Shuffle algebras

AUTHORS:

• Frédéric Chapoton (2013-03): Initial version

• Matthieu Deneufchatel (2013-07): Implemented dual PBW basis

class sage.algebras.shuffle_algebra.DualPBWBasis(R, names)

Bases: CombinatorialFreeModule

The basis dual to the Poincaré-Birkhoff-Witt basis of the free algebra.

We recursively define the dual PBW basis as the basis of the shuffle algebra given by

\[\begin{split}S_w = \begin{cases} w & |w| = 1, \\ x S_u & w = xu \text{ and } w \in \mathrm{Lyn}(X), \\ \displaystyle \frac{S_{\ell_{i_1}}^{\ast \alpha_1} \ast \cdots \ast S_{\ell_{i_k}}^{\ast \alpha_k}}{\alpha_1! \cdots \alpha_k!} & w = \ell_{i_1}^{\alpha_1} \cdots \ell_{i_k}^{\alpha_k} \text{ with } \ell_1 > \cdots > \ell_k \in \mathrm{Lyn}(X). \end{cases}\end{split}\]

where \(S \ast T\) denotes the shuffle product of \(S\) and \(T\) and \(\mathrm{Lyn}(X)\) is the set of Lyndon words in the alphabet \(X\).

The definition may be found in Theorem 5.3 of [Reu1993].

INPUT:

• R – ring

• names – names of the generators (string or an alphabet)

EXAMPLES:

sage: S = ShuffleAlgebra(QQ, 'ab').dual_pbw_basis()
sage: S
The dual Poincare-Birkhoff-Witt basis of Shuffle Algebra on 2 generators ['a', 'b'] over Rational Field
sage: S.one()
S[]
sage: S.one_basis()
word:
sage: T = ShuffleAlgebra(QQ, 'abcd').dual_pbw_basis(); T
The dual Poincare-Birkhoff-Witt basis of Shuffle Algebra on 4 generators ['a', 'b', 'c', 'd'] over Rational Field
sage: T.algebra_generators()
(S[a], S[b], S[c], S[d])

class Element

Bases: IndexedFreeModuleElement

An element in the dual PBW basis.

expand()

Expand self in words of the shuffle algebra.

EXAMPLES:

sage: S = ShuffleAlgebra(QQ, 'ab').dual_pbw_basis()
sage: f = S('ab') + S('bab')
sage: f.expand()
B[ab] + 2*B[abb] + B[bab]

algebra_generators()

Return the algebra generators of self.

EXAMPLES:

sage: S = ShuffleAlgebra(QQ, 'ab').dual_pbw_basis()
sage: S.algebra_generators()
(S[a], S[b])

antipode(elt)

Return the antipode of the element elt.

EXAMPLES:

sage: A = ShuffleAlgebra(QQ, 'ab')
sage: S = A.dual_pbw_basis()
sage: w = S('abaab').antipode(); w
S[abaab] - 2*S[ababa] - S[baaba]
 + 3*S[babaa] - 6*S[bbaaa]
sage: w.antipode()
S[abaab]

coproduct(elt)

Return the coproduct of the element elt.

EXAMPLES:

sage: A = ShuffleAlgebra(QQ, 'ab')
sage: S = A.dual_pbw_basis()
sage: S('ab').coproduct()
S[] # S[ab] + S[a] # S[b]
 + S[ab] # S[]
sage: S('ba').coproduct()
S[] # S[ba] + S[a] # S[b]
 + S[b] # S[a] + S[ba] # S[]

counit(S)

Return the counit of S.

EXAMPLES:

sage: F = ShuffleAlgebra(QQ,'ab').dual_pbw_basis()
sage: (3*F.gen(0)+5*F.gen(1)**2).counit()
0
sage: (4*F.one()).counit()
4

degree_on_basis(w)

Return the degree of the element w.

