Elements of Finite Algebras¶

class sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra_element.FiniteDimensionalAlgebraElement[source]¶

Bases: AlgebraElement

Create an element of a FiniteDimensionalAlgebra using a multiplication table.

INPUT:

A – a FiniteDimensionalAlgebra which will be the parent
elt – vector, matrix or element of the base field (default: None)
check – boolean (default: True); if False and elt is a matrix, assume that it is known to be the matrix of an element

If elt is a vector or a matrix consisting of a single row, it is interpreted as a vector of coordinates with respect to the given basis of A. If elt is a square matrix, it is interpreted as a multiplication matrix with respect to this basis.

EXAMPLES:

Sage

sage: A = FiniteDimensionalAlgebra(GF(3), [Matrix([[1,0], [0,1]]),
....:                                      Matrix([[0,1], [0,0]])])
sage: A(17)
2*e0
sage: A([1,1])
e0 + e1

Python

>>> from sage.all import *
>>> A = FiniteDimensionalAlgebra(GF(Integer(3)), [Matrix([[Integer(1),Integer(0)], [Integer(0),Integer(1)]]),
...                                      Matrix([[Integer(0),Integer(1)], [Integer(0),Integer(0)]])])
>>> A(Integer(17))
2*e0
>>> A([Integer(1),Integer(1)])
e0 + e1

Sage Live

A = FiniteDimensionalAlgebra(GF(3), [Matrix([[1,0], [0,1]]),
                                     Matrix([[0,1], [0,0]])])
A(17)
A([1,1])

characteristic_polynomial()[source]¶

Return the characteristic polynomial of self.

Note

This function just returns the characteristic polynomial of the matrix of right multiplication by self. This may not be a very meaningful invariant if the algebra is not unitary and associative.

EXAMPLES:

Sage

sage: B = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0,0], [0,1,0], [0,0,0]]),
....:                                   Matrix([[0,1,0], [0,0,0], [0,0,0]]),
....:                                   Matrix([[0,0,0], [0,0,0], [0,0,1]])])
sage: B(0).characteristic_polynomial()                                      # needs sage.libs.pari
x^3
sage: b = B.random_element()
sage: f = b.characteristic_polynomial(); f  # random                        # needs sage.libs.pari
x^3 - 8*x^2 + 16*x
sage: f(b) == 0                                                             # needs sage.libs.pari
True

Python

>>> from sage.all import *
>>> B = FiniteDimensionalAlgebra(QQ, [Matrix([[Integer(1),Integer(0),Integer(0)], [Integer(0),Integer(1),Integer(0)], [Integer(0),Integer(0),Integer(0)]]),
...                                   Matrix([[Integer(0),Integer(1),Integer(0)], [Integer(0),Integer(0),Integer(0)], [Integer(0),Integer(0),Integer(0)]]),
...                                   Matrix([[Integer(0),Integer(0),Integer(0)], [Integer(0),Integer(0),Integer(0)], [Integer(0),Integer(0),Integer(1)]])])
>>> B(Integer(0)).characteristic_polynomial()                                      # needs sage.libs.pari
x^3
>>> b = B.random_element()
>>> f = b.characteristic_polynomial(); f  # random                        # needs sage.libs.pari
x^3 - 8*x^2 + 16*x
>>> f(b) == Integer(0)                                                             # needs sage.libs.pari
True

Sage Live

B = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0,0], [0,1,0], [0,0,0]]),
                                  Matrix([[0,1,0], [0,0,0], [0,0,0]]),
                                  Matrix([[0,0,0], [0,0,0], [0,0,1]])])
B(0).characteristic_polynomial()                                      # needs sage.libs.pari
b = B.random_element()
f = b.characteristic_polynomial(); f  # random                        # needs sage.libs.pari
f(b) == 0                                                             # needs sage.libs.pari

inverse()[source]¶

Return the two-sided multiplicative inverse of self, if it exists.

This assumes that the algebra to which self belongs is associative.

