Heisenberg Group

In mathematics, the Heisenberg group, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form

\begin{pmatrix} 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end{pmatrix}

or its generalizations under the operation of matrix multiplication. Elements a, b and c, can be taken from some (arbitrary) commutative ring with identity, often taken to be the ring of real numbers or the ring of integers.

The real Heisenberg group arises in the description of one-dimensional quantum mechanical systems. More generally, one can consider groups associated to n-dimensional systems, and most generally, to any symplectic vector space.

Read more about Heisenberg Group:  Higher Dimensions, On Symplectic Vector Spaces, The Connection With The Weyl Algebra, Representation Theory, As A Sub-Riemannian Manifold

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