Hi,
I am currently working with Jacobi groups and I am not clear why $R^2 \cdot s_1$ is a central extension of $R^2$ by $s_1$
$R^2 \cdot s_1$ is the following constructed group:
The elements of $R^2 \cdot s_1$ are the pairs (x,s) with $x \in R^2 $ and $s \in s^1$
The product $(x, s) \cdot (x', s') := (x+x',ss'e^{2\pi i(det(|x | x'|))})$ where $det(|x | x'|) = \lambda \mu ' - \lambda ' \mu$ , $x = (\lambda, \mu)$
With calculation it's clear that $R^2 \cdot s_1$ is a group
- that's clear to me
$s_1$ is a subgroup of $R^2 \cdot s_1$ because of s->(0, s)
- that's clear to me
$s_1$ is the center of $R^2 \cdot s_1$
- that's clear to me
Now the unclear part comes and I don't know how this can be proofed.
$R^2 \cdot s_1$ is a central extension of $R^2$ by $s_1$
It tried properties of central groups, like
$1 -> s_1 -> R^2 \cdot s_1 -> R^2 -> 1$
calculating the direct product
$R^2 \cdot s_1 = s_1 \times R^2$
trying to proof surjectivity
$R^2 \cdot s_1 -> R^2$
and injectivity
$s_1 -> R^2 \cdot s_1$
but without success.
Do you have in clue how to proceed.
Thanks in advance.