# Quantum Oscillator Algebras#

AUTHORS:

• Travis Scrimshaw (2023-12): initial version

class sage.algebras.quantum_oscillator.FockSpaceRepresentation(oscillator_algebra)[source]#

The unique Fock space representation of the QuantumOscillatorAlgebra.

class Element[source]#
some_elements()[source]#

Return some elements of self.

EXAMPLES:

sage: O = algebras.QuantumOscillator()
sage: F = O.fock_space_representation()
sage: F.some_elements()
(|0>, |1>, |52>, |0> + 2*|1> + 3*|2> + |42>)

>>> from sage.all import *
>>> O = algebras.QuantumOscillator()
>>> F = O.fock_space_representation()
>>> F.some_elements()
(|0>, |1>, |52>, |0> + 2*|1> + 3*|2> + |42>)

vacuum()[source]#

Return the vacuum element $$|0\rangle$$ of self.

EXAMPLES:

sage: O = algebras.QuantumOscillator()
sage: F = O.fock_space_representation()
sage: F.vacuum()
|0>

>>> from sage.all import *
>>> O = algebras.QuantumOscillator()
>>> F = O.fock_space_representation()
>>> F.vacuum()
|0>

class sage.algebras.quantum_oscillator.QuantumOscillatorAlgebra(q, R)[source]#

The quantum oscillator algebra.

Let $$R$$ be a commutative algebra and $$q \in R$$ be a unit. The quantum oscillator algebra, or $$q$$-oscillator algebra, is the unital associative $$R$$-algebra with generators $$a^+$$, $$a^-$$ and $$k^{\pm 1}$$ satisfying the following relations:

$k a^{\pm} = q^{\pm 1} a^{\pm} k, \qquad a^- a^+ = 1 - q^2 k^2, \qquad a^+ a^- = 1 - k^2.$

INPUT:

• q – (optional) the parameter $$q$$

• R – (default: $$\QQ(q)$$) the base ring that contains q

EXAMPLES:

We construct the algebra and perform some basic computations:

sage: O = algebras.QuantumOscillator()
sage: ap, am, k, ki = O.algebra_generators()
sage: q = O.q()
sage: k^-3 * ap * ki * am^2 * k - q^3 * ap * k^3
q^5*a-*k^-3 - q^3*a-*k^-1 - q^3*a+*k^3

>>> from sage.all import *
>>> O = algebras.QuantumOscillator()
>>> ap, am, k, ki = O.algebra_generators()
>>> q = O.q()
>>> k**-Integer(3) * ap * ki * am**Integer(2) * k - q**Integer(3) * ap * k**Integer(3)
q^5*a-*k^-3 - q^3*a-*k^-1 - q^3*a+*k^3


We construct representations of the type $$A_1$$ quantum coordinate ring using the quantum oscillator algebra and verify the quantum determinant:

sage: pi = matrix([[am, k], [-q*k, ap]]); pi
[  a-    k]
[-q*k   a+]
sage: pi[0,0] * pi[1,1] - q * pi[0,1] * pi[1,0]
1

>>> from sage.all import *
>>> pi = matrix([[am, k], [-q*k, ap]]); pi
[  a-    k]
[-q*k   a+]
>>> pi[Integer(0),Integer(0)] * pi[Integer(1),Integer(1)] - q * pi[Integer(0),Integer(1)] * pi[Integer(1),Integer(0)]
1


Next, we use this to build representations for type $$A_2$$:

