1
$\begingroup$

When looking for a non linear Lie Group I always find the example of the Heisenberg Group $H$ modulo a normal Group $N$. Where the matrix of the two groups are of this form

$$ H = \begin{bmatrix} 1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1 \end{bmatrix} $$ $$ N = \begin{bmatrix} 1 & 0 & n \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$

Do you know any other simple example of non-linear Lie Group?

$\endgroup$
  • 4
    $\begingroup$ By nonlinear I suppose you mean having no faithful finite-dimensional representation. For $n \geq 2$ the universal cover $\widetilde{SL}(n, \Bbb R)$ of $SL(n, \Bbb R)$ is such an example. $\endgroup$ – Travis Jan 11 '16 at 13:18
1
$\begingroup$

As Travis said, the universal cover of the a special linear group is not a matrix group. Another classical example is the metaplectic group.

Also any nilpotent lie group wich is non simply connected is not a matrix group.

See Wolfgang Ziller lecture notes on lie groups and representations p40 as reference.

Edit: the reduced heissenber group isnt a linear group also

$\endgroup$
  • $\begingroup$ I've already downloaded them early this morning, but thanks anyway $\endgroup$ – Dac0 Jan 31 '16 at 20:02

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.