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The double shuffle Lie algebra Ecalle’s theory of Moulds A new proof of Racinet’s theorem

On the double shuffle Lie algebra structure: Ecalle’s approach

Adriana Salerno (joint work with Leila Schneps)

Bates College

December 2, 2014

Adriana Salerno (joint work with Leila Schneps) On the double shuffle Lie algebra structure: Ecalle’s approach

The double shuffle Lie algebra Ecalle’s theory of Moulds A new proof of Racinet’s theorem

Outline

The double shuffle Lie algebra Ecalle’s theory of Moulds A new proof of Racinet’s theorem

Adriana Salerno (joint work with Leila Schneps) On the double shuffle Lie algebra structure: Ecalle’s approach

The double shuffle Lie algebra Ecalle’s theory of Moulds A new proof of Racinet’s theorem

The shuffle product

◮ Suppose x and y are variables that don’t commute. ◮ For two words u, v in Qx, y, the shuffle product sh(u, v) is

the sum of permutations of the letters of u and v where the letters of each word remain ordered.

◮ Example: sh(y, xy) = yxy + 2xyy.

Adriana Salerno (joint work with Leila Schneps) On the double shuffle Lie algebra structure: Ecalle’s approach

The double shuffle Lie algebra Ecalle’s theory of Moulds A new proof of Racinet’s theorem

The stuffle product

If u, v are words in Qx, y both ending in y, we can write them uniquely as words in the letters yi = xi−1y. The stuffle product of u, v is defined by st(u, v) = u if v = 1 and v is u = 1, and st(yiu, yjv) = {yist(u, yjv)} ∪ {yjst(yiu, v)} ∪ {yi+jst(u, v)}, where yi and yj are respectively the first letters of the words u and v written in the yj. For example: st(xy, xy) = st(y2, y2) = y2y2 + y2y2 + y4 = 2xyxy + x3y.

Adriana Salerno (joint work with Leila Schneps) On the double shuffle Lie algebra structure: Ecalle’s approach

The double shuffle Lie algebra Ecalle’s theory of Moulds A new proof of Racinet’s theorem

The double shuffle space

We define the set ds as the set of polynomials in Qx, y such that

• w∈sh(u,v)

(f |w) = 0 and

• w∈st(u,v)

(f |w) = 0 for all words u, v not both powers of y.

Adriana Salerno (joint work with Leila Schneps) On the double shuffle Lie algebra structure: Ecalle’s approach

The double shuffle Lie algebra Ecalle’s theory of Moulds A new proof of Racinet’s theorem

Lie polynomials

FACT: A polynomial satisfies the shuffle property if and only if it is a Lie polynomial in x, y. The Lie bracket is defined as: [x, y] = xy − yx Lie polynomials come from linear combinations of consecutive applications of Lie brackets. For example: [x, [x, y]] + [[x, y], y] = x[x, y] − [x, y]x + [x, y]y − y[x, y] = x2y − xyx − xyx + yx2 + xy2 − yxy − yxy + y = x2y − 2xyx + yx2 + xy2 − 2yxy + y2x

Adriana Salerno (joint work with Leila Schneps) On the double shuffle Lie algebra structure: Ecalle’s approach

The double shuffle Lie algebra Ecalle’s theory of Moulds A new proof of Racinet’s theorem

The Poisson bracket

For every f ∈ Lie[x, y], define a derivation Df of Lie[x, y] by Df (x) = 0, Df (y) = [y, f ]

• n the generators. Define the Poisson bracket on Lie[x, y] by

{f , g} = [f , g] + Df (g) − Dg(f ). This definition corresponds naturally to the Lie bracket on the space of derivations of Lie[x, y]; indeed, it is easy to check that [Df , Dg] = Df ◦ Dg − Dg ◦ Df = D{f ,g}.

Adriana Salerno (joint work with Leila Schneps) On the double shuffle Lie algebra structure: Ecalle’s approach

The double shuffle Lie algebra Ecalle’s theory of Moulds A new proof of Racinet’s theorem

Racinet’s theorem

The space ds is a Lie algebra under the Poisson bracket. (Racinet proved this in his Ph.D. thesis - the proof is very difficult, technical, and hard to motivate).

Figure : Georges Racinet

See Furusho’s streamlined version of the proof for details.

Adriana Salerno (joint work with Leila Schneps) On the double shuffle Lie algebra structure: Ecalle’s approach

The double shuffle Lie algebra Ecalle’s theory of Moulds A new proof of Racinet’s theorem

New machinery: Ecalle’s point of view

Jean Ecalle has constructed a framework – machinery that yields Racinet’s theorem and other results easily. The machinery is big, but many results come out beautifully.

