Finite-Dimensional Algebras

class sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra.FiniteDimensionalAlgebra(k, table, names='e', assume_associative=False, category=None)

Bases: sage.rings.ring.Algebra

Create a finite-dimensional \(k\)-algebra from a multiplication table.

INPUT:

  • k – a field
  • table – a list of matrices
  • names – (default: 'e') string; names for the basis elements
  • assume_associative – (default: False) boolean; if True, then methods requiring associativity assume this without checking
  • category – (default: FiniteDimensionalAlgebrasWithBasis(k)) the category to which this algebra belongs

The list table must have the following form: there exists a finite-dimensional \(k\)-algebra of degree \(n\) with basis \((e_1, \ldots, e_n)\) such that the \(i\)-th element of table is the matrix of right multiplication by \(e_i\) with respect to the basis \((e_1, \ldots, e_n)\).

EXAMPLES:

sage: A = FiniteDimensionalAlgebra(GF(3), [Matrix([[1, 0], [0, 1]]), Matrix([[0, 1], [0, 0]])])
sage: A
Finite-dimensional algebra of degree 2 over Finite Field of size 3

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
Finite-dimensional algebra of degree 3 over Rational Field
Element

alias of FiniteDimensionalAlgebraElement

base_extend(F)

Return self base changed to F.

EXAMPLES:

sage: C = FiniteDimensionalAlgebra(GF(2), [Matrix([1])])
sage: k.<y> = GF(4)
sage: C.base_extend(k)
Finite-dimensional algebra of degree 1 over Finite Field in y of size 2^2
basis()

Return a list of the basis elements of self.

EXAMPLES:

sage: A = FiniteDimensionalAlgebra(GF(3), [Matrix([[1, 0], [0, 1]]), Matrix([[0, 1], [0, 0]])])
sage: A.basis()
[e0, e1]
cardinality()

Return the cardinality of self.

EXAMPLES:

sage: A = FiniteDimensionalAlgebra(GF(7), [Matrix([[1, 0], [0, 1]]), Matrix([[0, 1], [2, 3]])])
sage: A.cardinality()
49

sage: B = FiniteDimensionalAlgebra(RR, [Matrix([[1, 0], [0, 1]]), Matrix([[0, 1], [2, 3]])])
sage: B.cardinality()
+Infinity

sage: C = FiniteDimensionalAlgebra(RR, [])
sage: C.cardinality()
1
degree()

Return the number of generators of self, i.e., the degree of self over its base field.

EXAMPLES:

sage: A = FiniteDimensionalAlgebra(GF(3), [Matrix([[1, 0], [0, 1]]), Matrix([[0, 1], [0, 0]])])
sage: A.ngens()
2
from_base_ring(x)

TESTS:

sage: A = FiniteDimensionalAlgebra(QQ, [Matrix([0])])
sage: a = A(0)
sage: a.parent()
Finite-dimensional algebra of degree 1 over Rational Field
sage: A(1)
Traceback (most recent call last):
...
TypeError: algebra is not unitary

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(17)
17*e0 + 17*e2
gen(i)

Return the \(i\)-th basis element of self.

EXAMPLES:

sage: A = FiniteDimensionalAlgebra(GF(3), [Matrix([[1, 0], [0, 1]]), Matrix([[0, 1], [0, 0]])])
sage: A.gen(0)
e0
ideal(gens=None, given_by_matrix=False)

Return the right ideal of self generated by gens.

INPUT:

  • A – a FiniteDimensionalAlgebra
  • gens – (default: None) - either an element of A or a list of elements of A, given as vectors, matrices, or FiniteDimensionalAlgebraElements. If given_by_matrix is True, then gens should instead be a matrix whose rows form a basis of an ideal of A.
  • given_by_matrix – boolean (default: False) - if True, no checking is done

EXAMPLES:

sage: A = FiniteDimensionalAlgebra(GF(3), [Matrix([[1, 0], [0, 1]]), Matrix([[0, 1], [0, 0]])])
sage: A.ideal(A([1,1]))
Ideal (e0 + e1) of Finite-dimensional algebra of degree 2 over Finite Field of size 3
is_associative()

Return True if self is associative.

EXAMPLES:

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

sage: B = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0,0], [0,1,0], [0,0,1]]), Matrix([[0,1,0], [0,0,0], [0,0,0]]), Matrix([[0,0,1], [0,0,0], [1,0,0]])])
sage: B.is_associative()
False

sage: e = B.basis()
sage: (e[1]*e[2])*e[2]==e[1]*(e[2]*e[2])
False
is_commutative()

Return True if self is commutative.

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.is_commutative()
True

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.is_commutative()
False
is_finite()

Return True if the cardinality of self is finite.

EXAMPLES:

sage: A = FiniteDimensionalAlgebra(GF(7), [Matrix([[1, 0], [0, 1]]), Matrix([[0, 1], [2, 3]])])
sage: A.is_finite()
True

sage: B = FiniteDimensionalAlgebra(RR, [Matrix([[1, 0], [0, 1]]), Matrix([[0, 1], [2, 3]])])
sage: B.is_finite()
False

sage: C = FiniteDimensionalAlgebra(RR, [])
sage: C.is_finite()
True
is_unitary()

Return True if self has a two-sided multiplicative identity element.