Note

If an element of a finite-dimensional unitary associative algebra over a field admits a left inverse, then this is the unique left inverse, and it is also a right inverse.

EXAMPLES:

Sage

sage: C = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0], [0,1]]),
....:                                   Matrix([[0,1], [-1,0]])])
sage: C([1,2]).inverse()
1/5*e0 - 2/5*e1

Python

>>> from sage.all import *
>>> C = FiniteDimensionalAlgebra(QQ, [Matrix([[Integer(1),Integer(0)], [Integer(0),Integer(1)]]),
...                                   Matrix([[Integer(0),Integer(1)], [-Integer(1),Integer(0)]])])
>>> C([Integer(1),Integer(2)]).inverse()
1/5*e0 - 2/5*e1

Sage Live

C = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0], [0,1]]),
                                  Matrix([[0,1], [-1,0]])])
C([1,2]).inverse()

is_invertible()[source]¶

Return True if self has a two-sided multiplicative inverse.

This assumes that the algebra to which self belongs is associative.

Note

If an element of a unitary finite-dimensional algebra over a field admits a left inverse, then this is the unique left inverse, and it is also a right inverse.

EXAMPLES:

Sage

sage: C = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0], [0,1]]),
....:                                   Matrix([[0,1], [-1,0]])])
sage: C([1,2]).is_invertible()
True
sage: C(0).is_invertible()
False

Python

>>> from sage.all import *
>>> C = FiniteDimensionalAlgebra(QQ, [Matrix([[Integer(1),Integer(0)], [Integer(0),Integer(1)]]),
...                                   Matrix([[Integer(0),Integer(1)], [-Integer(1),Integer(0)]])])
>>> C([Integer(1),Integer(2)]).is_invertible()
True
>>> C(Integer(0)).is_invertible()
False

Sage Live

C = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0], [0,1]]),
                                  Matrix([[0,1], [-1,0]])])
C([1,2]).is_invertible()
C(0).is_invertible()

is_nilpotent()[source]¶

Return True if self is nilpotent.

EXAMPLES:

Sage

sage: C = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0], [0,1]]),
....:                                   Matrix([[0,1], [0,0]])])
sage: C([1,0]).is_nilpotent()
False
sage: C([0,1]).is_nilpotent()
True

sage: A = FiniteDimensionalAlgebra(QQ, [Matrix([0])])
sage: A([1]).is_nilpotent()
True

Python

>>> from sage.all import *
>>> C = FiniteDimensionalAlgebra(QQ, [Matrix([[Integer(1),Integer(0)], [Integer(0),Integer(1)]]),
...                                   Matrix([[Integer(0),Integer(1)], [Integer(0),Integer(0)]])])
>>> C([Integer(1),Integer(0)]).is_nilpotent()
False
>>> C([Integer(0),Integer(1)]).is_nilpotent()
True

>>> A = FiniteDimensionalAlgebra(QQ, [Matrix([Integer(0)])])
>>> A([Integer(1)]).is_nilpotent()
True

Sage Live

C = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0], [0,1]]),
                                  Matrix([[0,1], [0,0]])])
C([1,0]).is_nilpotent()
C([0,1]).is_nilpotent()
A = FiniteDimensionalAlgebra(QQ, [Matrix([0])])
A([1]).is_nilpotent()

is_zerodivisor()[source]¶

Return True if self is a left or right zero-divisor.

EXAMPLES:

Sage

sage: C = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0], [0,1]]),
....:                                   Matrix([[0,1], [0,0]])])
sage: C([1,0]).is_zerodivisor()
False
sage: C([0,1]).is_zerodivisor()
True

Python

>>> from sage.all import *
>>> C = FiniteDimensionalAlgebra(QQ, [Matrix([[Integer(1),Integer(0)], [Integer(0),Integer(1)]]),
...                                   Matrix([[Integer(0),Integer(1)], [Integer(0),Integer(0)]])])
>>> C([Integer(1),Integer(0)]).is_zerodivisor()
False
>>> C([Integer(0),Integer(1)]).is_zerodivisor()
True

Sage Live

C = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0], [0,1]]),
                                  Matrix([[0,1], [0,0]])])
C([1,0]).is_zerodivisor()
C([0,1]).is_zerodivisor()

left_matrix()[source]¶

Return the matrix for multiplication by self from the left.

