 Solution

ID: past-exam-of-the-lie-groups-and-differentiable-manifolds-course-of-the-university-of-oxford/2008/a/1/b/ii/solution

Differentiating the affine block form gives

L (Aff_{n} (C)) = {(B 0 u 0) : B \in M_{n} (C), u \in C^{n}} .

Conversely 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 to

[(B, u), (C, v)] = ([B, C], B v - C u) .

The 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

2 n^{2} + 2 n

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