Lie Groups Part 1

Problem 4.37 Prove that the following are Lie groups: (i) Each finite-dimensional real vector space with its structure of additive group. In particular $\mathbb{R}^{n}$. (ii) The set of non-zero complex numbers $\mathbb{C}^{*}$ with the multiplication of complex numbers. (iii) $G \times H$, where $G, H$ are Lie groups, with the product $(g, h)\left(g^{\prime}, h^{\prime}\right)=$ $\left(g g^{\prime}, h h^{\prime}\right), g, g^{\prime} \in G, h, h^{\prime} \in H$. In general, if $G_{i}, i=1, \ldots, n$, is a Lie group, then $G_{1} \times \cdots \times G_{n}$ is a Lie group. (iv) $T^{n}$ for $n \geqslant 1$ (toral group). (v) Aut $V$, where $V$ is a vector space of finite dimension over $\mathbb{R}$ or $\mathbb{C}$, with the composition product, and in particular $\operatorname{GL}(n, \mathbb{R})=\operatorname{Aut}_{\mathbb{R}} \mathbb{R}^{n}$ and $\operatorname{GL}(n, \mathbb{C})=$ Aut $_{\mathbb{C}} \mathbb{C}^{n}$ (vi) $K=\mathbb{R}^{n} \times \operatorname{GL}(n, \mathbb{R}), n>1$, with the group structure defined by $$ (x, A)\left(x^{\prime}, A^{\prime}\right)=\left(x+A x^{\prime}, A A^{\prime}\right) $$ The relevant theory is developed, for instance, in Warner [13].

Problem 4.38 Consider the product $T^{1} \times \mathbb{R}^{+}$of the one-dimensional torus by the multiplicative group of strictly positive numbers (that group is called the group of similarities of the plane). Let $(\theta, x)$ denote local coordinates. Show that the vector field $$ \frac{\partial}{\partial \theta}+x \frac{\partial}{\partial x} $$ is left-invariant.

Problem 4.39 Using the coordinate vector fields $\partial / \partial x_{j}^{i}, 1 \leqslant i, j \leqslant n$, on GL( $\left.n, \mathbb{R}\right)$, prove that the vector field $Y$ on $\operatorname{GL}(n, \mathbb{R})$ whose matrix of components at the identity is $A=\left(a_{j}^{i}\right)$ and whose matrix of components is equal to $B A$ at the element $B=\left(b_{j}^{i}\right)$ of $\operatorname{GL}(n, \mathbb{R})$ is a left-invariant vector field.