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

Table of contents
- Solution 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

.

 Ancestors (13)

 View article source

 Discussion (0) New discussion

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version