EXAMPLES:

Sage

sage: C = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0,0], [0,0,0], [0,0,0]]),
....:                                   Matrix([[0,1,0], [0,0,0], [0,0,0]]),
....:                                   Matrix([[0,0,0], [0,1,0], [0,0,1]])])
sage: C([1,2,0]).left_matrix()
[1 0 0]
[0 1 0]
[0 2 0]

Python

>>> from sage.all import *
>>> C = FiniteDimensionalAlgebra(QQ, [Matrix([[Integer(1),Integer(0),Integer(0)], [Integer(0),Integer(0),Integer(0)], [Integer(0),Integer(0),Integer(0)]]),
...                                   Matrix([[Integer(0),Integer(1),Integer(0)], [Integer(0),Integer(0),Integer(0)], [Integer(0),Integer(0),Integer(0)]]),
...                                   Matrix([[Integer(0),Integer(0),Integer(0)], [Integer(0),Integer(1),Integer(0)], [Integer(0),Integer(0),Integer(1)]])])
>>> C([Integer(1),Integer(2),Integer(0)]).left_matrix()
[1 0 0]
[0 1 0]
[0 2 0]

Sage Live

C = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0,0], [0,0,0], [0,0,0]]),
                                  Matrix([[0,1,0], [0,0,0], [0,0,0]]),
                                  Matrix([[0,0,0], [0,1,0], [0,0,1]])])
C([1,2,0]).left_matrix()

matrix()[source]¶

Return the matrix for multiplication by self from the right.

EXAMPLES:

Sage

sage: B = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0,0], [0,1,0], [0,0,0]]),
....:                                   Matrix([[0,1,0], [0,0,0], [0,0,0]]),
....:                                   Matrix([[0,0,0], [0,0,0], [0,0,1]])])
sage: B(5).matrix()
[5 0 0]
[0 5 0]
[0 0 5]

Python

>>> from sage.all import *
>>> B = FiniteDimensionalAlgebra(QQ, [Matrix([[Integer(1),Integer(0),Integer(0)], [Integer(0),Integer(1),Integer(0)], [Integer(0),Integer(0),Integer(0)]]),
...                                   Matrix([[Integer(0),Integer(1),Integer(0)], [Integer(0),Integer(0),Integer(0)], [Integer(0),Integer(0),Integer(0)]]),
...                                   Matrix([[Integer(0),Integer(0),Integer(0)], [Integer(0),Integer(0),Integer(0)], [Integer(0),Integer(0),Integer(1)]])])
>>> B(Integer(5)).matrix()
[5 0 0]
[0 5 0]
[0 0 5]

Sage Live

B = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0,0], [0,1,0], [0,0,0]]),
                                  Matrix([[0,1,0], [0,0,0], [0,0,0]]),
                                  Matrix([[0,0,0], [0,0,0], [0,0,1]])])
B(5).matrix()

minimal_polynomial()[source]¶

Return the minimal polynomial of self.

EXAMPLES:

Sage

sage: B = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0,0], [0,1,0], [0,0,0]]),
....:                                   Matrix([[0,1,0], [0,0,0], [0,0,0]]),
....:                                   Matrix([[0,0,0], [0,0,0], [0,0,1]])])
sage: B(0).minimal_polynomial()                                             # needs sage.libs.pari
x
sage: b = B.random_element()
sage: f = b.minimal_polynomial(); f  # random                               # needs sage.libs.pari
x^3 + 1/2*x^2 - 7/16*x + 1/16
sage: f(b) == 0                                                             # needs sage.libs.pari
True

