ON THE PRE-LIE ALGEBRA OF SPECIFIED FEYNMAN GRAPHS

ON THE PRE-LIE ALGEBRA OF SPECIFIED FEYNMAN GRAPHS MOHAMED BELHAJ MOHAMED Abstract. We study the pre-Lie algebra of specified Feynman graphs VeT and we define a pre-Lie structure on its doubling space FeT . We prove that FeT is pre-Lie module on VeT and we find some relations between the two pre-Lie structures. Also, we study the enveloping algebras f′ T , ⋆, Φ) and (H f′ T , ⋆, Ψ) and we prove that of two pre-Lie algebras denoted respectively by (D f′ T , ⋆, Φ) is a module-bialgebra on (H f′ T , ⋆, Ψ). (D

MSC Classification: 05C90, 81Q30, 16T05, 16T15, 16S30. Keywords: Bialgebra, Hopf algebra, Feynman graphs, Pre-Lie algebra, Enveloping algebra, Comodule-coalgebra, Module-bialgebra, Doubling bialgebra.

Contents 1. Introduction 2. Feynman graphs 2.1. Basic definitions 2.2. Quantum field theory and specified graphs 2.3. Hopf algebras of Feynman graphs 3. Pre-Lie structure on specified Feynman graphs 3.1. Pre-Lie algebras 3.2. Pre-Lie algebra of specified graphs 4. Doubling pre-Lie algebra of specified Feynman graphs 4.1. Doubling bialgebra of specified graphs 4.2. Doubling pre-Lie algebra of specified graphs 4.3. Pre-Lie module 4.4. Relation between → and ⊙ 5. Enveloping algebra of pre-Lie algebra 5.1. Enveloping algebra of the pre-Lie algebra of specified Feynman graphs 5.2. Enveloping algebra of doubling pre-Lie algebra of specified graphs f′ T , ⋆, Φ) and (H f′ T , ⋆, Ψ) 5.3. Relation between (D References Date: December 2018. 1

2 4 4 5 5 6 6 7 8 9 9 11 12 13 14 15 16 19

2

MOHAMED BELHAJ MOHAMED

1. Introduction Hopf algebras of Feynman graphs have been studied by A. Connes and D. Kreimer in [8], [9], [10]and [12] as a powerful tool to explain the combinatorics of renormalization in quantum field theory, and it appeared thereafter that in the works of K. Ebrahimi-Fard, D. Manchon [16, 17], van Suijlekom [21, 22] and many others. In [4], we have introduced the concept of doubling bialgebra of specified Feynman graphs to give a sense to some divergent integrals given by a Feynman graphs. It is given by the eT spanned by the pairs (Γ, ¯ γ¯ ) of locally 1P I specified graphs, with γ¯ ⊂ Γ ¯ and vector space D ¯ γ∈H eT . The product m is again given by juxtaposition: Γ/¯
¯ 1 , γ¯1 ) ⊗ (Γ ¯ 2 , γ¯2 ) = (Γ ¯ 1Γ ¯ 2 , γ¯1 γ¯2 ), m (Γ

and the coproduct ∆ is defined as follows: X ¯ ⊗ (Γ/ ¯ γ¯ /δ). ¯ ¯ δ) ¯ δ, ¯ γ¯ ) = (Γ, ∆(Γ, ¯ γ δ⊆¯ ¯ γ ¯ /δ∈T

We have also studied, in collaboration with Dominique Manchon [3], the notion of doubling bialgebra in the context of rooted trees, we have defined the doubling bialgebras of rooted trees given by extraction contraction and admissible cuts, and we have shown the existence of many relations between these two structures. Pre-Lie algebra of insertion was studied by A. Connes and D. Kreimer [8] in the context of Feynman graphs and F. Chapoton and M. Livernet in context of rooted trees [7] as being the space of primitive elements in the graded dual of a right-sided Hopf algebra. The Hopf algebra of specified Feynman graphs is also right-sided so we can construct a pre-Lie structure on its graded dual. In this article, we star by describing this pre-Lie structure. On the space of connected specified Feynman graphs VeT we define the pre-Lie product ⊲ by insertion of graphs. For all ¯ 1 = (Γ1 , i), Γ ¯ 2 = (Γ2 , j) ∈ VeT we have: Γ (Γ1 , i) ⊲ (Γ2 , j) :=

X

(Γ1 , i)⊲v (Γ2 , j),

v∈V(Γ2 )

where (Γ1 , i)⊲v (Γ2 , j) is the graph obtained by replacing the vertex v by the graph Γ1 in Γ2 . In the third section, we find a pre-Lie structure on the doubling space of connected specified ¯ 1 , γ¯1 ), (Γ ¯ 2 , γ¯2 ) in FeT by: graphs VeT noted FeT . The pre-Lie product is defined for all (Γ X ¯ 1 , γ¯1 ) ⊙ (Γ ¯ 2 , γ¯2 ) := ¯ 1 ⊲v Γ ¯ 2 , γ¯1 γ¯2 ), (Γ (Γ v∈V(Γ2 −γ2 ) ,v=res Γ1

where v ∈ V(Γ2 − γ2 ) denotes that v is a vertex of Γ2 but its not a vertex of γ2 and res Γ1 denotes the residue of the graph Γ1 . We also prove that FeT is a pre-Lie module on VeT where

ON THE PRE-LIE ALGEBRA OF SPECIFIED FEYNMAN GRAPHS

3

¯ 1 ∈ VeT and (Γ ¯ 2 , γ¯2 ) ∈ FeT by: the action is given for all Γ X ¯ 1 → (Γ ¯ 2 , γ¯2 ) := ¯ 1 ⊲v Γ ¯ 2, Γ ¯ 1 ⊲v γ¯2 ). Γ (Γ v∈V(γ2 ) ,v=res γ1

We also give some relations between the action → and the pre-Lie product ⊙. We show that ¯ 1 ∈ VeT and the action → is a derivation of the algebra (FeT , ⊙). In other words, for any Γ ¯ 2 , γ¯2 ), (Γ ¯ 3 , γ¯3 ) ∈ FeT , we have: (Γ


¯ 1 → (Γ ¯ 2 , γ¯2 ) ⊙ (Γ ¯ 3 , γ¯3 ) = Γ ¯ 1 → (Γ ¯ 2 , γ¯2 ) ⊙ (Γ ¯ 3 , γ¯3 ) + (Γ ¯ 2 , γ¯2 ) ⊙ Γ ¯ 1 → (Γ ¯ 3 , γ¯3 ) , Γ

and we show that the following diagram is commutative: →

VeT ⊗ FeT

eT / F

P2

I⊗P2





eT VeT ⊗ H

⊲

eT / H

which means that the projection on the second component P2 is a morphism of pre-Lie modules. In the last section, we use the method of Oudom and Guin [18] to find the enveloping algebra of the pre-Lie algebra. We start by finding the enveloping algebra of the pre-Lie algebra of specified Feynman graphs. Starting from the pre-Lie algebra (VeT , ⊲), we consider f′ T := S(VeT ) equipped with its usual unshuffling coproduct Ψ the Hopf symmetric algebra H and a product ⋆ coming from the pre-Lie structure: X ¯⋆Γ ¯′ = ¯ (1) (Γ ¯ (2) ⊲ Γ ¯ ′ ). Γ Γ (Γ)

