We study the Identity Problem, the problem of determining if a finitely
generated semigroup of matrices contains the identity matrix; see Problem 3
(Chapter 10.3) in ``Unsolved Problems in Mathematical Systems and Control
Theory'' by Blondel and Megretski (2004). This fundamental problem is known to
be undecidable for Z4×4 and decidable for Z2×2. The Identity Problem has been recently shown to be in polynomial
time by Dong for the Heisenberg group over complex numbers in any fixed
dimension with the use of Lie algebra and the Baker-Campbell-Hausdorff formula.
We develop alternative proof techniques for the problem making a step forward
towards more general problems such as the Membership Problem. We extend our
techniques to show that the fundamental problem of determining if a given set
of Heisenberg matrices generates a group, can also be decided in polynomial
time