Finite dimensional semisimple algebras with basis¶
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class
sage.categories.finite_dimensional_semisimple_algebras_with_basis.
FiniteDimensionalSemisimpleAlgebrasWithBasis
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring
The category of finite dimensional semisimple algebras with a distinguished basis
EXAMPLES:
sage: from sage.categories.finite_dimensional_semisimple_algebras_with_basis import FiniteDimensionalSemisimpleAlgebrasWithBasis sage: C = FiniteDimensionalSemisimpleAlgebrasWithBasis(QQ); C Category of finite dimensional semisimple algebras with basis over Rational Field
This category is best constructed as:
sage: D = Algebras(QQ).Semisimple().FiniteDimensional().WithBasis(); D Category of finite dimensional semisimple algebras with basis over Rational Field sage: D is C True
TESTS:
sage: TestSuite(C).run()
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Commutative
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring
TESTS:
sage: C = Modules(ZZ).FiniteDimensional(); C Category of finite dimensional modules over Integer Ring sage: type(C) <class 'sage.categories.modules.Modules.FiniteDimensional_with_category'> sage: type(C).__base__.__base__ <class 'sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring'> sage: TestSuite(C).run()
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class
ParentMethods
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central_orthogonal_idempotents
()¶ Return the central orthogonal idempotents of this semisimple commutative algebra.
Those idempotents form a maximal decomposition of the identity into primitive orthogonal idempotents.
OUTPUT:
A list of orthogonal idempotents of
self
.EXAMPLES:
sage: A4 = SymmetricGroup(4).algebra(QQ) sage: Z4 = A4.center() sage: idempotents = Z4.central_orthogonal_idempotents() sage: idempotents (1/24*B[0] + 1/24*B[1] + 1/24*B[2] + 1/24*B[3] + 1/24*B[4], 3/8*B[0] + 1/8*B[1] - 1/8*B[2] - 1/8*B[4], 1/6*B[0] + 1/6*B[2] - 1/12*B[3], 3/8*B[0] - 1/8*B[1] - 1/8*B[2] + 1/8*B[4], 1/24*B[0] - 1/24*B[1] + 1/24*B[2] + 1/24*B[3] - 1/24*B[4])
Lifting those idempotents from the center, we recognize among them the sum and alternating sum of all permutations:
sage: [e.lift() for e in idempotents] [1/24*() + 1/24*(3,4) + 1/24*(2,3) + 1/24*(2,3,4) + 1/24*(2,4,3) + 1/24*(2,4) + 1/24*(1,2) + 1/24*(1,2)(3,4) + 1/24*(1,2,3) + 1/24*(1,2,3,4) + 1/24*(1,2,4,3) + 1/24*(1,2,4) + 1/24*(1,3,2) + 1/24*(1,3,4,2) + 1/24*(1,3) + 1/24*(1,3,4) + 1/24*(1,3)(2,4) + 1/24*(1,3,2,4) + 1/24*(1,4,3,2) + 1/24*(1,4,2) + 1/24*(1,4,3) + 1/24*(1,4) + 1/24*(1,4,2,3) + 1/24*(1,4)(2,3), ..., 1/24*() - 1/24*(3,4) - 1/24*(2,3) + 1/24*(2,3,4) + 1/24*(2,4,3) - 1/24*(2,4) - 1/24*(1,2) + 1/24*(1,2)(3,4) + 1/24*(1,2,3) - 1/24*(1,2,3,4) - 1/24*(1,2,4,3) + 1/24*(1,2,4) + 1/24*(1,3,2) - 1/24*(1,3,4,2) - 1/24*(1,3) + 1/24*(1,3,4) + 1/24*(1,3)(2,4) - 1/24*(1,3,2,4) - 1/24*(1,4,3,2) + 1/24*(1,4,2) + 1/24*(1,4,3) - 1/24*(1,4) - 1/24*(1,4,2,3) + 1/24*(1,4)(2,3)]
We check that they indeed form a decomposition of the identity of \(Z_4\) into orthogonal idempotents:
sage: Z4.is_identity_decomposition_into_orthogonal_idempotents(idempotents) True
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class
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class
FiniteDimensionalSemisimpleAlgebrasWithBasis.
ParentMethods
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central_orthogonal_idempotents
()¶ Return a maximal list of central orthogonal idempotents of
self
.Central orthogonal idempotents of an algebra \(A\) are idempotents \((e_1, \dots, e_n)\) in the center of \(A\) such that \(e_i e_j = 0\) whenever \(i \neq j\).
With the maximality condition, they sum up to \(1\) and are uniquely determined (up to order).
INPUT:
self
– a semisimple algebra.
EXAMPLES:
For the algebra of the symmetric group \(S_3\), we recover the sum and alternating sum of all permutations, together with a third idempotent:
sage: A3 = SymmetricGroup(3).algebra(QQ) sage: idempotents = A3.central_orthogonal_idempotents() sage: idempotents (1/6*() + 1/6*(2,3) + 1/6*(1,2) + 1/6*(1,2,3) + 1/6*(1,3,2) + 1/6*(1,3), 2/3*() - 1/3*(1,2,3) - 1/3*(1,3,2), 1/6*() - 1/6*(2,3) - 1/6*(1,2) + 1/6*(1,2,3) + 1/6*(1,3,2) - 1/6*(1,3)) sage: A3.is_identity_decomposition_into_orthogonal_idempotents(idempotents) True
For the semisimple quotient of a quiver algebra, we recover the vertices of the quiver:
sage: A = FiniteDimensionalAlgebrasWithBasis(QQ).example(); A An example of a finite dimensional algebra with basis: the path algebra of the Kronecker quiver (containing the arrows a:x->y and b:x->y) over Rational Field sage: Aquo = A.semisimple_quotient() sage: Aquo.central_orthogonal_idempotents() (B['x'], B['y'])
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radical_basis
(**keywords)¶ Return a basis of the Jacobson radical of this algebra.
keywords
– for compatibility; ignored.
OUTPUT: the empty list since this algebra is semisimple.
EXAMPLES:
sage: A = SymmetricGroup(4).algebra(QQ) sage: A.radical_basis() ()
TESTS:
sage: A.radical_basis.__module__ 'sage.categories.finite_dimensional_semisimple_algebras_with_basis'
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