sage: def quantum_det(M):
....:     n = M.nrows()
....:     return sum((-q)**sigma.length()
....:                * prod(M[i,sigma[i]-1] for i in range(n))
....:                for sigma in Permutations(n))
sage: def build_repr(wd, gens):
....:     n = gens[0].nrows()
....:     ret = gens[wd[0]-1]
....:     for ind in wd[1:]:
....:         g = gens[ind-1]
....:         temp = [[None]*n for _ in range(n)]
....:         for i in range(n):
....:             for j in range(n):
....:                 temp[i][j] = sum(tensor([ret[i,k], g[k,j]]) for k in range(n))
....:         ret = matrix(temp)
....:     return ret
sage: pi1 = matrix.block_diagonal(pi, matrix.identity(1)); pi1
[  a-    k|   0]
[-q*k   a+|   0]
[---------+----]
[   0    0|   1]
sage: pi2 = matrix.block_diagonal(matrix.identity(1), pi); pi2
[   1|   0    0]
[----+---------]
[   0|  a-    k]
[   0|-q*k   a+]
sage: quantum_det(pi1) == 1
True
sage: quantum_det(pi2) == 1
True
sage: pi12 = build_repr([1,2], [pi1, pi2]); pi12
[  a- # 1   k # a-    k # k]
[-q*k # 1  a+ # a-   a+ # k]
[       0 -q*1 # k   1 # a+]
sage: quantum_det(pi12)
1 # 1
sage: pi121 = build_repr([1,2,1], [pi1, pi2]); pi121
[   a- # 1 # a- - q*k # a- # k      a- # 1 # k + k # a- # a+                     k # k # 1]
[-q*k # 1 # a- - q*a+ # a- # k   -q*k # 1 # k + a+ # a- # a+                    a+ # k # 1]
[                q^2*1 # k # k                 -q*1 # k # a+                    1 # a+ # 1]
sage: quantum_det(pi121)
1 # 1 # 1
sage: pi212 = build_repr([2,1,2], [pi1, pi2]); pi212
[                   1 # a- # 1                    1 # k # a-                     1 # k # k]
[                -q*a- # k # 1    a- # a+ # a- - q*k # 1 # k      a- # a+ # k + k # 1 # a+]
[                q^2*k # k # 1 -q*k # a+ # a- - q*a+ # 1 # k   -q*k # a+ # k + a+ # 1 # a+]
sage: quantum_det(pi212)
1 # 1 # 1

>>> from sage.all import *
>>> def quantum_det(M):
...     n = M.nrows()
...     return sum((-q)**sigma.length()
...                * prod(M[i,sigma[i]-Integer(1)] for i in range(n))
...                for sigma in Permutations(n))
>>> def build_repr(wd, gens):
...     n = gens[Integer(0)].nrows()
...     ret = gens[wd[Integer(0)]-Integer(1)]
...     for ind in wd[Integer(1):]:
...         g = gens[ind-Integer(1)]
...         temp = [[None]*n for _ in range(n)]
...         for i in range(n):
...             for j in range(n):
...                 temp[i][j] = sum(tensor([ret[i,k], g[k,j]]) for k in range(n))
...         ret = matrix(temp)
...     return ret
>>> pi1 = matrix.block_diagonal(pi, matrix.identity(Integer(1))); pi1
[  a-    k|   0]
[-q*k   a+|   0]
[---------+----]
[   0    0|   1]
>>> pi2 = matrix.block_diagonal(matrix.identity(Integer(1)), pi); pi2
[   1|   0    0]
[----+---------]
[   0|  a-    k]
[   0|-q*k   a+]
>>> quantum_det(pi1) == Integer(1)
True
>>> quantum_det(pi2) == Integer(1)
True
>>> pi12 = build_repr([Integer(1),Integer(2)], [pi1, pi2]); pi12
[  a- # 1   k # a-    k # k]
[-q*k # 1  a+ # a-   a+ # k]
[       0 -q*1 # k   1 # a+]
>>> quantum_det(pi12)
1 # 1
>>> pi121 = build_repr([Integer(1),Integer(2),Integer(1)], [pi1, pi2]); pi121
[   a- # 1 # a- - q*k # a- # k      a- # 1 # k + k # a- # a+                     k # k # 1]
[-q*k # 1 # a- - q*a+ # a- # k   -q*k # 1 # k + a+ # a- # a+                    a+ # k # 1]
[                q^2*1 # k # k                 -q*1 # k # a+                    1 # a+ # 1]
>>> quantum_det(pi121)
1 # 1 # 1
>>> pi212 = build_repr([Integer(2),Integer(1),Integer(2)], [pi1, pi2]); pi212
[                   1 # a- # 1                    1 # k # a-                     1 # k # k]
[                -q*a- # k # 1    a- # a+ # a- - q*k # 1 # k      a- # a+ # k + k # 1 # a+]
[                q^2*k # k # 1 -q*k # a+ # a- - q*a+ # 1 # k   -q*k # a+ # k + a+ # 1 # a+]
>>> quantum_det(pi212)
1 # 1 # 1


REFERENCES:

class Element[source]#
algebra_generators()[source]#

Return the algebra generators of self.