f′ T , ⋆, Ψ) is a Hopf algebra which is isomorphic to the enveloping By construction, the space (H f′ T , m, Ψ) is a comodule-coalgebra algebra U(VeLie ) of the pre-Lie algebra VeLie . We prove that (H eT , m, ∆), where H eT and H f′ T are isomorphic as algebras. on (H Similarly, we construct the enveloping algebra of doubling pre-Lie algebra of specified Feynman graphs. Starting from the pre-Lie algebra (FeT , ⊙), we consider the Hopf symmetric f′ T := S(FeT ) equipped with its usual unshuffling coproduct φ and the product ⋆ algebra D coming from the pre-Lie structure: X
¯ γ¯ )⋆(Γ ¯ ′ , γ¯ ′ ) = ¯ γ¯ )(1) (Γ, ¯ γ¯ )(2) ⊙ (Γ′ , γ ′ ) . (Γ, (Γ, (Γ,γ)

f′ T , ⋆, φ) is a Hopf algebra which is isomorphic to the envelopBy construction, the space (D f′ T , m, Φ) is ing Hopf algebra U(FeLie ) of the Lie algebra FeLie , and we finish by proving that (D eT , m, ∆), where D eT and D f′ T are isomorphic as algebras. a comodule-coalgebra on (D f′ T , ⋆, Φ) and (H f′ T , ⋆, Ψ), At the end, we give a relation between the two hopf structures (D f′ T , ⋆, Φ) is a module-bialgebra on (H f′ T , ⋆, Ψ). we prove that (D

4

MOHAMED BELHAJ MOHAMED

Acknowledgements: I would like to thank Dominique Manchon for support and advice.

2. Feynman graphs 2.1. Basic definitions. A Feynman graph is a graph with a finite number of vertices and edges, which can be internal or external. An internal edge is an edge connected at both ends to a vertex, an external edge is an edge with one open end, the other end being connected to a vertex. The edges are obtained by using half-edges. More precisely, let us consider two finite sets V and E. A graph Γ with V (resp. E) as set of vertices (resp. half-edges) is defined as follows: let σ : E −→ E be an involution and ∂ : E −→ V. For any vertex v ∈ V we denote by st(v) = {e ∈ E/∂(e) = v} the set of half-edges adjacent to v. The fixed points of σ are the external edges and the internal edges are given by the pairs {e, σ(e)} for e 6= σ(e). The graph Γ associated to these data is obtained by attaching half-edges e ∈ st(v) to any vertex v ∈ V, and joining the two half-edges e and σ(e) if σ(e) 6= e. Several types of half-edges will be considered later on: the set E is partitioned into several pieces Ei . In that case we ask that the involution σ respects the different types of half-edges, i.e. σ(Ei ) ⊂ Ei . We denote by I(Γ) the set of internal edges and by Ext(Γ) the set of external edges. The loop number of a graph Γ is given by: L(Γ) = |I(Γ)| − |V(Γ)| + |π0 (Γ)| , where π0 (Γ) is the set of connected components of Γ. A one-particle irreducible graph (in short, 1P I graph) is a connected graph which remains connected when we cut any internal edge. A disconnected graph is said to be locally 1P I if any of its connected components is 1P I. A covering subgraph of Γ is a Feynman graph γ (not necessarily connected), obtained from Γ by cutting internal edges. In other words: (1) (2) (3) (4)

V(γ) = V(Γ). E(γ) = E(Γ). σΓ (e) = e =⇒ σγ (e) = e. If σγ (e) 6= σΓ (e) then σγ (e) = e and σγ (σΓ (e)) = σΓ (e).

For any covering subgraph γ, the contracted graph Γ/γ is defined by shrinking all connected components of γ inside Γ onto a point. The residue of the graph Γ, denoted by res Γ, is the contracted graph Γ/Γ. The skeleton of a graph Γ denoted by sk Γ is the graph obtained by cutting all internal edges.

ON THE PRE-LIE ALGEBRA OF SPECIFIED FEYNMAN GRAPHS

5

2.2. Quantum field theory and specified graphs. We will work inside a physical theory T (ϕ3 , ϕ4 , QED, QCD etc). The particular form of the Lagrangian leads to consider certain types of vertices and edges. A difficulty appears: the type of half-edges of st(v) is not sufficient to determine the type of the vertex v. We denote by E(T ) the set of possible types of half-edges and by V(T ) the set of possible types of vertices. Example 1. E(ϕ3 ) = { V(ϕ3 ) = {

,

, 0

} ,

E(QED) = {

} ,

V(QED) = {

1

,

}.

, 0

, 1

}.

, 1

Definition 1. A specified graph of theory T is a couple (Γ, i) where: (1) Γ is a locally 1P I superficially divergent graph (the residue of Γ is an element of T ) with half-edges and vertices of the type prescribed in T . (2) i : π0 (Γ) −→ N, the values of i(γ) being prescribed by the possible types of vertex obtained by contracting the connected component γ on a point.
We will say that (γ, j) is a specified covering subgraph of (Γ, i), and we note (γ, j) ⊂ (Γ, i) if: (1) γ is a covering subgraph of Γ. (2) if γ0 is a full connected component of γ, i.e if γ0 is also a connected component of Γ, then j(γ0 ) = i(γ0 ). ¯ ¯ = (Γ, i) the specified graph, and we will write γ¯ ⊂ Γ Remark 1. Sometimes we denote by Γ for (γ, j) ⊂ (Γ, i). Definition 2. Let be (γ, j) ⊂ (Γ, i). The contracted specified subgraph is written: Γ/¯ γ = (Γ/¯ γ , i), ¯ γ is obtained by contracting each connected component of γ on a point, and specifying where Γ/¯ the vertices obtained with j. ¯ and the contracted graph Γ/¯ ¯ γ. Remark 2. The specification i is the same for the graph Γ 2.3. Hopf algebras of Feynman graphs. Let VeT be the vector space generated by 1P I connected specified graphs Γ with edges in E(T ) and vertices in V(T ) such that res Γ is a eT = S(VeT ) be the vector vertex in VT (condition of superficial divergence [1], [8], [12]). Let H space generated by the specified superficially divergent Feynman graphs of a field theory T . The product is given by disjoint union, the unit 1 is identified with the empty graph and the coproduct is defined by: X ¯ = ¯ γ, ∆(Γ) γ¯ ⊗ Γ/¯ ¯ γ ¯ ⊆Γ ¯ γ ∈T Γ/¯

¯ = (Γ, i), where the sum runs over all locally 1P I specified covering subgraphs γ¯ = (γ, j) of Γ such that the contracted subgraph (Γ/(γ, j), i) is in the theory T .

6

MOHAMED BELHAJ MOHAMED

¯ γ ∈ T is crucial, and means also that γ¯ is a ”superficially Remark 3. The condition Γ/¯ divergent” subgraph. For example, in ϕ3 : Γ= gives Γ/γ =

and γ =

,

by contraction, which must be eliminated because of the tetravalent vertex.

Example 2. In ϕ3 Theory: ∆(

, 0) = (

+

, 0) ⊗

⊗(

, 0)

0

+ (

, 0)

, 0) ⊗ ( 0

+ (

, 0).

