The Heisenberg group 
in 
complex variables, where 
is a positive integer,
is the group of all 
with 
and 
having multiplication
![(w,t)(z,t^')=(w+z,t+t^'+Im[w^*z]),](/images/equations/HeisenbergGroup/NumberedEquation1.svg) |
(1)
|
where 
is the conjugate transpose and 
denotes the imaginary part.
The Heisenberg group is isomorphic to the group of block matrices
![[1 z^* 1/2||z||^2+it; 0 I_n z; 0 0 1],](/images/equations/HeisenbergGroup/NumberedEquation2.svg) |
(2)
|
where 
is the 
identity matrix and 
is the squared l2-norm.
It satisfies
 |
(3)
|
Its group center is 
. Every finite-dimensional unitary
representation is trivial on 
and therefore factors to a group
representation of the quotient group 
(Knapp 1996, p. 544).
