Heisenberg group

and the central relations. Their flows are right translations. The dual frame of the right-invariant coframe is different and has the opposite Lie-bracket sign.

Modular Heisenberg group ( $H_{p} = U T_{3} (F_{p})$ )

 0  0

The modular Heisenberg group consists of upper-unitriangular

3 \times 3

matrices over the finite field

F_{p}

. In coordinates

M (x, y, z)

, multiplication is

M (x, y, z) M (x', y', z') = M (x + x', y + y' + x z', z + z')

, and its order is

p^{3}

. Its center of a group is

{M (0, y, 0)} ≅ C_{p}

; the quotient group by the centre is

C_{p} \times C_{p}

, represented by the

x, z

coordinates. The elementary generators

a = M (1, 0, 0)

b = M (0, 0, 1)

c = M (0, 1, 0)

satisfy

a^{p} = b^{p} = c^{p} = e

ab a^{- 1} b^{- 1} = c

, and

c

central. For

p = 2

(ab)^{2} = c

, so the whole group does not have exponent two.

Left-invariant frame of the real Heisenberg group

 0  0

For multiplication

(x, y, z) (a, b, c) = (x + a, y + b, z + c + x b)

, the left-invariant frame is

E_{1} = \partial_{x}, E_{2} = \partial_{y} + x \partial_{z}, E_{3} = \partial_{z} .

(1)

Its Lie brackets of vector fields satisfy

[E_{1}, E_{2}] = E_{3}

and the other basis brackets vanish.

Heisenberg horizontal distribution ( $D = span (E_{1}, E_{2})$ )

 0  0

The horizontal smooth distribution on the real Heisenberg group is spanned by

E_{1} = \partial_{x}

and

E_{2} = \partial_{y} + x \partial_{z}

. It is not an involutive distribution because their bracket is

\partial_{z}

, but their brackets span the full tangent space. A horizontal curve satisfies

\overset{x}{˙} = α

\overset{y}{˙} = β

, and

\overset{z}{˙} = x β

Smooth horizontal reachability in the real Heisenberg group

 0  0

Every point

(P_{x}, P_{y}, P_{z})

can be reached from the identity by a smooth curve tangent to the Heisenberg horizontal distribution. Put

a = 2 (P_{z} - P_{x} P_{y} /2 + P_{x})

and choose

α (t) = P_{x} + a cos (2 π t), β (t) = P_{y} + 2 π sin (2 π t) .

(1)

Their integrals give

x (1) = P_{x}

y (1) = P_{y}

, and

z (1) = \int_{0}^{1} x (t) β (t) d t = P_{x} P_{y} /2 - P_{x} + a /2 = P_{z}

. The third coordinate measures the effect of the noncommuting horizontal directions; integrability would instead confine horizontal curves to a two-dimensional leaf.

Integer Heisenberg group ( $H (Z)$ )

 0  0

The integer Heisenberg group consists of upper-unitriangular three-by-three integer matrices. In coordinates

(x, y, z)

its multiplication is

(x, y, z) (x^{'}, y^{'}, z^{'}) = (x + x^{'}, y + y^{'}, z + z^{'} + x y^{'}) .

(1)

It is the semidirect product

Z^{2} ⋊_{A} Z

for

A = (1101),

(2)

and its abelianization is

Z^{2}

Scaled Heisenberg lattice ( $N_{k} = {(x, y, z) : x, y, z \in k Z}$ )

 0  0

In the real Heisenberg group with multiplication

(x, y, z) (x^{'}, y^{'}, z^{'}) = (x + x^{'}, y + y^{'}, z + z^{'} + x y^{'})

, the displayed set is a discrete cocompact subgroup. Writing

a = (k, 0, 0)

b = (0, k, 0)

c = (0, 0, k)

gives

[a, b] = c^{k}

and

c

central. Hence its abelianization is

Z^{2} \oplus Z / k

. The right quotient maps to

R^{2} / k Z^{2}

by its first two coordinates; each fibre is the central circle

R / k Z

Commutator identity in the integer Heisenberg group ( $[a^{r}, b^{s}] = c^{rs}$ )

 0  0

Write the Integer Heisenberg group with central generator

c = [a, b]

, so

ab = c ba

. Moving

a

successively past powers of

b

gives

a^{r} b^{s} a^{- r} b^{- s} = c^{rs} (r, s \in Z) .

(1)

For positive exponents this follows by two inductions; inverses give the remaining signs. The representation

a = I + E_{12}

b = I + E_{23}

c = I + E_{13}

checks the identity and shows

c

has infinite order.

Schrödinger representation of the Heisenberg group

 0  0

L^{2} (R)

, translations and multiplication by phases give the Schrödinger representation of the Heisenberg group. The central coordinate acts by an overall phase.

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The Heisenberg group is a mathematical structure that arises in the context of group theory and analysis, particularly in the study of nilpotent Lie groups and geometric analysis. It is named after the physicist Werner Heisenberg, although its mathematical development is independent of his work in quantum mechanics. The Heisenberg group can be defined in various contexts, such as algebraically, geometrically, or analytically.

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