, 1) ⊗ ( 1

In QED: ∆(

⊗(

, 1) =

+ (

, 1) ⊗

, 1)

, 0) ⊗ (

+( 1

+ (

, 1) ⊗ (

1

0

, 1)

, 1).

Theorem 1. [2] The coproduct ∆ is coassociative. The Hopf algebra HT is given by identifying all graphs of degree zero (the residues) to unit 1: eT /J HT = H

(1)

¯ where Γ ¯ is an 1P I specified graph. One where J is the ideal generated by the elements 1−res Γ immediately checks that J is a bi-ideal. HT is a connected graded bialgebra, it is therefore a connected graded Hopf algebra. The coproduct then becomes: X ¯ =1⊗Γ ¯+Γ ¯⊗1+ ¯ γ, (2) ∆(Γ) γ¯ ⊗ Γ/¯ ¯ γ ¯ proper subgraph of Γ ¯ γ ∈T loc 1PI. Γ/¯

3. Pre-Lie structure on specified Feynman graphs 3.1. Pre-Lie algebras. Definition 3. A Lie algebra over a field k is a vector space V endowed with a bilinear bracket [., .] satisfying:

ON THE PRE-LIE ALGEBRA OF SPECIFIED FEYNMAN GRAPHS

7

(1) the antisymmetry: [x, y] = −[y, x]

∀x, y ∈ V.

(2) the Jacobi identity: [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0

∀x, y, z ∈ V.

Definition 4. [6, 15] A left pre-Lie algebra over a field k is a k-vector space A with a binary composition ⊲ that satisfies the left pre-Lie identity: (3)

(x ⊲ y) ⊲ z − x ⊲ (y ⊲ z) = (y ⊲ x) ⊲ z − y ⊲ (x ⊲ z),

for all x, y, z ∈ A. Analogously, a right pre-Lie algebra is a k-vector space A with a binary composition ⊳ that satisfies the right pre-Lie identity: (4)

(x ⊳ y) ⊳ z − x ⊳ (y ⊳ z) = (x ⊳ z) ⊳ y − x ⊳ (z ⊳ y).

As any right pre-Lie algebra (A, ⊳) is also a left pre-Lie algebra with product x ⊲ y := y ⊳ x, we will only consider left pre-Lie algebras for the moment. The left pre-Lie identity rewrites as: (5)

L[x,y] = [Lx , Ly ],

where Lx : A −→ A is defined by Lx y = a ⊲ b, and where the bracket on the left-hand side is defined by [a, b] := a ⊲ b − b ⊲ a. As a consequence this bracket satisfies the Jacobi identity. 3.2. Pre-Lie algebra of specified graphs. ¯ 1 = (Γ1 , i) and Γ ¯ 2 = (Γ2 , j) be two connected specified graphs, i.e Γ ¯1 = Definition 5. Let Γ ¯ 2 = (Γ2 , j) ∈ VeT . We defined the insertion of Γ ¯ 1 at v ∈ V(Γ ¯ 2 ) by: (Γ1 , i), Γ ( ¯1 (Γ1 ∪v Γ2 , j) if v = res Γ (6) (Γ1 , i)⊲v (Γ2 , j) := 0 ifnot, where Γ1 ∪v Γ2 is a sum of all possibles graphs obtained by replacing the vertex v by the graph Γ1 in Γ2 . ¯ 1 in Γ ¯ 2 by: We define then the insertion of Γ X (Γ1 , i)⊲v (Γ2 , j). (7) (Γ1 , i) ⊲ (Γ2 , j) := v∈V(Γ2 )

Example 3. In ϕ3 Theory: (

, 0) = 2(

, 1) ⊲ (

, 0).

1

In QED: (

, 1) ⊲ (

1

, 0) = 2(

Theorem 2. Equiped by ⊲, the space VeT is a pre-Lie algebra.

, 0).

8

MOHAMED BELHAJ MOHAMED

Proof. Let (Γ1 , i), (Γ2 , j) and (Γ3 , k) three elements of VeT , we have:

(Γ1 , i) ⊲ [(Γ2 , j) ⊲ (Γ3 , k)] − [(Γ1 , i) ⊲ (Γ2 , j)] ⊲ (Γ3 , k) X X [(Γ1 , i)⊲s (Γ2 , j)] ⊲ (Γ3 , k) = (Γ1 , i) ⊲ [(Γ2 , j)⊲v (Γ3 , k)] − ¯1 s∈V(Γ2 ) , s=res Γ

v∈V(Γ3 )

=

X

¯2 v∈V(Γ3 ); v=res Γ

=

X

(Γ1 , i) ⊲ (Γ2 ∪v Γ3 , k) −

X

(Γ1 , i)⊲s (Γ2 ∪v Γ3 , k) −

X

(Γ1 ∪s [Γ2 ∪v Γ3 ], k) −

v∈V(Γ3 ) , s∈V(Γ2 ∪v Γ3 ) ¯ 2 , s=res Γ ¯1 v=res Γ

= X

([Γ1 ∪s Γ2 ]⊲v Γ3 , k)

X

([Γ1 ∪s Γ2 ]∪v Γ3 , k)

v∈V(Γ3 ) , s∈V(Γ2 ) ¯ 2 , s=res Γ ¯1 v=res Γ

([Γ1 ∪s Γ2 ]∪v Γ3 , k) +

X

(Γ1 ∪s [Γ2 ∪v Γ3 ], k) −

s,v∈V(Γ3 ) , s6=v ¯ 2 , s=res Γ ¯1 v=res Γ

v∈V(Γ3 ) , s∈V(Γ2 ) ¯ 2 , s=res Γ ¯1 v=res Γ

=

X

v∈V(Γ3 ) , s∈V(Γ2 ) ¯ 2 , s=res Γ ¯1 v=res Γ

v∈V(Γ3 ) , s∈V(Γ2 ∪v Γ3 ) ¯ 2 , s=res Γ ¯1 v=res Γ

=

(Γ1 ∪s Γ2 , j) ⊲ (Γ3 , k)

¯1 s∈V(Γ2 ) , s=res Γ

X

(Γ1 ∪s [Γ2 ∪v Γ3 ], k)

X

(Γ2 ∪v [Γ1 ∪s Γ3 ], k)

X

([Γ1 ∪s Γ2 ]∪v Γ3 , k)

v∈V(Γ3 ) , s∈V(Γ2 ) ¯ 2 , s=res Γ ¯1 v=res Γ

s,v∈V(Γ3 ) , s6=v ¯ 2 , s=res Γ ¯1 v=res Γ

=

s,v∈V(Γ3 ) , s6=v ¯ 2 , s=res Γ ¯1 v=res Γ

= (Γ2 , i) ⊲ [(Γ1 , j) ⊲ (Γ3 , k)] − [(Γ2 , i) ⊲ (Γ1 , j)] ⊲ (Γ3 , k), which proves the theorem.



Remark 4. We see that the Hopf algebra HT is right-sided, i.e, we have: ∆(VeT ) ⊂ HT ⊗ VeT .

Then the dual graded of VeT is a left pre-Lie algebra [14].

