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 "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: Since the multiplication is not commutative, the group is non-abelian. The neutral element of the Heisenberg group is the identity matrix, and inverses are given by The group is a subgroup of the 2-dimensional affine group Aff(2): formula_2 acting on formula_6 corresponds to the affine transform formula_7. There are several prominent examples of the three-dimensional case. If , are real numbers (in the ring R) then one has the continuous Heisenberg group H(R). It is a nilpotent