EXAMPLES:

sage: O = algebras.QuantumOscillator()
sage: O.algebra_generators()
Finite family {'am': a-, 'ap': a+, 'k': k, 'ki': k^-1}

>>> from sage.all import *
>>> O = algebras.QuantumOscillator()
>>> O.algebra_generators()
Finite family {'am': a-, 'ap': a+, 'k': k, 'ki': k^-1}

fock_space_representation()[source]#

Return the Fock space representation of self.

EXAMPLES:

sage: O = algebras.QuantumOscillator()
sage: O.fock_space_representation()
Fock space representation of Quantum oscillator algebra with q=q
over Fraction Field of Univariate Polynomial Ring in q over Integer Ring

>>> from sage.all import *
>>> O = algebras.QuantumOscillator()
>>> O.fock_space_representation()
Fock space representation of Quantum oscillator algebra with q=q
over Fraction Field of Univariate Polynomial Ring in q over Integer Ring

gens()[source]#

Return the generators of self.

EXAMPLES:

sage: O = algebras.QuantumOscillator()
sage: O.gens()
(a+, a-, k, k^-1)

>>> from sage.all import *
>>> O = algebras.QuantumOscillator()
>>> O.gens()
(a+, a-, k, k^-1)

one_basis()[source]#

Return the index of the basis element of $$1$$.

EXAMPLES:

sage: O = algebras.QuantumOscillator()
sage: O.one_basis()
(0, 0)

>>> from sage.all import *
>>> O = algebras.QuantumOscillator()
>>> O.one_basis()
(0, 0)

product_on_basis(ml, mr)[source]#

Return the product of the basis elements indexed by ml and mr.

EXAMPLES:

sage: O = algebras.QuantumOscillator()
sage: ap, am, k, ki = O.algebra_generators()
sage: O.product_on_basis((-2, 3), (-4, 5))
1/q^12*a-^6*k^8
sage: O.product_on_basis((2, 3), (4, -5))
q^12*a+^6*k^-2
sage: O.product_on_basis((2, 3), (0, -3))
a+^2
sage: k^5 * ki^10
k^-5
sage: k^10 * ki^5
k^5
sage: ap^3 * k^5
a+^3*k^5
sage: am^3 * k^5
a-^3*k^5
sage: k^5 * ap^3
q^15*a+^3*k^5
sage: k^5 * am^3
1/q^15*a-^3*k^5
sage: ki^5 * ap^3
1/q^15*a+^3*k^-5
sage: ki^5 * am^3
q^15*a-^3*k^-5
sage: ap * am
1 - k^2
sage: am * ap
1 - q^2*k^2

sage: (ap + am + k + ki)^2
a-^2 + (q+1)*a-*k^-1 + ((q+1)/q)*a-*k + k^-2 + 4 - q^2*k^2
+ ((q+1)/q)*a+*k^-1 + (q+1)*a+*k + a+^2

sage: (ap)^3 * (am)^5
a-^2 + ((-q^4-q^2-1)/q^8)*a-^2*k^2 + ((q^4+q^2+1)/q^14)*a-^2*k^4 - 1/q^18*a-^2*k^6
sage: (ap)^5 * (am)^3
a+^2 + ((-q^4-q^2-1)/q^4)*a+^2*k^2 + ((q^4+q^2+1)/q^6)*a+^2*k^4 - 1/q^6*a+^2*k^6
sage: (am)^3 * (ap)^5
a+^2 + (-q^10-q^8-q^6)*a+^2*k^2 + (q^18+q^16+q^14)*a+^2*k^4 - q^24*a+^2*k^6
sage: (am)^5 * (ap)^3
a-^2 + (-q^6-q^4-q^2)*a-^2*k^2 + (q^10+q^8+q^6)*a-^2*k^4 - q^12*a-^2*k^6