¯ 1 = (Γ1 , i) and Γ ¯ 2 = (Γ2 , j) two connected specified graphs, and we define Corollary 1. Let be Γ   the bracket , by: (8)    is a Lie algebra. VeT , ,

¯ 1, Γ ¯ 2 ] := Γ ¯1 ⊲ Γ ¯2 − Γ ¯2 ⊲ Γ ¯ 1. [Γ

4. Doubling pre-Lie algebra of specified Feynman graphs

We have studied the concept of doubling bialgebra of specified Feynman graphs [4, §3] to give a sense to some divergent integrals given by Feynman graphs, We have also studied the notion of doubling bialgebra in the context of rooted trees, we have defined the doubling bialgebras of rooted trees given by extraction contraction and admissible cuts, and we have shown the existence of many relations between these two structures [3]. In this section we define a pre-Lie

ON THE PRE-LIE ALGEBRA OF SPECIFIED FEYNMAN GRAPHS

9

structure on the doubling space of connected specified graphs VeT noted FeT . We prove that FeT is pre-Lie module on VeT and we find some relations between the pre-Lie structures on FeT and VeT . 4.1. Doubling bialgebra of specified graphs.

eT be the vector space spanned by the pairs (Γ, ¯ γ¯ ) of locally 1P I specified Definition 6. [4] Let D ¯ and Γ/¯ ¯ γ ∈ H eT . This is the free commutative algebra generated by the graphs, with γ¯ ⊂ Γ corresponding connected objects. The product m is again given by juxtaposition:
¯ 1 , γ¯1 ) ⊗ (Γ ¯ 2 , γ¯2 ) = (Γ ¯ 1Γ ¯ 2 , γ¯1 γ¯2 ), (9) m (Γ and the coproduct ∆ is defined as follows: X ¯ ⊗ (Γ/ ¯ γ¯ /δ) ¯ ¯ δ) ¯ δ, ¯ γ¯ ) = (Γ, (10) ∆(Γ, ¯ γ δ⊆¯ ¯ γ ¯ /δ∈T

eT , m, ∆, u, ε) is a graded bialgebra, and Proposition 1. [4] (D eT −→ H eT P2 : D ¯ γ¯ ) 7−→ γ¯ (Γ,

is a bialgebra morphism. 4.2. Doubling pre-Lie algebra of specified graphs. Let FeT be the vector space spanned ¯ γ¯ ) where Γ is a 1P I connected specified graph, γ¯ ⊂ Γ ¯ and Γ/¯ ¯ γ ∈ VeT . by the pairs (Γ, ¯ 1 , γ¯1 ) and (Γ ¯ 2 , γ¯2 ) two elements of FeT , we defined the map ⊙ by: Definition 7. Let be (Γ X ¯ 1 , γ¯1 ) ⊙ (Γ ¯ 2 , γ¯2 ) := ¯ 1 ⊲v Γ ¯ 2 , γ¯1 γ¯2 ), (11) (Γ (Γ v∈V(Γ2 −γ2 ) ,v=res Γ1

where the notation v ∈ V(Γ2 − γ2 ) denotes that v is a vertex of Γ2 but is not a vertex of γ2 . Example 4. In QED: 

(

, 0), (

, 1)





⊙ (

0

0



= 2 (

, 1), (

0

0



+ 2 (

0

0

, 1), (

0

0

, 1), (

Theorem 3. Equiped by ⊙, the space FeT is a pre-Lie algebra.



, 0) , 1)(

, 1)(



, 0)

0

0



, 0)

10

MOHAMED BELHAJ MOHAMED

¯ 1 , γ¯1 ), (Γ ¯ 2 , γ¯2 ) and (Γ ¯ 3 , γ¯3 ) be three elements of FeT , we have: Proof. Let (Γ X  
¯ 1 , γ¯1 ) ⊙ (Γ ¯ 2 , γ¯2 ) ⊙ (Γ ¯ 3 , γ¯3 ) = (Γ ¯ 1 , γ¯1 ) ⊙ ¯ 2 ⊲v Γ ¯ 3 , γ¯2 γ¯3 ) (Γ (Γ v∈V(Γ3 −γ3 ) ,v=res Γ2

=

X

¯ 1 ⊲r (Γ ¯ 2 ⊲v Γ ¯ 3 ), γ¯1 γ¯2 γ¯3 Γ

v∈V(Γ3 −γ3 ) r∈V(Γ2 ⊲v Γ3 −¯ γ2 γ ¯3 ) v=res Γ2 , r=res Γ1

X

=

¯ 1 ⊲r (Γ ¯ 2 ⊲v Γ ¯ 3 ), γ¯1 γ¯2 γ¯3 Γ

r , v∈V(Γ3 −γ3 ) r6=v v=res Γ2 , r=res Γ1

+

X

v∈V(Γ3 −γ3 ) r∈V(Γ2 −¯ γ2 ) v=res Γ2 , r=res Γ1








¯ 1 ⊲r ( Γ ¯ 2 ⊲v Γ ¯ 3 ), γ¯1 γ¯2 γ¯3 . Γ

On the other hand we have: 

X

 ¯ 1 , γ¯1 ) ⊙ (Γ ¯ 2 , γ¯2 ) ⊙ (Γ ¯ 3 , γ¯3 ) = (Γ

¯ 1 ⊲r Γ ¯ 2 , γ¯1 γ¯2 ) ⊙ (Γ ¯ 3 , γ¯3 ) (Γ

r∈V(Γ2 −γ2 ) ,r=res Γ1

=

X

¯ 1 ⊲r Γ ¯ 2 )⊲v Γ ¯ 3 , γ¯1 γ¯2 γ¯2 (Γ

X


¯ 1 ⊲r Γ ¯ 2 )⊲v Γ ¯ 3 , γ¯1 γ¯2 γ¯3 . (Γ

v∈V(Γ3 −γ3 ) r∈V(Γ2 −¯ γ2 ) ¯ 1 ⊲r Γ ¯ 2 , r=res Γ1 v=res Γ

=

v∈V(Γ3 −γ3 ) r∈V(Γ2 −¯ γ2 ) ¯ 2 , r=res Γ1 v=res Γ




Then we have:     ¯ 1 , γ¯1 ) ⊙ (Γ ¯ 2 , γ¯2 ) ⊙ (Γ ¯ 3 , γ¯3 ) − (Γ ¯ 1 , γ¯1 ) ⊙ (Γ ¯ 2 , γ¯2 ) ⊙ (Γ ¯ 3 , γ¯3 ) (Γ X
¯ 1 ⊲r (Γ ¯ 2 ⊲v Γ ¯ 3 ), γ1 ∪ γ¯2 γ¯3 = Γ r , v∈V(Γ3 −γ3 ) r6=v v=res Γ2 , r=res Γ1

X

+

¯ 1 ⊲r ( Γ ¯ 2 ⊲v Γ ¯ 3 ), γ1 ∪ γ¯2 γ¯3 Γ

v∈V(Γ3 −γ3 ) r∈V(Γ2 −¯ γ2 ) v=res Γ2 , r=res Γ1

X

−

¯ 1 ⊲r Γ ¯ 2 )⊲v Γ ¯ 3 , γ¯1 γ¯2 γ¯3 (Γ

v∈V(Γ3 −γ3 ) r∈V(Γ2 −¯ γ2 ) ¯ 2 , r=res Γ1 v=res Γ

X

=

¯ 1 ⊲r (Γ ¯ 2 ⊲v Γ ¯ 3 ), γ1 γ¯2 γ¯3 Γ

r , v∈V(Γ3 −γ3 ) r6=v v=res Γ2 , r=res Γ1



¯ 2 , γ¯2 ) ⊙ (Γ ¯ 1 , γ¯1 ) ⊙ (Γ ¯ 3 , γ¯3 ) (Γ Consequently, ⊙ is pre-Lie.



