y^{n} f=Y\left(Y^{n-1} f\right)\).
Show that the groups \(S L(n, \mathbb{R})\) and \(S O(n)\) are closed subgroups of \(G L \cdot(N, \mathbb{R})\), and that \(U(n)\) and \(S U(n)\) are closed subgroups of \(G L(n, C)\). Show furthcrmore that \(S O(n)\) and \(U(n)\) are compact Lie subgroups.
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