Heisenberg group by 90°. The other reflections are "xyx" and "yxy", and rotation by 180° is "xyxy" (="yxyx"). The Lie algebra formula_16 of the Heisenberg group formula_1 (over the real numbers) is the space of formula_18 matrices of the form with formula_20. The following three elements form a basis for formula_16: The basis elements satisfy the commutation relations: The name "Heisenberg group" is motivated by the preceding relations, which have the same form as the canonical commutation relations in quantum mechanics: where formula_25 is the position operator, formula_26 is the momentum operator, and formula_27 is Planck's constant. The Heisenberg group formula_1 has
Heisenberg group group of a locally compact abelian group "K", equipped with a Haar measure. Such a group has a Pontrjagin dual formula_101, consisting of all continuous formula_102-valued characters on "K", which is also a locally compact abelian group if endowed with the compact-open topology. The Heisenberg group associated with the locally compact abelian group "K" is the subgroup of the unitary group of formula_103 generated by translations from "K" and multiplications by elements of formula_101. In more detail, the Hilbert space formula_103 consists of square-integrable complex-valued functions formula_106 on "K". The translations in "K" form a unitary representation of "K" as