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 real Lie group of dimension 3. In addition to the representation as real 3x3 matrices, the continuous Heisenberg group also has several different representations in terms of function spaces. By Stone–von Neumann theorem, there is, up to isomorphism, a unique irreducible unitary representation of H in which its centre acts by a given nontrivial character. This representation has several important realizations, or models. In the "Schrödinger model", the Heisenberg group acts on the space of square integrable functions. In the theta representation, it acts on the space of holomorphic functions on the upper half-plane; it is so named for its