Python

>>> from sage.all import *
>>> B = FiniteDimensionalAlgebra(QQ, [Matrix([[Integer(1),Integer(0),Integer(0)], [Integer(0),Integer(1),Integer(0)], [Integer(0),Integer(0),Integer(0)]]),
...                                   Matrix([[Integer(0),Integer(1),Integer(0)], [Integer(0),Integer(0),Integer(0)], [Integer(0),Integer(0),Integer(0)]]),
...                                   Matrix([[Integer(0),Integer(0),Integer(0)], [Integer(0),Integer(0),Integer(0)], [Integer(0),Integer(0),Integer(1)]])])
>>> B(Integer(0)).minimal_polynomial()                                             # needs sage.libs.pari
x
>>> b = B.random_element()
>>> f = b.minimal_polynomial(); f  # random                               # needs sage.libs.pari
x^3 + 1/2*x^2 - 7/16*x + 1/16
>>> f(b) == Integer(0)                                                             # needs sage.libs.pari
True

Sage Live

B = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0,0], [0,1,0], [0,0,0]]),
                                  Matrix([[0,1,0], [0,0,0], [0,0,0]]),
                                  Matrix([[0,0,0], [0,0,0], [0,0,1]])])
B(0).minimal_polynomial()                                             # needs sage.libs.pari
b = B.random_element()
f = b.minimal_polynomial(); f  # random                               # needs sage.libs.pari
f(b) == 0                                                             # needs sage.libs.pari

monomial_coefficients(copy=True)[source]¶

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

sage: B = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0], [0,1]]),
....:                                   Matrix([[0,1], [-1,0]])])
sage: elt = B(Matrix([[1,1], [-1,1]]))
sage: elt.monomial_coefficients()
{0: 1, 1: 1}

Python

>>> from sage.all import *
>>> B = FiniteDimensionalAlgebra(QQ, [Matrix([[Integer(1),Integer(0)], [Integer(0),Integer(1)]]),
...                                   Matrix([[Integer(0),Integer(1)], [-Integer(1),Integer(0)]])])
>>> elt = B(Matrix([[Integer(1),Integer(1)], [-Integer(1),Integer(1)]]))
>>> elt.monomial_coefficients()
{0: 1, 1: 1}

Sage Live

B = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0], [0,1]]),
                                  Matrix([[0,1], [-1,0]])])
elt = B(Matrix([[1,1], [-1,1]]))
elt.monomial_coefficients()

vector()[source]¶

Return self as a vector.

EXAMPLES:

Sage

sage: B = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0,0], [0,1,0], [0,0,0]]),
....:                                   Matrix([[0,1,0], [0,0,0], [0,0,0]]),
....:                                   Matrix([[0,0,0], [0,0,0], [0,0,1]])])
sage: B(5).vector()
(5, 0, 5)

Python

>>> from sage.all import *
>>> B = FiniteDimensionalAlgebra(QQ, [Matrix([[Integer(1),Integer(0),Integer(0)], [Integer(0),Integer(1),Integer(0)], [Integer(0),Integer(0),Integer(0)]]),
...                                   Matrix([[Integer(0),Integer(1),Integer(0)], [Integer(0),Integer(0),Integer(0)], [Integer(0),Integer(0),Integer(0)]]),
...                                   Matrix([[Integer(0),Integer(0),Integer(0)], [Integer(0),Integer(0),Integer(0)], [Integer(0),Integer(0),Integer(1)]])])
>>> B(Integer(5)).vector()
(5, 0, 5)

Sage Live

B = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0,0], [0,1,0], [0,0,0]]),
                                  Matrix([[0,1,0], [0,0,0], [0,0,0]]),
                                  Matrix([[0,0,0], [0,0,0], [0,0,1]])])
B(5).vector()

sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra_element.unpickle_FiniteDimensionalAlgebraElement(A, vec, mat)[source]¶: Helper for unpickling of finite dimensional algebra elements.