# Finite dimensional algebras with basis¶

class sage.categories.finite_dimensional_algebras_with_basis.FiniteDimensionalAlgebrasWithBasis(base_category)

The category of finite dimensional algebras with a distinguished basis.

EXAMPLES:

sage: C = FiniteDimensionalAlgebrasWithBasis(QQ); C
Category of finite dimensional algebras with basis over Rational Field
sage: C.super_categories()
[Category of algebras with basis over Rational Field,
Category of finite dimensional modules with basis over Rational Field]


TESTS:

sage: TestSuite(C).run()
sage: C is Algebras(QQ).FiniteDimensional().WithBasis()
True
sage: C is Algebras(QQ).WithBasis().FiniteDimensional()
True

class ElementMethods
on_left_matrix(new_BR=None)

Returns the matrix of the action of self on the algebra my multiplication on the left

If new_BR is specified, then the matrix will be over new_BR.

TODO: split into to parts
• build the endomorphism of multiplication on the left
• build the matrix of an endomorphism

EXAMPLES:

sage: QS3 = SymmetricGroupAlgebra(QQ, 3)
sage: a = QS3([2,1,3])
sage: a.on_left_matrix()
[0 0 1 0 0 0]
[0 0 0 0 1 0]
[1 0 0 0 0 0]
[0 0 0 0 0 1]
[0 1 0 0 0 0]
[0 0 0 1 0 0]
sage: a.on_left_matrix(RDF)
[0.0 0.0 1.0 0.0 0.0 0.0]
[0.0 0.0 0.0 0.0 1.0 0.0]
[1.0 0.0 0.0 0.0 0.0 0.0]
[0.0 0.0 0.0 0.0 0.0 1.0]
[0.0 1.0 0.0 0.0 0.0 0.0]
[0.0 0.0 0.0 1.0 0.0 0.0]


AUTHOR: Mike Hansen

to_matrix(new_BR=None)

Returns the matrix of the action of self on the algebra my multiplication on the left

If new_BR is specified, then the matrix will be over new_BR.

TODO: split into to parts
• build the endomorphism of multiplication on the left
• build the matrix of an endomorphism

EXAMPLES:

sage: QS3 = SymmetricGroupAlgebra(QQ, 3)
sage: a = QS3([2,1,3])
sage: a.on_left_matrix()
[0 0 1 0 0 0]
[0 0 0 0 1 0]
[1 0 0 0 0 0]
[0 0 0 0 0 1]
[0 1 0 0 0 0]
[0 0 0 1 0 0]
sage: a.on_left_matrix(RDF)
[0.0 0.0 1.0 0.0 0.0 0.0]
[0.0 0.0 0.0 0.0 1.0 0.0]
[1.0 0.0 0.0 0.0 0.0 0.0]
[0.0 0.0 0.0 0.0 0.0 1.0]
[0.0 1.0 0.0 0.0 0.0 0.0]
[0.0 0.0 0.0 1.0 0.0 0.0]


AUTHOR: Mike Hansen

class FiniteDimensionalAlgebrasWithBasis.ParentMethods

Finite Crystals

#### Next topic

Finite dimensional bialgebras with basis