Examples of Com-PreLie Hopf algebras Loïc Foissy Fédération de Recherche Mathématique du Nord Pas de Calais FR 2956 Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville Université du Littoral Côte dOpale-Centre Universitaire de la Mi-Voix 50, rue Ferdinand Buisson, CS 80699, 62228 Calais Cedex, France email: [email protected] 5 1 0 2 ABSTRACT. We gives examples of Com-PreLie bialgebras, that is to say bialgebras with a n preLie product satisfying certain compatibilities. Three families are defined on shuffle algebras: a one associated to linear endomorphisms, one associated to linear form, one associated to preLie J algebras. We also give all graded preLie product on K[X], making this bialgebra a Com-PreLie 6 bialgebra, and classify all connected cocommutative Com-PreLie bialgebras. 2 ] A KEYWORDS.Com-PreLiebialgebras;PreLiealgebras;connectedcocommutativebialgebras. R . AMS CLASSIFICATION. 17D25 h t a m [ Contents 1 v 1 Com-PreLie and Zinbiel-PreLie algebras 4 5 7 1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 1.2 Linear endomorphism on primitive elements . . . . . . . . . . . . . . . . . . . . . 5 6 0 . 2 Examples on shuffle algebras 6 1 0 2.1 Com-PreLie algebra attached to a linear endomorphism . . . . . . . . . . . . . . 7 5 2.2 Com-PreLie algebra attached to a linear form . . . . . . . . . . . . . . . . . . . . 8 1 : 2.3 Com-PreLie algebra associated to a preLie algebra . . . . . . . . . . . . . . . . . 14 v i X 3 Examples on K[X] 16 r a 3.1 Graded preLie products on K[X] . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2 Classification of graded preLie products on K[X] . . . . . . . . . . . . . . . . . . 20 4 Cocommutative Com-PreLie bialgebras 22 4.1 First case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2 Second case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Introduction The composition of Fliess operators [6] gives a group structure on set of noncommutative formal series Khhx ,x ii in two variables x and x . For example, let us consider the following formal 0 1 0 1 1 series: A = a +a x +a x +a x2+a x x +a x x +a x2+..., ∅ 0 0 1 1 00 0 01 0 1 10 1 0 11 1 B = b +b x +b x +b x2+b x x +b x x +b x2+..., ∅ 0 0 1 1 00 0 01 0 1 10 1 0 11 1 B = c +c x +c x +c x2+c x x +c x x +c x2+...; ∅ 0 0 1 1 00 0 01 0 1 10 1 0 11 1 if C = A.B, then: c = a +b , ∅ ∅ ∅ c = a +b +a b , 0 0 0 1 ∅ c = a +b +a b +a b +a b2+a b , 00 00 00 01 ∅ 10 ∅ 11 ∅ 1 0 c = a +b +a b +a b , 01 01 01 11 ∅ 1 1 c = a +b +a b , 10 10 10 11 ∅ c = a +b . 11 11 11 This quite complicated structure can be more easily described with the help of the Hopf algebra of coordinates of this group; this leads to a Lie algebra structure on the algebra Khx ,x i 0 1 of noncommutative polynomials in two variables, which is in a certain sense the infinitesimal structure associated to the group of Fliess operators. As explained in [3], this Lie bracket comes from a nonassociative, preLie product •. For example: x x •x = 0, x x •x = 0, 0 0 0 0 0 1 x x •x = x x x , x x •x = x x x , 0 1 0 0 0 0 0 1 1 0 0 1 x x •x = 2x x x , x x •x = x x x +x x x , 1 0 0 0 0 0 1 0 1 0 0 1 0 1 0 x x •x = x x x +x x x +x x x , x x •x = x x x +2x x x . 