Heisenberg Algebras

AUTHORS:

• Travis Scrimshaw (2013-08-13): Initial version

class sage.algebras.lie_algebras.heisenberg.HeisenbergAlgebra(R, n)

Bases: HeisenbergAlgebra_fd, HeisenbergAlgebra_abstract, LieAlgebraWithGenerators

A Heisenberg algebra defined using structure coefficients.

The \(n\)-th Heisenberg algebra (where \(n\) is a nonnegative integer or infinity) is the Lie algebra with basis \(\{p_i\}_{1 \leq i \leq n} \cup \{q_i\}_{1 \leq i \leq n} \cup \{z\}\) with the following relations:

\[[p_i, q_j] = \delta_{ij} z, \quad [p_i, z] = [q_i, z] = [p_i, p_j] = [q_i, q_j] = 0.\]

This Lie algebra is also known as the Heisenberg algebra of rank \(n\).

Note

The relations \([p_i, q_j] = \delta_{ij} z\), \([p_i, z] = 0\), and \([q_i, z] = 0\) are known as canonical commutation relations. See Wikipedia article Canonical_commutation_relations.

Warning

The \(n\) in the above definition is called the “rank” of the Heisenberg algebra; it is not, however, a rank in any of the usual meanings that this word has in the theory of Lie algebras.

INPUT:

• R – the base ring

• n – the rank of the Heisenberg algebra

REFERENCES:

• Wikipedia article Heisenberg_algebra

EXAMPLES:

sage: L = lie_algebras.Heisenberg(QQ, 2)

class sage.algebras.lie_algebras.heisenberg.HeisenbergAlgebra_abstract(I)

Bases: IndexedGenerators

The common methods for the (non-matrix) Heisenberg algebras.

class Element

Bases: LieAlgebraElement

bracket_on_basis(x, y)

Return the bracket of basis elements indexed by x and y where x < y.

The basis of a Heisenberg algebra is ordered in such a way that the \(p_i\) come first, the \(q_i\) come next, and the \(z\) comes last.

EXAMPLES:

sage: H = lie_algebras.Heisenberg(QQ, 3)
sage: p1 = ('p', 1)
sage: q1 = ('q', 1)
sage: H.bracket_on_basis(p1, q1)
z

p(i)

The generator \(p_i\) of the Heisenberg algebra.

EXAMPLES:

sage: L = lie_algebras.Heisenberg(QQ, oo)
sage: L.p(2)
p2

q(i)

The generator \(q_i\) of the Heisenberg algebra.

EXAMPLES:

sage: L = lie_algebras.Heisenberg(QQ, oo)
sage: L.q(2)
q2

step()

Return the nilpotency step of self.

EXAMPLES:

sage: h = lie_algebras.Heisenberg(ZZ, 10)
sage: h.step()
2

sage: h = lie_algebras.Heisenberg(ZZ, oo)
sage: h.step()
2

z()

Return the basis element \(z\) of the Heisenberg algebra.

The element \(z\) spans the center of the Heisenberg algebra.

EXAMPLES:

sage: L = lie_algebras.Heisenberg(QQ, oo)
sage: L.z()
z

class sage.algebras.lie_algebras.heisenberg.HeisenbergAlgebra_fd(n)

Bases: object

Common methods for finite-dimensional Heisenberg algebras.

basis()

Return the basis of self.

EXAMPLES:

sage: H = lie_algebras.Heisenberg(QQ, 1)
sage: H.basis()
Finite family {'p1': p1, 'q1': q1, 'z': z}

gen(i)

Return the i-th generator of self.

EXAMPLES:

sage: H = lie_algebras.Heisenberg(QQ, 2)
sage: H.gen(0)
p1
sage: H.gen(3)
q2

gens()

Return the Lie algebra generators of self.

EXAMPLES:

sage: H = lie_algebras.Heisenberg(QQ, 2)
sage: H.gens()
(p1, p2, q1, q2)
sage: H = lie_algebras.Heisenberg(QQ, 0)
sage: H.gens()
(z,)

lie_algebra_generators()

Return the Lie algebra generators of self.

EXAMPLES:

sage: H = lie_algebras.Heisenberg(QQ, 1)
sage: H.lie_algebra_generators()
Finite family {'p1': p1, 'q1': q1}
sage: H = lie_algebras.Heisenberg(QQ, 0)
sage: H.lie_algebra_generators()
Finite family {'z': z}

n()

Return the rank of the Heisenberg algebra self.

This is the n such that self is the \(n\)-th Heisenberg algebra. The dimension of this Heisenberg algebra is then \(2n + 1\).

EXAMPLES:

sage: H = lie_algebras.Heisenberg(QQ, 3)
sage: H.n()
3
sage: H = lie_algebras.Heisenberg(QQ, 3, representation="matrix")
sage: H.n()
3

class sage.algebras.lie_algebras.heisenberg.HeisenbergAlgebra_matrix(R, n)

Bases: HeisenbergAlgebra_fd, LieAlgebraFromAssociative

A Heisenberg algebra represented using matrices.