= −



 ¯ 2 , γ¯2 ) ⊙ (Γ ¯ 1 , γ¯1 ) ⊙ (Γ ¯ 3 , γ¯3 ). (Γ












ON THE PRE-LIE ALGEBRA OF SPECIFIED FEYNMAN GRAPHS

11

4.3. Pre-Lie module. Definition 8. Let (A, ◦) be pre-Lie algebra. A left A-module is a Vector space M provided with a bilinear law noted ≻: A ⊗ M −→ M such that for all a, b ∈ A and m ∈ M, we have: (12)

a ≻ (b ≻ m) − (a ◦ b) ≻ m = b ≻ (a ≻ m) − (b ◦ a) ≻ m.

¯ 1 ∈ VeT and (Γ ¯ 2 , γ¯2 ) ∈ FeT , we define the map → by: Definition 9. Let Γ (13)

X

¯ 1 → (Γ ¯ 2 , γ¯2 ) := Γ

¯ 1 ⊲v Γ ¯ 2, Γ ¯ 1 ⊲v γ¯2 ). (Γ

v∈V(γ2 ) ,v=res Γ1

Theorem 4. Equiped by →, the space FeT is a pre-Lie module on VeT . In other words for any ¯ 1, Γ ¯ 2 ∈ VeT and (Γ ¯ 3 , γ¯3 ) ∈ FeT , we have: Γ h i h i ¯1 → Γ ¯ 2 → (Γ ¯ 3 , γ¯3 ) − (Γ ¯1 ⊲ Γ ¯ 2 ) → (Γ ¯ 3 , γ¯3 ) = Γ ¯2 → Γ ¯ 1 → (Γ ¯ 3 , γ¯3 ) − (Γ ¯2 ⊲ Γ ¯ 1 ) → (Γ ¯ 3 , γ¯3 ). Γ

¯ 1, Γ ¯ 2 be two elements of VeT and (Γ ¯ 3 , γ¯3 ) an element of FeT , we have: Proof. Let Γ h i h ¯1 → Γ ¯ 2 → (Γ ¯ 3 , γ¯3 ) = Γ ¯1 → Γ

X

s∈V(γ3 ) ,v=res Γ2

X

=

¯ 2 ⊲s Γ ¯ 3, Γ ¯ 2 ⊲s γ¯3 ) (Γ

i

¯ 1 ⊲l ( Γ ¯ 2 ⊲s Γ ¯ 3 ), Γ ¯ 1 ⊲l (Γ ¯ 2 ⊲s γ¯3 )] [Γ

s∈V(γ3 ) l∈V(Γ2 ⊲s γ3 ) s=res Γ2 , l=res Γ1

X

=

¯ 3 ), Γ ¯ 1 ⊲l (Γ ¯ 2 ⊲s γ¯3 )] ¯ 1 ⊲l ( Γ ¯ 2 ⊲s Γ [Γ

l , s∈V(γ3 ) v6=s s=res Γ2 , l=res Γ1

+

X

¯ 1 ⊲l ( Γ ¯ 2 ⊲s Γ ¯ 3 ), Γ ¯ 1 ⊲l (Γ ¯ 2 ⊲s γ¯3 )]. [Γ

l∈V(Γ2 ) , s∈V(γ3 ) s=res Γ2 , l=res Γ1

On the other hand we have: ¯1 ⊲ Γ ¯ 2 ) → (Γ ¯ 3 , γ¯3 ) = (Γ

X

¯ 1 ⊲l Γ ¯ 2 → (Γ ¯ 3 , γ¯3 ) Γ

l∈V(γ2 ) ,l=res Γ1

=

X

l∈V(Γ2 ) s∈V(γ3 ) ¯ 1 ⊲l Γ ¯2 l=res Γ1 , s=res Γ

=

X

l∈V(Γ2 ) s∈V(γ3 ) ¯2 l=res Γ1 , s=res Γ





¯ 1 ⊲l Γ ¯ 2 )⊲s Γ ¯ 3 , (Γ ¯ 1 ⊲l Γ ¯ 2 )⊲s γ¯3 (Γ

 ¯ 1 ⊲l Γ ¯ 2 )⊲s Γ ¯ 3 , (Γ ¯ 1 ⊲l Γ ¯ 2 )⊲s γ¯3 . (Γ



12

MOHAMED BELHAJ MOHAMED

Then, we have: h i h i ¯1 → Γ ¯ 2 → (Γ ¯ 3 , γ¯3 ) − Γ ¯1 ⊲ Γ ¯ 2 → (Γ ¯ 3 , γ¯3 ) Γ X ¯ 1 ⊲l (Γ ¯ 2 ⊲s Γ ¯ 3 ), Γ ¯ 1 ⊲l ( Γ ¯ 2 ⊲s γ¯3 )] = [Γ l , s∈V(γ3 ) v6=s s=res Γ2 , l=res Γ1

X

+

¯ 1 ⊲l (Γ ¯ 2 ⊲s Γ ¯ 3 ), Γ ¯ 1 ⊲l ( Γ ¯ 2 ⊲s γ¯3 )] [Γ

l∈V(Γ2 ) , s∈V(γ3 ) s=res Γ2 , l=res Γ1

X

−

l∈V(Γ2 ) s∈V(γ3 ) ¯2 l=res Γ1 , s=res Γ

=

X



¯ 1 ⊲l Γ ¯ 2 )⊲s Γ ¯ 3 , (Γ ¯ 1 ⊲l Γ ¯ 2 )⊲s γ¯3 (Γ



¯ 1 ⊲l (Γ ¯ 2 ⊲s Γ ¯ 3 ), Γ ¯ 1 ⊲l ( Γ ¯ 2 ⊲s γ¯3 )], [Γ

l , s∈V(γ3 ) v6=s s=res Γ2 , l=res Γ1

¯ 1 and Γ ¯ 2 . Therefore: which is symmetric in Γ h i h i ¯1 → Γ ¯ 2 → (Γ ¯ 3 , γ¯3 ) − (Γ ¯1 ⊲ Γ ¯ 2 ) → (Γ ¯ 3 , γ¯3 ) = Γ ¯2 → Γ ¯ 1 → (Γ ¯ 3 , γ¯3 ) − (Γ ¯2 ⊲ Γ ¯ 1 ) → (Γ ¯ 3 , γ¯3 ), Γ which proves the theorem.



4.4. Relation between → and ⊙. In this section , we prove that there exist relations between the action → and the pre-Lie product ⊙ defined on FeT .