Figure : Jean Ecalle

Adriana Salerno (joint work with Leila Schneps) On the double shuffle Lie algebra structure: Ecalle’s approach

The double shuffle Lie algebra Ecalle’s theory of Moulds A new proof of Racinet’s theorem

New point of view: Moulds

A mould is a family of functions M(u1, u2, . . . , ur) for each r ≥ 0. We will restrict our attention to rational functions with coefficients in Q. We will also consider moulds on the alphabet {v1, v2, . . . }.

Adriana Salerno (joint work with Leila Schneps) On the double shuffle Lie algebra structure: Ecalle’s approach

The double shuffle Lie algebra Ecalle’s theory of Moulds A new proof of Racinet’s theorem

New point of view: Moulds

A mould is a family of functions M(u1, u2, . . . , ur) for each r ≥ 0. We will restrict our attention to rational functions with coefficients in Q. We will also consider moulds on the alphabet {v1, v2, . . . }. ARI (resp. ARI) will denote the vector space (for obvious addition and scalar multiplication laws) of moulds M in the ui (resp. vi) satisfying M0(∅) = 0.

Adriana Salerno (joint work with Leila Schneps) On the double shuffle Lie algebra structure: Ecalle’s approach

The double shuffle Lie algebra Ecalle’s theory of Moulds A new proof of Racinet’s theorem

From ARI to ARI

We have the following map from moulds in A in ARI to moulds in ARI: swap(A)(u1, . . . , ur) = A(vr, vr−1 − vr, vr−2 − vr−1, . . . , v1 − v2) And its inverse (also called swap) is as follows: swap(A)(v1, . . . , vr) = A(u1+· · ·+ur, u1+· · ·+ur−1, . . . , u1+u2, u1)

Adriana Salerno (joint work with Leila Schneps) On the double shuffle Lie algebra structure: Ecalle’s approach

The double shuffle Lie algebra Ecalle’s theory of Moulds A new proof of Racinet’s theorem

Moulds and polynomials

One can associate moulds to polynomials in Qx, y.

◮ Let πy(f ) be the projection of f onto monomials ending in y. ◮ Let the depth r of a monomial denote the number of y’s. ◮ Let f r denote the depth r part of f , so f = f r. ◮ Let πy(f r) = cai,...,ar xa1−1y · · · xar−1y. ◮ Define a mould mif by

mif (v1, . . . , vr) =

• cai,...,ar va1−1

1

· · · var−1

r

for r ≥ 0 and maf (u1, . . . , ur) = swap(mif )(v1, . . . , vr) = mif (u1 + · · · + ur, . . . , u1 + u2, u1)

Adriana Salerno (joint work with Leila Schneps) On the double shuffle Lie algebra structure: Ecalle’s approach

The double shuffle Lie algebra Ecalle’s theory of Moulds A new proof of Racinet’s theorem

Example

For example, the Lie polynomial given by f = [x, [x, y]] + [[x, y], y] = x2y − 2xyx + yx2 + xy2 − 2yxy + y2x, which projects to x2y + xy2 − 2yxy, is associated to the mould M = maf given by M(∅) = M(u1) = u2

1

M(u1, u2) = −u1 + u2 M(u1, · · · , ur) = 0 if r > 2.

Adriana Salerno (joint work with Leila Schneps) On the double shuffle Lie algebra structure: Ecalle’s approach

The double shuffle Lie algebra Ecalle’s theory of Moulds A new proof of Racinet’s theorem

A mould is alternal if

• w∈sh((u1,...,ui),(ui+1,...,ur))

M(w) = 0, for 1 ≤ i ≤ ⌊ r

2⌋.

ARIal = {alternal moulds} ARIal/al = {alternal moulds A such that swap(A) is alternal}

Adriana Salerno (joint work with Leila Schneps) On the double shuffle Lie algebra structure: Ecalle’s approach

The double shuffle Lie algebra Ecalle’s theory of Moulds A new proof of Racinet’s theorem

Example

For M as in the previous example, M(∅) = M(u1) = u2

1

M(u1, u2) = −u1 + u2 M(u1, · · · , ur) = 0 if r > 2. Notice M(u1, u2) + M(u2, u1) = 0, so M is alternal.