The \(n\)-th Heisenberg algebra over \(R\) is a Lie algebra which is defined as the Lie algebra of the \((n+2) \times (n+2)\)-matrices:

\[\begin{split}\begin{bmatrix} 0 & p^T & k \\ 0 & 0_n & q \\ 0 & 0 & 0 \end{bmatrix}\end{split}\]

where \(p, q \in R^n\) and \(0_n\) in the \(n \times n\) zero matrix. It has a basis consisting of

\[\begin{split}\begin{aligned} p_i & = \begin{bmatrix} 0 & e_i^T & 0 \\ 0 & 0_n & 0 \\ 0 & 0 & 0 \end{bmatrix} \qquad \text{for } 1 \leq i \leq n , \\ q_i & = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0_n & e_i \\ 0 & 0 & 0 \end{bmatrix} \qquad \text{for } 1 \leq i \leq n , \\ z & = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 0_n & 0 \\ 0 & 0 & 0 \end{bmatrix}, \end{aligned}\end{split}\]

where \(\{e_i\}\) is the standard basis of \(R^n\). In other words, it has the basis \((p_1, p_2, \ldots, p_n, q_1, q_2, \ldots, q_n, z)\), where \(p_i = E_{1, i+1}\), \(q_i = E_{i+1, n+2}\) and \(z = E_{1, n+2}\) are elementary matrices.

This Lie algebra is isomorphic to the \(n\)-th Heisenberg algebra constructed in HeisenbergAlgebra; the bases correspond to each other.

INPUT:

• R – the base ring

• n – the nonnegative integer \(n\)

EXAMPLES:

sage: L = lie_algebras.Heisenberg(QQ, 1, representation="matrix")
sage: p = L.p(1)
sage: q = L.q(1)
sage: z = L.bracket(p, q); z
[0 0 1]
[0 0 0]
[0 0 0]
sage: z == L.z()
True
sage: L.dimension()
3

sage: L = lie_algebras.Heisenberg(QQ, 2, representation="matrix")
sage: sorted(dict(L.basis()).items())
[(
      [0 1 0 0]
      [0 0 0 0]
      [0 0 0 0]
'p1', [0 0 0 0]
),
 (
      [0 0 1 0]
      [0 0 0 0]
      [0 0 0 0]
'p2', [0 0 0 0]
),
 (
      [0 0 0 0]
      [0 0 0 1]
      [0 0 0 0]
'q1', [0 0 0 0]
),
 (
      [0 0 0 0]
      [0 0 0 0]
      [0 0 0 1]
'q2', [0 0 0 0]
),
 (
     [0 0 0 1]
     [0 0 0 0]
     [0 0 0 0]
'z', [0 0 0 0]
)]

sage: L = lie_algebras.Heisenberg(QQ, 0, representation="matrix")
sage: sorted(dict(L.basis()).items())
[(
     [0 1]
'z', [0 0]
)]
sage: L.gens()
(
[0 1]
[0 0]
)
sage: L.lie_algebra_generators()
Finite family {'z': [0 1]
[0 0]}

class Element

Bases: LieAlgebraMatrixWrapper, Element

monomial_coefficients(copy=True)

Return a dictionary whose keys are indices of basis elements in the support of self and whose values are the corresponding coefficients.

INPUT:

• copy – ignored

EXAMPLES:

sage: L = lie_algebras.Heisenberg(QQ, 3, representation="matrix")
sage: elt = L(Matrix(QQ, [[0, 1, 3, 0, 3], [0, 0, 0, 0, 0], [0, 0, 0, 0, -3],
....:                     [0, 0, 0, 0, 7], [0, 0, 0, 0, 0]]))
sage: elt
[ 0  1  3  0  3]
[ 0  0  0  0  0]
[ 0  0  0  0 -3]
[ 0  0  0  0  7]
[ 0  0  0  0  0]
sage: sorted(elt.monomial_coefficients().items())
[('p1', 1), ('p2', 3), ('q2', -3), ('q3', 7), ('z', 3)]

p(i)

Return the generator \(p_i\) of the Heisenberg algebra.

EXAMPLES:

sage: L = lie_algebras.Heisenberg(QQ, 1, representation="matrix")
sage: L.p(1)
[0 1 0]
[0 0 0]
[0 0 0]

q(i)

Return the generator \(q_i\) of the Heisenberg algebra.

EXAMPLES:

sage: L = lie_algebras.Heisenberg(QQ, 1, representation="matrix")
sage: L.q(1)
[0 0 0]
[0 0 1]
[0 0 0]

step()

Return the nilpotency step of self.

EXAMPLES:

sage: h = lie_algebras.Heisenberg(ZZ, 2, representation="matrix")
sage: h.step()
2

z()

Return the basis element \(z\) of the Heisenberg algebra.

The element \(z\) spans the center of the Heisenberg algebra.

EXAMPLES:

sage: L = lie_algebras.Heisenberg(QQ, 1, representation="matrix")
sage: L.z()
[0 0 1]
[0 0 0]
[0 0 0]

class sage.algebras.lie_algebras.heisenberg.InfiniteHeisenbergAlgebra(R)

Bases: HeisenbergAlgebra_abstract, LieAlgebraWithGenerators

The infinite Heisenberg algebra.

This is the Heisenberg algebra on an infinite number of generators. In other words, this is the Heisenberg algebra of rank \(\infty\). See HeisenbergAlgebra for more information.

basis()

Return the basis of self.

EXAMPLES:

sage: L = lie_algebras.Heisenberg(QQ, oo)
sage: L.basis()
Lazy family (basis map(i))_{i in Disjoint union of Family ({'z'},
 The Cartesian product of (Positive integers, {'p', 'q'}))}
sage: L.basis()['z']
z
sage: L.basis()[(12, 'p')]
p12

lie_algebra_generators()

Return the generators of self as a Lie algebra.

EXAMPLES:

sage: L = lie_algebras.Heisenberg(QQ, oo)
sage: L.lie_algebra_generators()
Lazy family (generator map(i))_{i in The Cartesian product of
                                (Positive integers, {'p', 'q'})}