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# Finite dimensional algebras with basis

class `sage.categories.finite_dimensional_algebras_with_basis.``FiniteDimensionalAlgebrasWithBasis`(base_category)

  Bases: `sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring`

  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`

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