Past exam of the lie Groups and Differentiable Manifolds course of the University of Oxford / 2008 / A / 1

For a Matrix Lie group, $T_{g}G$ consists of derivatives ${\gamma }^{\prime }(0)$ of differentiable curves in $G$ with $\gamma (0)=g$, viewed inside $M_n(\mathbb{C})$ as a real vector space. At the identity, multiplying two such curves adds their derivatives, and reparametrizing a curve multiplies its derivative by any real scalar. Thus $L(G)=T_{I}G$ is a real vector subspace.

If $Y\in L(G)$ and $g\in G$, conjugating a curve with derivative $Y$ proves $gY{g}^{-1}\in L(G)$. Let $g(s)$ have $g(0)=I$ and $g^{\prime }(0)=X\in L(G)$. Differentiating the curve $g(s)Yg(s{)}^{-1}$ inside the finite-dimensional, hence closed, vector subspace $L(G)$ givesThe matrix commutator is real bilinear, antisymmetric, and satisfies the Jacobi identity by associativity of matrix multiplication. This proves that $L(G)$ is a real Lie algebra.

The exponential-map theorem for a linear Lie group says $\exp X\in G$ whenever $X\in L(G)$. Applying it to every real multiple of $X$ gives $\exp (tX)\in G$ for every real $t$. Conversely that curve has derivative $X$ at zero, so $X\in L(G)$. Hence

Here the orthogonal group is the real group $O_n(\mathbb{R})$. Differentiating $g(t{)}^{T}g(t)=I$ at zero gives $X^T+X=0$. Conversely a real skew-symmetric matrix satisfies $(e^{tX}{)}^T{e}^{tX}=e^{-tX}{e}^{tX}=I$. ThereforeFor $n\ge 2$ this is not a complex Lie subalgebra of $M_n(\mathbb{C})$: multiplying a nonzero real member by $i$ leaves this real subspace. For $n=1$ the zero algebra is the trivial complex Lie algebra as well. The complex orthogonal group would be a different group from the real orthogonal group used here.

Differentiating the affine block form givesConversely the exponential of such a block matrix has bottom row $(0,1)$ and invertible upper-left block $e^{tB}$, so it remains in the affine group. Its commutator corresponds toThe space is closed under multiplication by $i$ and its bracket is complex bilinear. Thus it is a complex Lie algebra of dimension $n^2+n$, and its real dimension is $2n^2+2n$.