Elements of Finite Algebras#

class sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra_element.FiniteDimensionalAlgebraElement#

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: 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

characteristic_polynomial()#

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: 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

inverse()#

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

is_invertible()#

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: 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

is_nilpotent()#

Return True if self is nilpotent.

EXAMPLES:

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

is_zerodivisor()#

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

EXAMPLES:

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

left_matrix()#

Return the matrix for multiplication by self from the left.

EXAMPLES:

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]

matrix()#

Return the matrix for multiplication by self from the right.

EXAMPLES:

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]

minimal_polynomial()#

Return the minimal polynomial of self.

EXAMPLES:

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

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: 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}

vector()#

Return self as a vector.

EXAMPLES:

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)

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