EXAMPLES:

sage: S = ShuffleAlgebra(QQ, 'ab').dual_pbw_basis()
sage: [S.degree_on_basis(x.leading_support()) for x in S.some_elements() if x != 0]
[0, 1, 1, 2]

expansion()

Return the morphism corresponding to the expansion into words of the shuffle algebra.

EXAMPLES:

sage: S = ShuffleAlgebra(QQ, 'ab').dual_pbw_basis()
sage: f = S('ab') + S('aba')
sage: S.expansion(f)
2*B[aab] + B[ab] + B[aba]

expansion_on_basis(w)

Return the expansion of \(S_w\) in words of the shuffle algebra.

INPUT:

• w – a word

EXAMPLES:

sage: S = ShuffleAlgebra(QQ, 'ab').dual_pbw_basis()
sage: S.expansion_on_basis(Word())
B[]
sage: S.expansion_on_basis(Word()).parent()
Shuffle Algebra on 2 generators ['a', 'b'] over Rational Field
sage: S.expansion_on_basis(Word('abba'))
2*B[aabb] + B[abab] + B[abba]
sage: S.expansion_on_basis(Word())
B[]
sage: S.expansion_on_basis(Word('abab'))
2*B[aabb] + B[abab]

gen(i)

Return the i-th generator of self.

EXAMPLES:

sage: S = ShuffleAlgebra(QQ, 'ab').dual_pbw_basis()
sage: S.gen(0)
S[a]
sage: S.gen(1)
S[b]

gens()

Return the algebra generators of self.

EXAMPLES:

sage: S = ShuffleAlgebra(QQ, 'ab').dual_pbw_basis()
sage: S.algebra_generators()
(S[a], S[b])

one_basis()

Return the indexing element of the basis element \(1\).

EXAMPLES:

sage: S = ShuffleAlgebra(QQ, 'ab').dual_pbw_basis()
sage: S.one_basis()
word:

product(u, v)

Return the product of two elements u and v.

EXAMPLES:

sage: S = ShuffleAlgebra(QQ, 'ab').dual_pbw_basis()
sage: a,b = S.gens()
sage: S.product(a, b)
S[ba]
sage: S.product(b, a)
S[ba]
sage: S.product(b^2*a, a*b*a)
36*S[bbbaaa]

shuffle_algebra()

Return the associated shuffle algebra of self.

EXAMPLES:

sage: S = ShuffleAlgebra(QQ, 'ab').dual_pbw_basis()
sage: S.shuffle_algebra()
Shuffle Algebra on 2 generators ['a', 'b'] over Rational Field

some_elements()

Return some typical elements.

EXAMPLES:

sage: F = ShuffleAlgebra(QQ,'xyz').dual_pbw_basis()
sage: F.some_elements()
[0, S[], S[x], S[y], S[z], S[zx]]

class sage.algebras.shuffle_algebra.ShuffleAlgebra(R, names, prefix)

Bases: CombinatorialFreeModule

The shuffle algebra on some generators over a base ring.

Shuffle algebras are commutative and associative algebras, with a basis indexed by words. The product of two words \(w_1 \cdot w_2\) is given by the sum over the shuffle product of \(w_1\) and \(w_2\).

See also

For more on shuffle products, see shuffle_product and shuffle().

REFERENCES:

• Wikipedia article Shuffle algebra

INPUT:

• R – ring

• names – generator names (string or an alphabet)

