Finite dimensional semisimple algebras with basis

class sage.categories.finite_dimensional_semisimple_algebras_with_basis.FiniteDimensionalSemisimpleAlgebrasWithBasis(base_category)

Bases: 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

class Commutative(base_category)

Bases: CategoryWithAxiom_over_base_ring

class ParentMethods

Bases: object

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: # needs sage.groups sage.modules
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]                               # needs sage.groups sage.modules
[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)     # needs sage.groups sage.modules
True

class ParentMethods

Bases: object

central_orthogonal_idempotents()

Return a maximal list of central orthogonal idempotents of self.

Central orthogonal idempotents of an algebra \(A\) are idempotents \((e_1, \ldots, e_n)\) 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).

EXAMPLES:

For the algebra of the (abelian) alternating group \(A_3\), we recover three idempotents corresponding to the three one-dimensional representations \(V_i\) on which \((1,2,3)\) acts on \(V_i\) as multiplication by the \(i\)-th power of a cube root of unity:

sage: # needs sage.groups sage.rings.number_field
sage: R = CyclotomicField(3)
sage: A3 = AlternatingGroup(3).algebra(R)
sage: idempotents = A3.central_orthogonal_idempotents()
sage: idempotents
(1/3*() + 1/3*(1,2,3) + 1/3*(1,3,2),
 1/3*() - (1/3*zeta3+1/3)*(1,2,3) - (-1/3*zeta3)*(1,3,2),
 1/3*() - (-1/3*zeta3)*(1,2,3) - (1/3*zeta3+1/3)*(1,3,2))
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: # needs sage.graphs sage.modules
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'])

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)                                 # needs sage.groups sage.modules
sage: A.radical_basis()                                                 # needs sage.groups sage.modules
()