1 1 0 1 0 0 0 1 0 0 0 1 1 1 1 1 0 1 0 1 1 Moreover, Khx ,x iis naturally aHopf algebra with theshuffle product and thedeconcatena- 0 1 tion coproduct ∆, and it turns out that there exists compatibilities between this Hopf-algebraic structure and the preLie product •: • For all a,b,c ∈ A, (a b)•c = (a•c) b+a (b•c). • For all a,b ∈ A, ∆(a•b) =a(1)⊗a(2)•b+a(1)•b(1)⊗a(2) b(2), with Sweedler’s notation. this is a Com-PreLie bialgebra (definition 1). Moreover, the shuffle bracket can be induced by the half-shuffle product ≺, and there is also a compatibility between ≺ and •: • For all a,b,c ∈ A, (a ≺ b)•c = (a•c) ≺ b+a ≺ (b•c). we obtain a Zinbiel-PreLie bialgebra. Our aim in the present text is to give examples of other Com-PreLie algebras or bialgebras. We first introduce three families, all based on the shuffle Hopf algebra T(V) associated to a vector space V. 1. The first family T(V,f), introduced in [4], is parametrized by linear endomorphism of V. For example, if x ,x ,x ∈V, w ∈ T(V): 1 2 3 x •w = f(x )w, 1 1 x x •w = x f(x )w+f(x )(x w), 1 2 1 2 1 2 x x x •w = x x f(x )w+x f(x )(x w)+f(x )(x x w). 1 2 3 1 2 3 1 2 3 1 2 3 In particular, if V = Vect(x ,x ), f(x ) = 0 and f(x ) = x , we recover in this way the 0 1 0 1 0 Com-PreLie bialgebra of Fliess operators. 2 2. The second family T(V,f,λ) is indexed by pairs (f,λ), where f is a linear form on V and λ is a scalar. For example, if x,y ,y ,y ∈ V and w ∈ T(V): 1 2 3 xw•y = f(x)w y , 1 1 xw•y y = f(x)(w y y +λf(y )w y ), 1 2 1 2 1 2 xw•y y y = f(x)(w y y y +λf(y )w y y +λ2f(y )f(y )w y ). 1 2 3 1 2 3 1 2 3 1 2 3 We obtain a Com-PreLie algebra, but generally not a Com-PreLie bialgebra. Nevertheless, the subalgebra coS(V) generated by V is a Com-PreLie bialgebra. Up to an isomorphism, the symmetric algebra becomes a Com-PreLie bialgebra, denoted by S(V,f,λ). 3. If ⋆ is a preLie product on V, then it can be extended in a product on T(V), making it a Com-PreLie bialgebra denoted by T(V,⋆). For example, if x ,x ,x ,y ∈ V, w ∈ T(V). 1 2 3 x •yw = (x ⋆y)w, 1 1 x x •yw = (x ⋆y)(x w)+x (x ⋆y)w, 1 2 1 2 1 2 x x x •yw = (x ⋆y)(x x w)+x (x ⋆y)(x w)+x x (x ⋆y)w. 1 2 3 1 2 3 1 2 3 1 2 3 These examples answer some questions on Com-PreLie bialgebras. According to proposition 4, if A is a Com-PreLie bialgebra, the map f defined by f (x) = x•1 is an endomorphism of A A A Prim(A); if f = 0, then Prim(A) is a PreLie subalgebra of A. Then: A • If A = T(V,f), then f = f, which proves that any linear endomorphim can be obtained A in this way. • If A = T(V,⋆), then f = 0 and the preLie product on Prim(A) is ⋆, which proves that A any preLie product can be obtained in this way. Thenextsectionisdevoted tothealgebra K[X]. WefirstclassifypreLie productsmakingita gradedCom-PreLie algebra: thisgivesfourfamiliesofCom-PreLie algebrasdescribedintheorem 18, including certain cases of T(V,f). Only a few of them are compatible with the coproduct of K[X] (proposition 23). The last paragraph gives a classification of all connected, cocommu- tative Com-PreLie bialgebras (theorem 24): up to an isomorphism these are the S(V,f,λ) and examples on K[X]. Aknowledgment. The research leading these results was partially supported by the French National Research Agency under the reference ANR-12-BS01-0017. Notations. 