EXAMPLES:

sage: F = ShuffleAlgebra(QQ, 'xyz'); F
Shuffle Algebra on 3 generators ['x', 'y', 'z'] over Rational Field

sage: mul(F.gens())
B[xyz] + B[xzy] + B[yxz] + B[yzx] + B[zxy] + B[zyx]

sage: mul([ F.gen(i) for i in range(2) ]) + mul([ F.gen(i+1) for i in range(2) ])
B[xy] + B[yx] + B[yz] + B[zy]

sage: S = ShuffleAlgebra(ZZ, 'abcabc'); S
Shuffle Algebra on 3 generators ['a', 'b', 'c'] over Integer Ring
sage: S.base_ring()
Integer Ring

sage: G = ShuffleAlgebra(S, 'mn'); G
Shuffle Algebra on 2 generators ['m', 'n'] over Shuffle Algebra on 3 generators ['a', 'b', 'c'] over Integer Ring
sage: G.base_ring()
Shuffle Algebra on 3 generators ['a', 'b', 'c'] over Integer Ring

Shuffle algebras commute with their base ring:

sage: K = ShuffleAlgebra(QQ,'ab')
sage: a,b = K.gens()
sage: K.is_commutative()
True
sage: L = ShuffleAlgebra(K,'cd')
sage: c,d = L.gens()
sage: L.is_commutative()
True
sage: s = a*b^2 * c^3; s
(12*B[abb]+12*B[bab]+12*B[bba])*B[ccc]
sage: parent(s)
Shuffle Algebra on 2 generators ['c', 'd'] over Shuffle Algebra on 2 generators ['a', 'b'] over Rational Field
sage: c^3 * a * b^2
(12*B[abb]+12*B[bab]+12*B[bba])*B[ccc]

Shuffle algebras are commutative:

sage: c^3 * b * a * b == c * a * c * b^2 * c
True

We can also manipulate elements in the basis and coerce elements from our base field:

sage: F = ShuffleAlgebra(QQ, 'abc')
sage: B = F.basis()
sage: B[Word('bb')] * B[Word('ca')]
B[bbca] + B[bcab] + B[bcba] + B[cabb]
 + B[cbab] + B[cbba]
sage: 1 - B[Word('bb')] * B[Word('ca')] / 2
B[] - 1/2*B[bbca] - 1/2*B[bcab] - 1/2*B[bcba]
 - 1/2*B[cabb] - 1/2*B[cbab] - 1/2*B[cbba]

algebra_generators()

Return the generators of this algebra.

EXAMPLES:

sage: A = ShuffleAlgebra(ZZ,'fgh'); A
Shuffle Algebra on 3 generators ['f', 'g', 'h'] over Integer Ring
sage: A.algebra_generators()
Family (B[f], B[g], B[h])

sage: A = ShuffleAlgebra(QQ, ['x1','x2'])
sage: A.algebra_generators()
Family (B[x1], B[x2])

antipode_on_basis(w)

Return the antipode on the basis element w.

EXAMPLES:

sage: A = ShuffleAlgebra(QQ,'abc')
sage: W = A.basis().keys()
sage: A.antipode_on_basis(W("acb"))
-B[bca]

coproduct_on_basis(w)

Return the coproduct of the element of the basis indexed by the word w.

The coproduct is given by deconcatenation.

INPUT:

• w – a word

EXAMPLES:

sage: F = ShuffleAlgebra(QQ,'ab')
sage: F.coproduct_on_basis(Word('a'))
B[] # B[a] + B[a] # B[]
sage: F.coproduct_on_basis(Word('aba'))
B[] # B[aba] + B[a] # B[ba]
 + B[ab] # B[a] + B[aba] # B[]
sage: F.coproduct_on_basis(Word())
B[] # B[]

counit(S)

Return the counit of S.

EXAMPLES:

sage: F = ShuffleAlgebra(QQ,'ab')
sage: S = F.an_element(); S
B[] + 2*B[a] + 3*B[b] + B[bab]
sage: F.counit(S)
1

degree_on_basis(w)

Return the degree of the element w.

EXAMPLES:

sage: A = ShuffleAlgebra(QQ, 'ab')
sage: [A.degree_on_basis(x.leading_support()) for x in A.some_elements() if x != 0]
[0, 1, 1, 2]

dual_pbw_basis()

Return the dual PBW of self.