Adriana Salerno (joint work with Leila Schneps) On the double shuffle Lie algebra structure: Ecalle’s approach

The double shuffle Lie algebra Ecalle’s theory of Moulds A new proof of Racinet’s theorem

Alternal and alternil

A mould is alternil if it satisfies a similar symmetry but with respect to the stuffle product. Thus for example, since the stuffle for depth 2 is: st(a, b) = (a, b) + (b, a) + (a + b), the alternility condition in depth 2 is given by 0 = M(v1, v2) + M(v2, v1) + 1 v1 − v2 M(v1) + 1 v2 − v1 M(v2).

Lemma

Let f a polynomial.

• 1. f satisfies shuffle iff maf is alternal.
• 2. f satisfies stuffle iff mif is alternil.

Adriana Salerno (joint work with Leila Schneps) On the double shuffle Lie algebra structure: Ecalle’s approach

The double shuffle Lie algebra Ecalle’s theory of Moulds A new proof of Racinet’s theorem

The ari bracket

Let ARIpol = {polynomial-valued moulds in ARI} Ecalle defines a Lie bracket on ARI, denoted ari.

Theorem (Racinet)

Restricted to alternal polynomial moulds ARIpol

al , the ari bracket

corresponds to the Poisson bracket. In other words, ma : Lie[x, y] → ARIpol

al

f → maf is a Lie algebra isomorphism.

Adriana Salerno (joint work with Leila Schneps) On the double shuffle Lie algebra structure: Ecalle’s approach

The double shuffle Lie algebra Ecalle’s theory of Moulds A new proof of Racinet’s theorem

The Lie algebra ARI

Theorem (Ecalle, S., Schneps)

ARIal and ARIal/al are Lie algebras under the ari bracket. ARIal/il = {alternal moulds A such that swap(A) is alternil} ARIpol

al/il = {polynomial-valued moulds in ARIal/il}

By definition ARIpol

al/il ≃ ds as vector spaces.

Adriana Salerno (joint work with Leila Schneps) On the double shuffle Lie algebra structure: Ecalle’s approach

The double shuffle Lie algebra Ecalle’s theory of Moulds A new proof of Racinet’s theorem

The main theorem

In mould language, proving ds is a Lie algebra under the Poisson bracket (Racinet’s Theorem) is then equivalent to proving that ARIpol

al/il

is a Lie algebra under the ari bracket. Since ARIpol is easily seen to be a Lie algebra under the ari bracket, the main theorem follows immediately from

Theorem

ARIal/il is a Lie algebra.

Adriana Salerno (joint work with Leila Schneps) On the double shuffle Lie algebra structure: Ecalle’s approach

The double shuffle Lie algebra Ecalle’s theory of Moulds A new proof of Racinet’s theorem

The Lie group GARI

We denote by expari the standard exponential map on the Lie algebra ARI. Thus, expari is an isomorphism from ARI to GARI, the associated group of all moulds with constant term 1. GARI acts on its Lie algebra ARI by the standard adjoint action, denoted adari.

Adriana Salerno (joint work with Leila Schneps) On the double shuffle Lie algebra structure: Ecalle’s approach

The double shuffle Lie algebra Ecalle’s theory of Moulds A new proof of Racinet’s theorem

The tools

Ecalle defines a mould pal ∈ ARI explicitly in terms of binomial coefficients and Bernoulli numbers. Let pic in GARI be the mould defined by pic(v1, . . . , vr) = 1 v1 · · · vr . For every mould M ∈ GARI Ecalle defines an automorphism ganitM : ARI → ARI.

Theorem

If A is alternal then ganit(pic)(A) is alternil.

Adriana Salerno (joint work with Leila Schneps) On the double shuffle Lie algebra structure: Ecalle’s approach

The double shuffle Lie algebra Ecalle’s theory of Moulds A new proof of Racinet’s theorem

The big idea

To prove that ARIal/il is a Lie algebra we prove ARIal/il = adari(pal).ARIal/al.

Adriana Salerno (joint work with Leila Schneps) On the double shuffle Lie algebra structure: Ecalle’s approach

The double shuffle Lie algebra Ecalle’s theory of Moulds A new proof of Racinet’s theorem

Proof

swap(adari(pal)(A)) = ganit(pic) · adari(pil).swap(A)

Adriana Salerno (joint work with Leila Schneps) On the double shuffle Lie algebra structure: Ecalle’s approach

The double shuffle Lie algebra Ecalle’s theory of Moulds A new proof of Racinet’s theorem

Thank you!

Adriana Salerno (joint work with Leila Schneps) On the double shuffle Lie algebra structure: Ecalle’s approach