Heisenberg group

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

under the operation of matrix multiplication. Elements a, b and c can be taken from any commutative ring with identity, often taken to be the ring of real numbers (resulting in the "continuous Heisenberg group") or the ring of integers (resulting in the "discrete Heisenberg group").

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

In the three-dimensional case, the product of two Heisenberg matrices is given by:

The neutral element of the Heisenberg group is the identity matrix, and inverses are given by

It is a subgroup of 2-dimensional affine group ${\displaystyle \mathrm {Aff} (2)}$. ${\displaystyle {\begin{pmatrix}1&a&c\\0&1&b\\0&0&1\\\end{pmatrix}}}$ corresponds to the affine transform ${\displaystyle {\begin{pmatrix}1&a\\0&1\end{pmatrix}}x+{\begin{pmatrix}c\\b\end{pmatrix}}}$.

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