Theorem 5. The law → is a derivation of the algebra (FeT , ⊙). In other words, for any ¯ 1 ∈ VeT and (Γ ¯ 2 , γ¯2 ), (Γ ¯ 3 , γ¯3 ) ∈ FeT , we have: Γ


¯ 1 → (Γ ¯ 2 , γ¯2 ) ⊙ (Γ ¯ 3 , γ¯3 ) = Γ ¯ 1 → (Γ ¯ 2 , γ¯2 ) ⊙ (Γ ¯ 3 , γ¯3 ) + (Γ ¯ 2 , γ¯2 ) ⊙ Γ ¯ 1 → (Γ ¯ 3 , γ¯3 ) . Γ ¯ 1 ∈ VeT and (Γ ¯ 2 , γ¯2 ), (Γ ¯ 3 , γ¯3 ) ∈ FeT , we have: Proof. Let be Γ X

¯ 2 ⊲v Γ ¯ 3 , γ2 γ3 ) ¯ 1 → (Γ ¯ 2 , γ¯2 ) ⊙ (Γ ¯ 3 , γ¯3 ) = Γ ¯1 → (Γ Γ v∈V(Γ3 −γ3 ) ¯2 v=res Γ

=

X

¯ 1 ⊲s (Γ ¯ 2 ⊲v Γ ¯ 3 ), Γ ¯ 1 ⊲s (γ2 γ3 ) Γ

s∈V(γ2 γ3 ) v∈V(Γ3 −γ3 ) ¯ 1 v=res Γ ¯2 s=res Γ

=

X

¯ 1 ⊲s Γ ¯ 2 )⊲v Γ ¯ 3 , (Γ ¯ 1 ⊲s γ2 )γ3 (Γ

s∈V(γ2 ) v∈V(Γ3 −γ3 ) ¯ 1 v=res Γ ¯2 s=res Γ

+

X

s∈V(γ3 ) v∈V(Γ3 −γ3 ) ¯ 1 v=res Γ ¯2 s=res Γ

=

¯ 2 ⊲v (Γ ¯ 1 ⊲s Γ ¯ 3 ), γ¯2 (Γ1 ⊲s γ3 ) Γ












¯ 1 → (Γ ¯ 2 , γ¯2 ) ⊙ (Γ ¯ 3 , γ¯3 ) + (Γ ¯ 2 , γ¯2 ) ⊙ Γ ¯ 1 → (Γ ¯ 3 , γ¯3 ) . Γ



ON THE PRE-LIE ALGEBRA OF SPECIFIED FEYNMAN GRAPHS

13

Theorem 6. The following diagram is commutative: →

VeT ⊗ FeT

eT / F

P2

I⊗P2





eT VeT ⊗ H

eT / H

⊲

In other words, the projection on the second component P2 is a morphism of pre-Lie modules. ¯ 1 ∈ VeT and (Γ ¯ 2 , γ¯2 ) ∈ FeT , we have: Proof. Let be Γ X

¯ 1 → (Γ ¯ 2 , γ¯2 ) = P2 ¯ 1 ⊲v Γ ¯ 2, Γ ¯ 1 ⊲v γ2 ) P2 Γ (Γ v∈V(γ2 ) v=res γ ¯1

X

=

¯ 1 ⊲v γ¯2 Γ

v∈V(γ2 ) ¯1 v=res Γ

¯ 1 ⊲ γ¯2 = Γ ¯ 1 ⊲ P2 (Γ ¯ 2 , γ¯2 ) = Γ


¯ 1 ⊗ (Γ ¯ 2 , γ¯2 ) , = (I ⊗ P2 ) Γ

which proves the theorem.



5. Enveloping algebra of pre-Lie algebra In this section We use the method of Oudom and Guin [18] to describe the enveloping algebra of the pre-Lie algebra. Definition 10. [18] Let (A, ⊲) be a pre-Lie algebra. We consider the Hopf symmetric algebra S(A) equipped with its usual coproduct ∆. We extend the product ⊲ to S(A). Let a, b and c ∈ S(A), and x ∈ A. We put: 1⊲a = a a ⊲ 1 = ε(a)1 (xa) ⊲ b = x ⊲ (a ⊲ b) − (x ⊲ a) ⊲ b X a ⊲ (bc) = (a(1) ⊲ b)(a(2) ⊲ c). a

On S(A), we define a product ⋆ by:

a⋆b =

X

a(1) (a(2) ⊲ b).

a

Theorem 7. The space (S(A), ⋆, ∆)is a Hopf algebra which is isomorphic to the enveloping Hopf algebra U(ALie ) of the Lie algebra ALie . Proof. This theorem was proved by Oudom and Guin in [18].



14

MOHAMED BELHAJ MOHAMED

5.1. Enveloping algebra of the pre-Lie algebra of specified Feynman graphs. We f′ T := S(VeT ) of the pre-Lie algebra (VeT , ⊲), equipped consider the Hopf symmetric algebra H f′ by the same method with its usual unshuffling coproduct Ψ. We extend the product ⊲ to H f′ T by: used in Defintion 10 and we define a product ⋆ on H X ¯⋆Γ ¯′ = ¯ (1) (Γ ¯ (2) ⊲ Γ ¯ ′ ). Γ Γ (Γ)

f′ T , ⋆, Ψ) is a Hopf algebra. By construction, the space (H

Example 5. In QED:

, 1) ⊲ (

(

(

(

, 1)(

1

1


, 0) ⊲ (

, 1)(


, 0) ⋆ (

, 0)

, 0)(

1

0

1




= 2(

1

+ 2(

, 0) = 1 ⊲ (

1

+ (

, 0) ⊲ (

+ ( +

, 0).

1

, 1) ⊲ (

= (

1

0

, 0)(

+ 4( + 2(

, 0) , 1)

, 0)(

0

1

, 1)(

(

, 1)(

, 0)(

0

, 0).

, 0)(

1

, 0) = 4(

0

, 0)

, 0)(

, 0)(

0


, 0) ⊲ (

1

, 1)(

, 0) 0

, 0)

, 0)

, 0) 0

, 0)(

, 1) + 2(

1

, 0)(

f′ T , m, Ψ) is a comodule-coalgebra on (H eT , m, ∆), where H eT and H f′ T are Theorem 8. (H isomorphic as algebras. f′ T −→ H eT ⊗ H f′ T is a coaction, that means that ∆ is coassocitive. Proof. It is clear that ∆ : H f′ T , we have: Let (Γ, i) ∈ H X ¯ = (I ⊗ ∆)( ¯ γ) (I ⊗ ∆) ◦ ∆(Γ) γ¯ ⊗ Γ/¯ =

X

γ⊂Γ

¯ γ δ¯ ⊗ γ¯ /δ¯ ⊗ Γ/¯

δ⊂γ⊂Γ

¯ = (∆ ⊗ I) ◦ ∆(Γ).

, 0).

ON THE PRE-LIE ALGEBRA OF SPECIFIED FEYNMAN GRAPHS

15

f′ T -comodules. This amounts to Second, we prove that the coproduct Ψ is morphism of left H the commutativity of the following diagram: ∆

f′ T H

Ψ

eT ⊗ H f′ T / H I⊗Ψ





f′ T ⊗ H f′ T H

eT ⊗ H f′ T ⊗ H f′ T H O

∆⊗∆

m⊗I



eT ⊗ H f′ T ⊗ H eT ⊗ H f′ T H

eT ⊗ H eT ⊗ H f′ T ⊗ H f′ T / H

τ 23

We use the shorthand notation: m13 := (m ⊗ I) ◦ τ 23 . X ¯ = (I ⊗ Ψ)( ¯ γ) γ¯ ⊗ Γ/¯ (I ⊗ Ψ) ◦ ∆(Γ) =

X

γ⊂Γ

¯ γ) γ¯ ⊗ Ψ(Γ/¯

γ⊂Γ

=

X

¯ γ )(1) ⊗ (Γ/¯ ¯ γ )(2) γ¯ ⊗ (Γ/¯

γ⊂Γ,(Γ/γ)

=

X

¯ (1) /¯ ¯ (2) /¯ γ¯1 γ¯2 ⊗ Γ γ1 ⊗ Γ γ2 .

