Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2010 • Question 10

Question

Math 6510 • Differentiable Manifolds Spring 2010 • Question 10

Let \(G\cong \SO (2)\ltimes \mathbb {R}^4\) be the Lie group given by matrices \[ \begin {pmatrix} \cos (\theta )&-\sin (\theta )&x&y\\ \sin (\theta )&\cos (\theta )&z&w\\ 0&0&\cos (\theta )&-\sin (\theta )\\ 0&0&\sin (\theta )&\cos (\theta ) \end {pmatrix} \] where \(\theta ,x,y,z,w\in \mathbb {R}\). Find the linear subspace of \(\mathfrak {gl}_4(\mathbb {R})\) that is the Lie algebra of \(G\).

Raw LaTeX

Let \(G\cong\SO(2)\ltimes\R^4\) be the Lie group given by matrices
\[
  \begin{pmatrix}
    \cos(\theta)&-\sin(\theta)&x&y\\
    \sin(\theta)&\cos(\theta)&z&w\\
    0&0&\cos(\theta)&-\sin(\theta)\\
    0&0&\sin(\theta)&\cos(\theta)
  \end{pmatrix}
\]
where \(\theta,x,y,z,w\in\R\). Find the linear subspace of
\(\mathfrak{gl}_4(\R)\) that is the Lie algebra of \(G\).

Topic Tags

Lie Algebras and Exponential Maps Lie Groups and Matrix Lie Groups