Bialgebras with basis#

class sage.categories.bialgebras_with_basis.BialgebrasWithBasis(base_category)#

Bases: CategoryWithAxiom_over_base_ring

The category of bialgebras with a distinguished basis.

EXAMPLES:

sage: C = BialgebrasWithBasis(QQ); C
Category of bialgebras with basis over Rational Field

sage: sorted(C.super_categories(), key=str)
[Category of algebras with basis over Rational Field,
 Category of bialgebras over Rational Field,
 Category of coalgebras with basis over Rational Field]
class ElementMethods#

Bases: object

adams_operator(n)#

Compute the \(n\)-th convolution power of the identity morphism \(\mathrm{Id}\) on self.

INPUT:

  • n – a nonnegative integer

OUTPUT:

  • the image of self under the convolution power \(\mathrm{Id}^{*n}\)

Note

In the literature, this is also called a Hopf power or Sweedler power, cf. [AL2015].

See also

sage.categories.bialgebras.ElementMethods.convolution_product()

Todo

Remove dependency on modules_with_basis methods.

EXAMPLES:

sage: h = SymmetricFunctions(QQ).h()
sage: h[5].adams_operator(2)
2*h[3, 2] + 2*h[4, 1] + 2*h[5]
sage: h[5].plethysm(2*h[1])
2*h[3, 2] + 2*h[4, 1] + 2*h[5]
sage: h([]).adams_operator(0)
h[]
sage: h([]).adams_operator(1)
h[]
sage: h[3,2].adams_operator(0)
0
sage: h[3,2].adams_operator(1)
h[3, 2]

sage: S = NonCommutativeSymmetricFunctions(QQ).S()
sage: S[4].adams_operator(5)
5*S[1, 1, 1, 1] + 10*S[1, 1, 2] + 10*S[1, 2, 1] + 10*S[1, 3] + 10*S[2, 1, 1] + 10*S[2, 2] + 10*S[3, 1] + 5*S[4]

sage: m = SymmetricFunctionsNonCommutingVariables(QQ).m()
sage: m[[1,3],[2]].adams_operator(-2)
3*m{{1}, {2, 3}} + 3*m{{1, 2}, {3}} + 6*m{{1, 2, 3}} - 2*m{{1, 3}, {2}}
convolution_product(*maps)#

Return the image of self under the convolution product (map) of the maps.

Let \(A\) and \(B\) be bialgebras over a commutative ring \(R\). Given maps \(f_i : A \to B\) for \(1 \leq i < n\), define the convolution product

\[(f_1 * f_2 * \cdots * f_n) := \mu^{(n-1)} \circ (f_1 \otimes f_2 \otimes \cdots \otimes f_n) \circ \Delta^{(n-1)},\]

where \(\Delta^{(k)} := \bigl(\Delta \otimes \mathrm{Id}^{\otimes(k-1)}\bigr) \circ \Delta^{(k-1)}\), with \(\Delta^{(1)} = \Delta\) (the ordinary coproduct in \(A\)) and \(\Delta^{(0)} = \mathrm{Id}\); and with \(\mu^{(k)} := \mu \circ \bigl(\mu^{(k-1)} \otimes \mathrm{Id})\) and \(\mu^{(1)} = \mu\) (the ordinary product in \(B\)). See [Swe1969].

(In the literature, one finds, e.g., \(\Delta^{(2)}\) for what we denote above as \(\Delta^{(1)}\). See [KMN2012].)

INPUT:

  • maps – any number \(n \geq 0\) of linear maps \(f_1, f_2, \ldots, f_n\) on self.parent(); or a single list or tuple of such maps

OUTPUT:

  • the convolution product of maps applied to self

AUTHORS:

  • Amy Pang - 12 June 2015 - Sage Days 65

Todo

Remove dependency on modules_with_basis methods.

EXAMPLES:

We compute convolution products of the identity and antipode maps on Schur functions:

sage: Id = lambda x: x
sage: Antipode = lambda x: x.antipode()
sage: s = SymmetricFunctions(QQ).schur()
sage: s[3].convolution_product(Id, Id)
2*s[2, 1] + 4*s[3]
sage: s[3,2].convolution_product(Id) == s[3,2]
True

The method accepts multiple arguments, or a single argument consisting of a list of maps:

sage: s[3,2].convolution_product(Id, Id)
2*s[2, 1, 1, 1] + 6*s[2, 2, 1] + 6*s[3, 1, 1] + 12*s[3, 2] + 6*s[4, 1] + 2*s[5]
sage: s[3,2].convolution_product([Id, Id])
2*s[2, 1, 1, 1] + 6*s[2, 2, 1] + 6*s[3, 1, 1] + 12*s[3, 2] + 6*s[4, 1] + 2*s[5]

