Free Pre-Lie Algebras

AUTHORS:

  • Florent Hivert, Frédéric Chapoton (2011)
class sage.combinat.free_prelie_algebra.FreePreLieAlgebra(R, names=None)

Bases: sage.combinat.free_module.CombinatorialFreeModule

The free pre-Lie algebra.

Pre-Lie algebras are non-associative algebras, where the product \(*\) satisfies

\[(x * y) * z - x * (y * z) = (x * z) * y - x * (z * y).\]

We use here the convention where the associator

\[(x, y, z) := (x * y) * z - x * (y * z)\]

is symmetric in its two rightmost arguments. This is sometimes called a right pre-Lie algebra.

They have appeared in numerical analysis and deformation theory.

The free Pre-Lie algebra on a given set \(E\) has an explicit description using rooted trees, just as the free associative algebra can be described using words. The underlying vector space has a basis indexed by finite rooted trees endowed with a map from their vertices to \(E\). In this basis, the product of two (decorated) rooted trees \(S * T\) is the sum over vertices of \(S\) of the rooted tree obtained by adding one edge from the root of \(T\) to the given vertex of \(S\). The root of these trees is taken to be the root of \(S\). The free pre-Lie algebra can also be considered as the free algebra over the PreLie operad.

Warning

The usual binary operator * can be used for the pre-Lie product. Beware that it but must be parenthesized properly, as the pre-Lie product is not associative. By default, a multiple product will be taken with left parentheses.

EXAMPLES:

sage: F = algebras.FreePreLie(ZZ, 'xyz')
sage: x,y,z = F.gens()
sage: (x * y) * z
B[x[y[z[]]]] + B[x[y[], z[]]]
sage: (x * y) * z - x * (y * z) == (x * z) * y - x * (z * y)
True

The free pre-Lie algebra is non-associative:

sage: x * (y * z) == (x * y) * z
False

The default product is with left parentheses:

sage: x * y * z == (x * y) * z
True
sage: x * y * z * x == ((x * y) * z) * x
True

The NAP product as defined in [Liv] is also implemented on the same vector space:

sage: N = F.nap_product
sage: N(x*y,z*z)
B[x[y[], z[z[]]]]

When there is only one generator, unlabelled trees are used instead:

sage: F1 = algebras.FreePreLie(QQ, 'w')
sage: w = F1.gen(0); w
B[[]]
sage: w * w * w * w
B[[[[[]]]]] + B[[[[], []]]] + 3*B[[[], [[]]]] + B[[[], [], []]]

REFERENCES:

[ChLi]F. Chapoton and M. Livernet, Pre-Lie algebras and the rooted trees operad, International Math. Research Notices (2001) no 8, pages 395-408.
[Liv]M. Livernet, A rigidity theorem for pre-Lie algebras, J. Pure Appl. Algebra 207 (2006), no 1, pages 1-18.
algebra_generators()

Return the generators of this algebra.

These are the rooted trees with just one vertex.

EXAMPLES:

sage: A = algebras.FreePreLie(ZZ, 'fgh'); A
Free PreLie algebra on 3 generators ['f', 'g', 'h']
 over Integer Ring
sage: list(A.algebra_generators())
[B[f[]], B[g[]], B[h[]]]

sage: A = algebras.FreePreLie(QQ, ['x1','x2'])
sage: list(A.algebra_generators())
[B[x1[]], B[x2[]]]
an_element()

Return an element of self.

EXAMPLES:

sage: A = algebras.FreePreLie(QQ, 'xy')
sage: A.an_element()
B[x[x[x[x[]]]]] + B[x[x[], x[x[]]]]
degree_on_basis(t)

Return the degree of a rooted tree in the free Pre-Lie algebra.

This is the number of vertices.

EXAMPLES:

sage: A = algebras.FreePreLie(QQ,'@')
sage: RT = A.basis().keys()
sage: A.degree_on_basis(RT([RT([])]))
2
gen(i)

Return the i-th generator of the algebra.

INPUT:

  • i – an integer

EXAMPLES:

sage: F = algebras.FreePreLie(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 self (as an algebra).

EXAMPLES:

sage: A = algebras.FreePreLie(ZZ, 'fgh')
sage: A.gens()
(B[f[]], B[g[]], B[h[]])
nap_product()

Return the NAP product.

EXAMPLES:

sage: A = algebras.FreePreLie(QQ, '@')
sage: RT = A.basis().keys()
sage: x = A(RT([RT([])]))
sage: A.nap_product(x, x)
B[[[], [[]]]]
nap_product_on_basis(x, y)

Return the NAP product of two trees.

This is the grafting of the root of \(y\) over the root of \(x\). The root of the resulting tree is the root of \(x\).

See also

nap_product()

EXAMPLES:

sage: A = algebras.FreePreLie(QQ, '@')
sage: RT = A.basis().keys()
sage: x = RT([RT([])])
sage: A.nap_product_on_basis(x, x)
B[[[], [[]]]]
pre_Lie_product()

Return the pre-Lie product.

EXAMPLES:

sage: A = algebras.FreePreLie(QQ, '@')
sage: RT = A.basis().keys()
sage: x = A(RT([RT([])]))
sage: A.pre_Lie_product(x, x)
B[[[[[]]]]] + B[[[], [[]]]]
pre_Lie_product_on_basis(x, y)

Return the pre-Lie product of two trees.

This is the sum over all graftings of the root of \(y\) over a vertex of \(x\). The root of the resulting trees is the root of \(x\).

EXAMPLES:

sage: A = algebras.FreePreLie(QQ, '@')
sage: RT = A.basis().keys()
sage: x = RT([RT([])])
sage: A.product_on_basis(x, x)
B[[[[[]]]]] + B[[[], [[]]]]
product_on_basis(x, y)

Return the pre-Lie product of two trees.

This is the sum over all graftings of the root of \(y\) over a vertex of \(x\). The root of the resulting trees is the root of \(x\).

EXAMPLES:

sage: A = algebras.FreePreLie(QQ, '@')
sage: RT = A.basis().keys()
sage: x = RT([RT([])])
sage: A.product_on_basis(x, x)
B[[[[[]]]]] + B[[[], [[]]]]
some_elements()

Return some elements of the free pre-Lie algebra.

EXAMPLES:

sage: A = algebras.FreePreLie(QQ,'@')
sage: A.some_elements()
[B[[]], B[[[]]], B[[[[[]]]]] + B[[[], [[]]]], B[[[[]]]] + B[[[], []]], B[[[]]]]

With several generators:

sage: A = algebras.FreePreLie(QQ, 'xy')
sage: A.some_elements()
[B[x[]],
 B[x[x[]]],
 B[x[x[x[x[]]]]] + B[x[x[], x[x[]]]],
 B[x[x[x[]]]] + B[x[x[], x[]]],
 B[x[x[y[]]]] + B[x[x[], y[]]]]
variable_names()

Return the names of the variables.

EXAMPLES:

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