>>> from sage.all import *
>>> O = algebras.QuantumOscillator()
>>> ap, am, k, ki = O.algebra_generators()
>>> O.product_on_basis((-Integer(2), Integer(3)), (-Integer(4), Integer(5)))
1/q^12*a-^6*k^8
>>> O.product_on_basis((Integer(2), Integer(3)), (Integer(4), -Integer(5)))
q^12*a+^6*k^-2
>>> O.product_on_basis((Integer(2), Integer(3)), (Integer(0), -Integer(3)))
a+^2
>>> k**Integer(5) * ki**Integer(10)
k^-5
>>> k**Integer(10) * ki**Integer(5)
k^5
>>> ap**Integer(3) * k**Integer(5)
a+^3*k^5
>>> am**Integer(3) * k**Integer(5)
a-^3*k^5
>>> k**Integer(5) * ap**Integer(3)
q^15*a+^3*k^5
>>> k**Integer(5) * am**Integer(3)
1/q^15*a-^3*k^5
>>> ki**Integer(5) * ap**Integer(3)
1/q^15*a+^3*k^-5
>>> ki**Integer(5) * am**Integer(3)
q^15*a-^3*k^-5
>>> ap * am
1 - k^2
>>> am * ap
1 - q^2*k^2

>>> (ap + am + k + ki)**Integer(2)
a-^2 + (q+1)*a-*k^-1 + ((q+1)/q)*a-*k + k^-2 + 4 - q^2*k^2
+ ((q+1)/q)*a+*k^-1 + (q+1)*a+*k + a+^2

>>> (ap)**Integer(3) * (am)**Integer(5)
a-^2 + ((-q^4-q^2-1)/q^8)*a-^2*k^2 + ((q^4+q^2+1)/q^14)*a-^2*k^4 - 1/q^18*a-^2*k^6
>>> (ap)**Integer(5) * (am)**Integer(3)
a+^2 + ((-q^4-q^2-1)/q^4)*a+^2*k^2 + ((q^4+q^2+1)/q^6)*a+^2*k^4 - 1/q^6*a+^2*k^6
>>> (am)**Integer(3) * (ap)**Integer(5)
a+^2 + (-q^10-q^8-q^6)*a+^2*k^2 + (q^18+q^16+q^14)*a+^2*k^4 - q^24*a+^2*k^6
>>> (am)**Integer(5) * (ap)**Integer(3)
a-^2 + (-q^6-q^4-q^2)*a-^2*k^2 + (q^10+q^8+q^6)*a-^2*k^4 - q^12*a-^2*k^6

q()[source]#

Return the $$q$$ of self.

EXAMPLES:

sage: O = algebras.QuantumOscillator()
sage: O.q()
q
sage: O = algebras.QuantumOscillator(q=QQ(-5))
sage: O.q()
-5

>>> from sage.all import *
>>> O = algebras.QuantumOscillator()
>>> O.q()
q
>>> O = algebras.QuantumOscillator(q=QQ(-Integer(5)))
>>> O.q()
-5

some_elements()[source]#

Return some elements of self.

EXAMPLES:

sage: O = algebras.QuantumOscillator()
sage: O.some_elements()
(a+, a-, k, k^-1, 1, a+^3, a-^4, k^2, k^-5, a+*k,
a-^4*k^-3, 1 + 3*k + 2*a+ + a+*k)

>>> from sage.all import *
>>> O = algebras.QuantumOscillator()
>>> O.some_elements()
(a+, a-, k, k^-1, 1, a+^3, a-^4, k^2, k^-5, a+*k,
a-^4*k^-3, 1 + 3*k + 2*a+ + a+*k)