EXAMPLES:

sage: A = FiniteDimensionalAlgebra(QQ, [])
sage: A.is_unitary()
True

sage: B = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0], [0,1]]), Matrix([[0,1], [-1,0]])])
sage: B.is_unitary()
True

sage: C = FiniteDimensionalAlgebra(QQ, [Matrix([[0,0], [0,0]]), Matrix([[0,0], [0,0]])])
sage: C.is_unitary()
False

sage: D = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0], [0,1]]), Matrix([[1,0], [0,1]])])
sage: D.is_unitary()
False

Note

If a finite-dimensional algebra over a field admits a left identity, then this is the unique left identity, and it is also a right identity.

is_zero()

Return True if self is the zero ring.

EXAMPLES:

sage: A = FiniteDimensionalAlgebra(QQ, [])
sage: A.is_zero()
True

sage: B = FiniteDimensionalAlgebra(GF(7), [Matrix([0])])
sage: B.is_zero()
False
left_table()

Return the list of matrices for left multiplication by the basis elements.

EXAMPLES:

sage: B = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0], [0,1]]), Matrix([[0,1],[-1,0]])])
sage: B.left_table()
[
[1 0]  [ 0  1]
[0 1], [-1  0]
]
maximal_ideal()

Compute the maximal ideal of the local algebra self.

Note

self must be unitary, commutative, associative and local (have a unique maximal ideal).

OUTPUT:

EXAMPLES:

sage: A = FiniteDimensionalAlgebra(GF(3), [Matrix([[1, 0], [0, 1]]), Matrix([[0, 1], [0, 0]])])
sage: A.maximal_ideal()
Ideal (0, e1) of Finite-dimensional algebra of degree 2 over Finite Field of size 3

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.maximal_ideal()
Traceback (most recent call last):
...
ValueError: algebra is not local
maximal_ideals()

Return a list consisting of all maximal ideals of self.

EXAMPLES:

sage: A = FiniteDimensionalAlgebra(GF(3), [Matrix([[1, 0], [0, 1]]), Matrix([[0, 1], [0, 0]])])
sage: A.maximal_ideals()
[Ideal (e1) of Finite-dimensional algebra of degree 2 over Finite Field of size 3]

sage: B = FiniteDimensionalAlgebra(QQ, [])
sage: B.maximal_ideals()
[]
ngens()

Return the number of generators of self, i.e., the degree of self over its base field.

EXAMPLES:

sage: A = FiniteDimensionalAlgebra(GF(3), [Matrix([[1, 0], [0, 1]]), Matrix([[0, 1], [0, 0]])])
sage: A.ngens()
2
one()

Return the multiplicative identity element of self, if it exists.

EXAMPLES:

sage: A = FiniteDimensionalAlgebra(QQ, [])
sage: A.one()
0

sage: B = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0], [0,1]]), Matrix([[0,1], [-1,0]])])
sage: B.one()
e0

sage: C = FiniteDimensionalAlgebra(QQ, [Matrix([[0,0], [0,0]]), Matrix([[0,0], [0,0]])])
sage: C.one()
Traceback (most recent call last):
...
TypeError: algebra is not unitary
primary_decomposition()

Return the primary decomposition of self.

Note

self must be unitary, commutative and associative.

OUTPUT:

  • a list consisting of the quotient maps self -> \(A\), with \(A\) running through the primary factors of self

EXAMPLES:

sage: A = FiniteDimensionalAlgebra(GF(3), [Matrix([[1, 0], [0, 1]]), Matrix([[0, 1], [0, 0]])])
sage: A.primary_decomposition()
[Morphism from Finite-dimensional algebra of degree 2 over Finite Field of size 3 to Finite-dimensional algebra of degree 2 over Finite Field of size 3 given by matrix [1 0]
[0 1]]

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.primary_decomposition()
[Morphism from Finite-dimensional algebra of degree 3 over Rational Field to Finite-dimensional algebra of degree 1 over Rational Field given by matrix [0]
[0]
[1], Morphism from Finite-dimensional algebra of degree 3 over Rational Field to Finite-dimensional algebra of degree 2 over Rational Field given by matrix [1 0]
[0 1]
[0 0]]
quotient_map(ideal)

Return the quotient of self by ideal.

INPUT:

  • ideal – a FiniteDimensionalAlgebraIdeal

OUTPUT:

EXAMPLES:

sage: A = FiniteDimensionalAlgebra(GF(3), [Matrix([[1, 0], [0, 1]]), Matrix([[0, 1], [0, 0]])])
sage: q0 = A.quotient_map(A.zero_ideal())
sage: q0
Morphism from Finite-dimensional algebra of degree 2 over Finite Field of size 3 to Finite-dimensional algebra of degree 2 over Finite Field of size 3 given by matrix
[1 0]
[0 1]
sage: q1 = A.quotient_map(A.ideal(A.gen(1)))
sage: q1
Morphism from Finite-dimensional algebra of degree 2 over Finite Field of size 3 to Finite-dimensional algebra of degree 1 over Finite Field of size 3 given by matrix
[1]
[0]
random_element(*args, **kwargs)

Return a random element of self.

Optional input parameters are propagated to the random_element method of the underlying VectorSpace.

EXAMPLES:

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

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.random_element(num_bound=1000)  # random
215/981*e0 + 709/953*e1 + 931/264*e2
table()

Return the multiplication table of self, as a list of matrices for right multiplication by the basis elements.

EXAMPLES:

sage: A = FiniteDimensionalAlgebra(GF(3), [Matrix([[1, 0], [0, 1]]), Matrix([[0, 1], [0, 0]])])
sage: A.table()
[
[1 0]  [0 1]
[0 1], [0 0]
]

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