γ1 ⊂Γ(1) ,γ2 ⊂Γ(2)

In the last passage, we used the fact that a graph and any of its contracted have the same number of connected components. X
¯ = (m13 ⊗ I) ¯ (1) ) ⊗ ∆(Γ ¯ (2) ) (m13 ⊗ I) ◦ (∆ ⊗ ∆) ◦ Ψ(Γ) ∆(Γ (Γ)

= (m13 ⊗ I) =

X

X

¯ (1) /¯ ¯ (2) /¯ γ¯1 ⊗ γ¯2 ⊗ Γ γ1 ⊗ Γ γ2

γ1 ⊂Γ(1) ,γ2 ⊂Γ(2)

¯ (1) /¯ ¯ (2) /¯ γ¯1 γ¯2 ⊗ Γ γ1 ⊗ Γ γ2 ,




γ1 ⊂Γ(1) ,γ2 ⊂Γ(2)

which proves the theorem.



5.2. Enveloping algebra of doubling pre-Lie algebra of specified graphs. Let FeT be ¯ γ¯ ) where Γ is an 1P I connected specified graph, the vector space spanned by the pairs (Γ, ¯ and Γ/¯ ¯ γ ∈ VeT . We use the method of Oudom and Guin [18] to find the enveloping γ¯ ⊂ Γ doubling pre-Lie algebra of specified Feynman graphs. We have (FeT , ⊙) is a pre-Lie algebra,so f′ T := S(FeT ) equipped with its usual unshuffling we consider the Hopf symmetric algebra D f′ T by using Definition 10 and we define a product coproduct Φ. We extend the product ⊙ to D f′ T by: ⋆ on D X
¯ γ¯ )⋆(Γ ¯ ′ , γ¯ ′ ) = ¯ γ¯ )(1) (Γ, ¯ γ¯ )(2) ⊙ (Γ′ , γ ′ ) . (Γ, (Γ, (Γ,γ)

16

MOHAMED BELHAJ MOHAMED

f′ T , ⋆, Φ) is a Hopf algebra. By construction, the space (D

f′ T , m, Φ) is a comodule-coalgebra on (D eT , m, ∆), where D eT and D f′ T are isoTheorem 9. (D morphic as algebras.

f′ T −→ D eT ⊗ D f′ T is a coaction, that means that ∆ is coassocitive. Proof. It is clear taht ∆ : D f′ T -comodules. This amounts Second, we prove that the coproduct Φ is morphism of left D to the commutativity of the following diagram: ∆

f′ T D

Φ

eT ⊗ D f′ T / D I⊗Φ





f′ T ⊗ D f′ T D

eT ⊗ D f′ T ⊗ D f′ T D O

∆⊗∆

m⊗I



eT ⊗ D f′ T ⊗ D eT ⊗ D f′ T D

eT ⊗ D eT ⊗ D f′ T ⊗ D f′ T / D

τ 23



¯ γ¯ ) = (I ⊗ Φ)  (I ⊗ Φ) ◦ ∆(Γ,

X ¯ γ δ⊆¯



¯ ⊗ (Γ/ ¯ γ¯ /δ) ¯ ¯ δ) ¯ δ, (Γ,

X ¯ ⊗ φ(Γ/ ¯ γ¯ /δ) ¯ ¯ δ) ¯ δ, (Γ, = ¯ γ δ⊆¯

=

X

¯ ⊗ (Γ/ ¯ γ¯ /δ) ¯ (1) ⊗ (Γ/ ¯ γ¯ /δ) ¯ (2) . ¯ δ) ¯ δ, ¯ δ, (Γ,

¯ γ ¯ γ ) , δ⊆¯ (Γ,¯



¯ γ¯ ) = (m13 ⊗ I)  (m13 ⊗ I) ◦ (∆ ⊗ ∆) ◦ Φ(Γ,

X

¯ γ) (Γ,¯

= (m13 ⊗ I) =

X





¯ (Γ, ¯ γ¯ )(2)  ¯ γ¯ )(1) ⊗ ∆ ∆ (Γ,

X

¯ (1) ⊗ (Γ/ ¯ γ¯ /δ) ¯ (1) ⊗ (Γ, ¯ (2) ⊗ (Γ/ ¯ γ¯ /δ) ¯ (2) ¯ δ) ¯ δ, ¯ δ) ¯ δ, (Γ,

¯ γ ¯ γ ),δ⊂¯ (Γ,¯

¯ (1) (Γ, ¯ (2) ⊗ (Γ/ ¯ γ¯ /δ) ¯ (1) ⊗ (Γ/ ¯ γ¯ /δ) ¯ (2) ¯ δ) ¯ δ) ¯ δ, ¯ δ, (Γ,

¯ γ ¯ γ ),δ⊂¯ (Γ,¯

=

X

¯ ⊗ (Γ/ ¯ γ¯ /δ) ¯ (1) ⊗ (Γ/ ¯ γ¯ /δ) ¯ (2) , ¯ δ) ¯ δ, ¯ δ, (Γ,

¯ γ ¯ γ ),δ⊂¯ (Γ,¯

which proves the theorem.



f′ T , ⋆, Φ) and (H f′ T , ⋆, Ψ). 5.3. Relation between (D

¯1 ∈ H f′ T and (Γ ¯ 2 , γ¯2 ) ∈ D f′ T , we define two products ⊲g and ⋆g by: Definition 11. Let Γ X ¯ 1 ⊲g Γ ¯ 2 := ¯ 1 ⊲v Γ ¯ 2, Γ (14) Γ v∈V(γ2 ) ,v=res Γ1




ON THE PRE-LIE ALGEBRA OF SPECIFIED FEYNMAN GRAPHS

(15)

¯ 1 ⋆g Γ ¯ 2 := Γ

X

17

¯ 1 )(1) (Γ ¯ 1 )(2) ⊲g Γ ¯ 2. (Γ

(Γ1 )

