The Heisenberg group 
in 
complex variables is the group
of all 
with 
and 
having multiplication
![(w,t)(z,t^')=(w+z,t+t^'+I[w^*z])](/images/equations/HeisenbergGroup/NumberedEquation1.svg) |
(1)
|
where 
is the adjoint. The Heisenberg group is isomorphic
to the group of matrices
![[1 z^* 1/2|z|^2+it; 0 1 z; 0 0 1],](/images/equations/HeisenbergGroup/NumberedEquation2.svg) |
(2)
|
and satisfies
 |
(3)
|
Every finite-dimensional unitary representation is trivial on 
and therefore factors to a group
representation of the quotient 
.
