Heisenberg Group

The Heisenberg group is the group of three by three upper triangular matrices with ones on every diagonal entry, under the operation of matrix multiplication, named after the physicist Werner Heisenberg. Its three off diagonal entries can be drawn from any commutative ring with identity, most often the real numbers, which gives the continuous Heisenberg group, or the integers, which gives the discrete Heisenberg group. The continuous Heisenberg group arises in describing one dimensional quantum mechanical systems, particularly through the Stone von Neumann theorem, and the construction generalizes to systems of any dimension and, most generally, to any symplectic vector space.