Remark 5. We remark that ⊲g and ⋆g are respectively the restrictions of ⊲ and ⋆ on the set of vertices of subgraph γ2 , so they are respectively pre-Lie and associative. f′ T , ⋆, Φ) is a module-bialgebra on (H f′ T , ⋆, Ψ). Theorem 10. (D

f′ T ⊗ H f′ T −→ D f′ T defined for all (Γ, γ) ∈ D f′ T and Γ′ ∈ H f′ T Proof. We consider the map: α : D by: ¯ γ¯ ) ⊗ Γ ¯ ′ ) = (Γ ¯⋆Γ ¯ ′ , γ¯ ). α((Γ, To prove this theorem, first we will show that α is an action which results in the following commutative diagram: α⊗I

f′ T ⊗ H f′ T ⊗ H f′ T D

f′ T ⊗ H f′ T / D α

I⊗⋆





f′ T ⊗ H f′ T D

f′ T / D

α

f′ T and Γ ¯ 1, Γ ¯2 ∈ H f′ T , we have: Let (Γ, γ) ∈ D

¯ γ¯ ) ⊗ Γ ¯1 ⊗ Γ ¯ 2 ] = α[(Γ ¯⋆Γ ¯ 1 , γ¯ ) ⊗ Γ ¯ 2] α ◦ (α ⊗ I)[(Γ,
¯⋆Γ ¯ 1) ⋆ Γ ¯ 2 , γ¯ = (Γ
¯ ⋆ (Γ ¯1 ⋆ Γ ¯ 2 ), γ¯ = Γ ¯ γ¯ ) ⊗ Γ ¯1 ⋆ Γ ¯ 2] = α[(Γ, ¯ γ¯ ) ⊗ Γ ¯1 ⊗ Γ ¯ 2 ]. = α ◦ (I ⊗ ⋆)[(Γ,

Second, we show that the following diagram is commutative: f′ T ⊗ D f′ T ⊗ H f′ T D

⋆⊗I

α

I⊗I⊗Ψ





f′ T ⊗ D f′ T ⊗ H f′ T ⊗ H f′ T D τ 23

f′ T D O

⋆



f′ T ⊗ H f′ T ⊗ D f′ T ⊗ H f′ T D

f′ T ⊗ H f′ T / D

α⊗α

f′ T ⊗ D f′ T / D

18

MOHAMED BELHAJ MOHAMED

We denoted by: Ψ23 = τ 23 ◦ (I ⊗ I ⊗ Ψ). ¯ 1 , γ¯1 ) ⊗ (Γ ¯ 2 , γ¯2 ) ⊗ Γ ¯ ⋆ ◦ (α ⊗ α) ◦ Ψ23 (Γ




=

X (Γ)

=

X



¯ 1 , γ¯1 ) ⊗ Γ ¯ (1) ⋆α (Γ ¯ 2 , γ¯2 ) ⊗ Γ ¯ (2) α (Γ

¯1 ⋆ Γ ¯ (1) , γ¯1 )⋆(Γ ¯2 ⋆ Γ ¯ (2) , γ¯2 ) (Γ

(Γ)

X

=

¯1 ⋆ Γ ¯ (1) )(1) (Γ ¯1 ⋆ Γ ¯ (1) )(2) ⊲v (Γ ¯2 ⋆ Γ ¯ (2) ), γ¯1 γ¯2 (Γ

(Γ) v∈Γ2 −γ2

=

X

¯1 ⋆ Γ ¯ (1) ) ⋆ (Γ ¯2 ⋆ Γ ¯ (2) ) − (Γ ¯1 ⋆ Γ ¯ (1) )⋆g (Γ ¯2 ⋆ Γ ¯ (2) ), γ¯1 γ¯2 (Γ

(Γ)


¯1 ⋆ Γ ¯ 2) ⋆ Γ ¯ − (Γ ¯ 1 ⋆g Γ ¯ 2 ) ⋆ Γ, ¯ γ¯1 γ¯2 . (Γ

= On the other hand: ¯ 1 , γ¯1 ) ⊗ (Γ ¯ 2 , γ¯2 ) ⊗ Γ ¯ α ◦ (⋆ ⊗ I) (Γ





¯ 1 , γ¯1 )⋆(Γ ¯ 2 , γ¯2 ) ⊗ Γ ¯ = α (Γ
¯1 ⋆ Γ ¯2 − Γ ¯1 ⋆ Γ ¯ 2 , γ¯1 γ¯2 ) ⊗ Γ ¯ = α (Γ
¯1 ⋆ Γ ¯2 − Γ ¯ 1 ⋆g Γ ¯ 2 ) ⋆ Γ, γ¯1 γ¯2 = (Γ
¯1 ⋆ Γ ¯ 2) ⋆ Γ ¯ − (Γ ¯ 1 ⋆g Γ ¯ 2 ) ⋆ Γ, ¯ γ¯1 γ¯2 . = (Γ

Finally, we prove that the coproduct Φ is morphism of modules. This amounts to the commutativity of the following diagram: α

f′ T ⊗ H f′ T D

eT / D

I⊗Ψ





f′ T ⊗ H f′ T ⊗ H f′ T D

α⊗α

Φ⊗I⊗I

f′ T ⊗ D f′ T ⊗ H f′ T ⊗ H f′ T D

f′ T ⊗ H f′ T ⊗ D f′ T ⊗ H f′ T / D

τ 23

f′ T and Γ ¯2 ∈ H f′ T , we have: Let (Γ1 , γ1 ) ∈ D ¯ 1 , γ¯1 ) ⊗ Γ ¯2 Φ ◦ α (Γ




Φ

eT ⊗ D f′ T D O




¯1 ⋆ Γ ¯ 2 , γ¯1 = Φ Γ X
¯ (1) ¯ (2) ¯ ¯ = Φ Γ 1 (Γ1 ⊲ Γ2 ), γ (Γ1 )

=

X

(11)

(12)

(1)

(1)

¯ 1 (Γ ¯1 (Γ

(1)

(21)

(22)

¯ 2 ), γ¯ (1) ) ⊗ (Γ ¯ 1 (Γ ¯1 ⊲Γ

¯ 1 ,¯ ¯2) (Γ γ 1 ) , (Γ

=

X

¯2) ¯ 1 ,¯ γ 1 ) , (Γ (Γ




(2)

(2)

¯1 ⋆ Γ ¯ 2 , γ¯ (1) ) ⊗ (Γ ¯1 ⋆ Γ ¯ 2 ), γ¯ (2) ). (Γ

(2)

¯ 2 ), γ¯ (2) ) ⊲Γ




ON THE PRE-LIE ALGEBRA OF SPECIFIED FEYNMAN GRAPHS

We use the notation: (Φ ⊗ I ⊗ I) ◦ (I ⊗ Ψ) = Φ ⊗ Ψ.
¯ 1 , γ¯1 ) ⊗ Γ ¯ 2 = (α ⊗ α) (α ⊗ α) ◦ τ 23 ◦ (Φ ⊗ Ψ) (Γ =

X

X

(1)

(1)

X

(1)


¯ 1 , γ¯1 ) ⊗ Γ ¯2 α (Γ

¯ 1 ,¯ ¯2) (Γ γ 1 ) , (Γ

=

(1)

(1)

(2)

(2)

(1)

(2)

(2)

(2)

¯ 1 , γ¯1 ) ⊗ Γ ¯2 ) ⊗ α (Γ (2)

(2)

¯1 ⋆ Γ ¯ 2 , γ¯ (1) ) ⊗ (Γ ¯1 ⋆ Γ ¯ 2 ), γ¯ (2) ), (Γ

¯ 1 ,¯ ¯2) (Γ γ 1 ) , (Γ

which proves the theorem.

(2)


¯ 1 , γ¯1 ) ⊗ Γ ¯ 2 ⊗ (Γ ¯ 1 , γ¯1 ) ⊗ Γ ¯2 (Γ

¯ 1 ,¯ ¯2) (Γ γ 1 ) , (Γ (1)

(1)

19

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