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 Q � x , 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 + 2 xyy . 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 Q � x , y � both ending in y , we can write them uniquely as words in the letters y i = x i − 1 y . The stuffle product of u , v is defined by st ( u , v ) = u if v = 1 and v is u = 1, and st ( y i u , y j v ) = { y i st ( u , y j v ) } ∪ { y j st ( y i u , v ) } ∪ { y i + j st ( u , v ) } , where y i and y j are respectively the first letters of the words u and v written in the y j . For example: st ( xy , xy ) = st ( y 2 , y 2 ) = y 2 y 2 + y 2 y 2 + y 4 = 2 xyxy + x 3 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 The double shuffle space We define the set d s as the set of polynomials in Q � x , y � such that � ( f | w ) = 0 w ∈ sh ( u , v ) and � ( f | w ) = 0 w ∈ st ( u , v ) 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 ] x 2 y − xyx − xyx + yx 2 + xy 2 − yxy − yxy + y = x 2 y − 2 xyx + yx 2 + xy 2 − 2 yxy + y 2 x = 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 D f of Lie [ x , y ] by D f ( x ) = 0 , D f ( y ) = [ y , f ] on the generators. Define the Poisson bracket on Lie [ x , y ] by { f , g } = [ f , g ] + D f ( g ) − D g ( 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 [ D f , D g ] = D f ◦ D g − D g ◦ D f = 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 d s 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 ( u 1 , u 2 , . . . , u r ) for each r ≥ 0. We will restrict our attention to rational functions with coefficients in Q . We will also consider moulds on the alphabet { v 1 , v 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 New point of view: Moulds A mould is a family of functions M ( u 1 , u 2 , . . . , u r ) for each r ≥ 0. We will restrict our attention to rational functions with coefficients in Q . We will also consider moulds on the alphabet { v 1 , v 2 , . . . } . ARI (resp. ARI ) will denote the vector space (for obvious addition and scalar multiplication laws) of moulds M in the u i (resp. v i ) satisfying M 0 ( ∅ ) = 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 )( u 1 , . . . , u r ) = A ( v r , v r − 1 − v r , v r − 2 − v r − 1 , . . . , v 1 − v 2 ) And its inverse (also called swap) is as follows: swap ( A )( v 1 , . . . , v r ) = A ( u 1 + · · · + u r , u 1 + · · · + u r − 1 , . . . , u 1 + u 2 , u 1 ) 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 Q � x , y � . ◮ Let π y ( 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 π y ( f r ) = � c a i ,..., a r x a 1 − 1 y · · · x a r − 1 y . ◮ Define a mould mi f by c a i ,..., a r v a 1 − 1 · · · v a r − 1 � mi f ( v 1 , . . . , v r ) = 1 r for r ≥ 0 and ma f ( u 1 , . . . , u r ) = swap ( mi f )( v 1 , . . . , v r ) = mi f ( u 1 + · · · + u r , . . . , u 1 + u 2 , u 1 ) 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 ] = x 2 y − 2 xyx + yx 2 + xy 2 − 2 yxy + y 2 x , which projects to x 2 y + xy 2 − 2 yxy , is associated to the mould M = ma f given by M ( ∅ ) = 0 u 2 M ( u 1 ) = 1 M ( u 1 , u 2 ) = − u 1 + u 2 M ( u 1 , · · · , u r ) = 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 � M ( w ) = 0 , w ∈ sh (( u 1 ,..., u i ) , ( u i +1 ,..., u r )) for 1 ≤ i ≤ ⌊ r 2 ⌋ . ARI al = { alternal moulds } ARI al / 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 ( ∅ ) = 0 u 2 M ( u 1 ) = 1 M ( u 1 , u 2 ) = − u 1 + u 2 M ( u 1 , · · · , u r ) = 0 if r > 2. Notice M ( u 1 , u 2 ) + M ( u 2 , u 1 ) = 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 1 1 0 = M ( v 1 , v 2 ) + M ( v 2 , v 1 ) + M ( v 1 ) + M ( v 2 ) . v 1 − v 2 v 2 − v 1 Lemma Let f a polynomial. 1. f satisfies shuffle iff ma f is alternal. 2. f satisfies stuffle iff mi f is alternil. Adriana Salerno (joint work with Leila Schneps) On the double shuffle Lie algebra structure: Ecalle’s approach
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