EXAMPLES:

sage: A = ShuffleAlgebra(QQ, 'ab')
sage: A.dual_pbw_basis()
The dual Poincare-Birkhoff-Witt basis of Shuffle Algebra on 2 generators ['a', 'b'] over Rational Field

gen(i)

Return the i-th generator of the algebra.

INPUT:

• i – an integer

EXAMPLES:

sage: F = ShuffleAlgebra(ZZ,'xyz')
sage: F.gen(0)
B[x]

sage: F.gen(4)
Traceback (most recent call last):
...
IndexError: argument i (= 4) must be between 0 and 2

gens()

Return the generators of this algebra.

EXAMPLES:

sage: A = ShuffleAlgebra(ZZ,'fgh'); A
Shuffle Algebra on 3 generators ['f', 'g', 'h'] over Integer Ring
sage: A.algebra_generators()
Family (B[f], B[g], B[h])

sage: A = ShuffleAlgebra(QQ, ['x1','x2'])
sage: A.algebra_generators()
Family (B[x1], B[x2])

one_basis()

Return the empty word, which index of \(1\) of this algebra, as per AlgebrasWithBasis.ParentMethods.one_basis().

EXAMPLES:

sage: A = ShuffleAlgebra(QQ,'a')
sage: A.one_basis()
word:
sage: A.one()
B[]

product_on_basis(w1, w2)

Return the product of basis elements w1 and w2, as per AlgebrasWithBasis.ParentMethods.product_on_basis().

INPUT:

• w1, w2 – Basis elements

EXAMPLES:

sage: A = ShuffleAlgebra(QQ,'abc')
sage: W = A.basis().keys()
sage: A.product_on_basis(W("acb"), W("cba"))
B[acbacb] + B[acbcab] + 2*B[acbcba]
 + 2*B[accbab] + 4*B[accbba] + B[cabacb]
 + B[cabcab] + B[cabcba] + B[cacbab]
 + 2*B[cacbba] + 2*B[cbaacb] + B[cbacab]
 + B[cbacba]

sage: (a,b,c) = A.algebra_generators()
sage: a * (1-b)^2 * c
2*B[abbc] - 2*B[abc] + 2*B[abcb] + B[ac]
 - 2*B[acb] + 2*B[acbb] + 2*B[babc]
 - 2*B[bac] + 2*B[bacb] + 2*B[bbac]
 + 2*B[bbca] - 2*B[bca] + 2*B[bcab]
 + 2*B[bcba] + B[ca] - 2*B[cab] + 2*B[cabb]
 - 2*B[cba] + 2*B[cbab] + 2*B[cbba]

some_elements()

Return some typical elements.

EXAMPLES:

sage: F = ShuffleAlgebra(ZZ,'xyz')
sage: F.some_elements()
[0, B[], B[x], B[y], B[z], B[xz] + B[zx]]

to_dual_pbw_element(w)

Return the element \(w\) of self expressed in the dual PBW basis.

INPUT:

• w – an element of the shuffle algebra

EXAMPLES:

sage: A = ShuffleAlgebra(QQ, 'ab')
sage: f = 2 * A(Word()) + A(Word('ab')); f
2*B[] + B[ab]
sage: A.to_dual_pbw_element(f)
2*S[] + S[ab]
sage: A.to_dual_pbw_element(A.one())
S[]
sage: S = A.dual_pbw_basis()
sage: elt = S.expansion_on_basis(Word('abba')); elt
2*B[aabb] + B[abab] + B[abba]
sage: A.to_dual_pbw_element(elt)
S[abba]
sage: A.to_dual_pbw_element(2*A(Word('aabb')) + A(Word('abab')))
S[abab]
sage: S.expansion(S('abab'))
2*B[aabb] + B[abab]

variable_names()

Return the names of the variables.

EXAMPLES:

sage: R = ShuffleAlgebra(QQ,'xy')
sage: R.variable_names()
{'x', 'y'}