We test the defining property of the antipode morphism; namely, that the antipode is the inverse of the identity map in the convolution algebra whose identity element is the composition of the counit and unit:

sage: s[3,2].convolution_product() == s[3,2].convolution_product(Antipode, Id) == s[3,2].convolution_product(Id, Antipode)
True

sage: Psi = NonCommutativeSymmetricFunctions(QQ).Psi()
sage: Psi[2,1].convolution_product(Id, Id, Id)
3*Psi[1, 2] + 6*Psi[2, 1]
sage: (Psi[5,1] - Psi[1,5]).convolution_product(Id, Id, Id)
-3*Psi[1, 5] + 3*Psi[5, 1]

sage: G = SymmetricGroup(3)
sage: QG = GroupAlgebra(G,QQ)
sage: x = QG.sum_of_terms([(p,p.length()) for p in Permutations(3)]); x
[1, 3, 2] + [2, 1, 3] + 2*[2, 3, 1] + 2*[3, 1, 2] + 3*[3, 2, 1]
sage: x.convolution_product(Id, Id)
5*[1, 2, 3] + 2*[2, 3, 1] + 2*[3, 1, 2]
sage: x.convolution_product(Id, Id, Id)
4*[1, 2, 3] + [1, 3, 2] + [2, 1, 3] + 3*[3, 2, 1]
sage: x.convolution_product([Id]*6)
9*[1, 2, 3]
class ParentMethods#

Bases: object

convolution_product(*maps)#

Return the convolution product (a map) of the given maps.

Let \(A\) and \(B\) be bialgebras over a commutative ring \(R\). Given maps \(f_i : A \to B\) for \(1 \leq i < n\), define the convolution product

\[(f_1 * f_2 * \cdots * f_n) := \mu^{(n-1)} \circ (f_1 \otimes f_2 \otimes \cdots \otimes f_n) \circ \Delta^{(n-1)},\]

where \(\Delta^{(k)} := \bigl(\Delta \otimes \mathrm{Id}^{\otimes(k-1)}\bigr) \circ \Delta^{(k-1)}\), with \(\Delta^{(1)} = \Delta\) (the ordinary coproduct in \(A\)) and \(\Delta^{(0)} = \mathrm{Id}\); and with \(\mu^{(k)} := \mu \circ \bigl(\mu^{(k-1)} \otimes \mathrm{Id})\) and \(\mu^{(1)} = \mu\) (the ordinary product in \(B\)). See [Swe1969].

(In the literature, one finds, e.g., \(\Delta^{(2)}\) for what we denote above as \(\Delta^{(1)}\). See [KMN2012].)

INPUT:

  • maps – any number \(n \geq 0\) of linear maps \(f_1, f_2, \ldots, f_n\) on self; or a single list or tuple of such maps

OUTPUT:

  • the new map \(f_1 * f_2 * \cdots * f_2\) representing their convolution product

See also

sage.categories.bialgebras.ElementMethods.convolution_product()

AUTHORS:

  • Aaron Lauve - 12 June 2015 - Sage Days 65

Todo

Remove dependency on modules_with_basis methods.

EXAMPLES:

We construct some maps: the identity, the antipode and projection onto the homogeneous component of degree 2:

sage: Id = lambda x: x
sage: Antipode = lambda x: x.antipode()
sage: Proj2 = lambda x: x.parent().sum_of_terms([(m, c) for (m, c) in x if m.size() == 2])

Compute the convolution product of the identity with itself and with the projection Proj2 on the Hopf algebra of non-commutative symmetric functions:

sage: R = NonCommutativeSymmetricFunctions(QQ).ribbon()
sage: T = R.convolution_product([Id, Id])
sage: [T(R(comp)) for comp in Compositions(3)]
[4*R[1, 1, 1] + R[1, 2] + R[2, 1],
 2*R[1, 1, 1] + 4*R[1, 2] + 2*R[2, 1] + 2*R[3],
 2*R[1, 1, 1] + 2*R[1, 2] + 4*R[2, 1] + 2*R[3],
 R[1, 2] + R[2, 1] + 4*R[3]]
sage: T = R.convolution_product(Proj2, Id)
sage: [T(R([i])) for i in range(1, 5)]
[0, R[2], R[2, 1] + R[3], R[2, 2] + R[4]]

Compute the convolution product of no maps on the Hopf algebra of symmetric functions in non-commuting variables. This is the composition of the counit with the unit:

sage: m = SymmetricFunctionsNonCommutingVariables(QQ).m()
sage: T = m.convolution_product()
sage: [T(m(lam)) for lam in SetPartitions(0).list() + SetPartitions(2).list()]
[m{}, 0, 0]

Compute the convolution product of the projection Proj2 with the identity on the Hopf algebra of symmetric functions in non-commuting variables:

sage: T = m.convolution_product(Proj2, Id)
sage: [T(m(lam)) for lam in SetPartitions(3)]
[0,
 m{{1, 2}, {3}} + m{{1, 2, 3}},
 m{{1, 2}, {3}} + m{{1, 2, 3}},
 m{{1, 2}, {3}} + m{{1, 2, 3}},
 3*m{{1}, {2}, {3}} + 3*m{{1}, {2, 3}} + 3*m{{1, 3}, {2}}]

Compute the convolution product of the antipode with itself and the identity map on group algebra of the symmetric group:

sage: G = SymmetricGroup(3)
sage: QG = GroupAlgebra(G, QQ)
sage: x = QG.sum_of_terms([(p,p.number_of_peaks() + p.number_of_inversions()) for p in Permutations(3)]); x
2*[1, 3, 2] + [2, 1, 3] + 3*[2, 3, 1] + 2*[3, 1, 2] + 3*[3, 2, 1]
sage: T = QG.convolution_product(Antipode, Antipode, Id)
sage: T(x)
2*[1, 3, 2] + [2, 1, 3] + 2*[2, 3, 1] + 3*[3, 1, 2] + 3*[3, 2, 1]