1. K is a commutative field of characteristic zero. All the objects (vector spaces, algebras, coalgebras, preLie algebras...) in this text will be taken over K. 2. Let A be a bialgebra. (a) We shall use Swwedler’s notation ∆(a)= a(1) ⊗a(2) for all a ∈ A. (b) We denote by A the augmentation ideal of A, and by ∆˜ the coassociative coproduct + defined by: A −→ A ⊗A ∆˜ : + + + a −→ ∆(a)−a⊗1 −1 ⊗a. A A (cid:26) We shall use Sweedler’s notation ∆˜(a) = a′⊗a′′ for all a ∈ A . + 3 1 Com-PreLie and Zinbiel-PreLie algebras 1.1 Definitions Definition 1 1. A Com-PreLie algebra [8] is a family A =(A, ,•), where A is a vector space and and • are bilinear products on A, such that: (a) (A, ) is an associative, commutative algebra. (b) (A,•) is a (right) preLie algebra, that is to say, for all a,b,c ∈ A: (a•b)•c−a•(b•c) = (a•c)•b−a•(c•b). (c) For all a,b,c ∈ A, (a b)•c= (a•c) b+a (b•c). 2. A Com-PreLie bialgebra is a family (A, ,•,∆), such that: (a) (A, ,•) is a unitary Com-PreLie algebra. (b) (A, ,∆) is a bialgebra. (c) For all a,b ∈ A, ∆(a•b) = a(1) ⊗a(2)•b+a(1)•b(1) ⊗a(2) b(2). We shall say that A is unitary if the associative algebra (A, ) has a unit. 3. A Zinbiel-PreLie algebra is a family A = (A,≺,•), where A is a vector space and ≺ and • are bilinear products on A, such that: (a) (A,≺) isaZinbielalgebra (orshufflealgebra, [9,7,5])that istosay, foralla,b,c ∈ A: (a ≺ b) ≺c = a ≺ (b ≺ c+c ≺ b). (b) (A,•) is a preLie algebra. (c) For all a,b,c ∈ A, (a ≺ b)•c = (a•c) ≺ b+a ≺ (b•c). 4. A Zinbiel-PreLie bialgebra is a family (A, ,≺,•,∆) such that: (a) (A, ,•,∆) is a Com-PreLie bialgebra. (b) (A ,≺,•) is a Zinbiel-PreLie algebra, and for all x,y ∈ A , x ≺y+y ≺x = x y. + + (c) For all a,b ∈ A : + ∆˜(a ≺ b) =a′ ≺ b′⊗a′′ b′′+a′ ≺ b⊗a′′+a′⊗a′′ b+a ≺ b′⊗b′′+a⊗b. Remarks. 1. If (A, ,•,∆) is a Com-PreLie bialgebra, then for any λ ∈ K, (A, ,λ•,∆) also is. 2. If A is a Zinbiel-preLie algebra, then the product defined by a b = a ≺ b+b ≺ a is associative and commutative, and (A, ,•) is a Com-PreLie algebra. Moreover, if A is a Zinbiel-PreLie bialgebra, it is also a Com-PreLie bialgebra. 3. If A is a Zinbiel-PreLie bialgebra, the product is entirely determined by ≺: we can omit in the description of a Zinbiel-PreLie bialgebra. 4. If A is a Zinbiel-PreLie bialgebra, we extend ≺ by a ≺ 1 = a and 1 ≺ a = 0 for all A A a ∈A . Note that 1 ≺ 1 is not defined. + A A 4 5. If A is a Com-Prelie bialgebra, if a,b ∈ A : + ∆˜(a•1 )= a′⊗a′′•1 +a′•1 ⊗a′′, A A A ∆˜(a•b)= a′⊗a′′•b+a•1 ⊗b+a•b′⊗b′′ A +a′•1 ⊗a′′ b+a′•b⊗a′′+a′•b′⊗a′′ b′′, A as we shall prove later (lemma 3) that 1 •c= 0 for all c∈ A. A Associative algebras are preLie. However, Com-PreLie algebras are rarely associative: Proposition 2 LetA= (A, ,•)beaCom-PreLiealgebra, suchthatforallx ∈ A,x x = 0 if, and only if, x = 0. If • is associative, then it is zero. Proof. Let x,y ∈ A. ((x x)•y)•y = 2((x•y) x)•y = 2((x•y)•y) x+2(x•y) (x•y) = 2(x•(y•y)) x+2(x•y) (x•y) = (x x)•(y•y)+2(x•y) (x•y). Hence, (x•y) (x•y)= 0. As A is a domain, x•y = 0. (cid:3) Hence, in our examples below, which are integral domains (shuffle algebras or symmetric algebras), the preLie product is associative if, and only if, it is zero. Here is another example, where • is associative. We take A = Vect(1,x), with the products defined by: 1 x • 1 x 1 1 x 1 0 0 x x 0 x 0 x If the characteristic of the base field K is 2, this is a Com-PreLie bialgebra, with the coproduct defined by ∆(x) = x⊗1+1⊗x. 1.2 Linear endomorphism on primitive elements Lemma 3 1. Let A be a Com-PreLie algebra. For all a ∈ A, 1 •a = 0. A 2. Let A be a Com-PreLie bialgebra, with counit ε. For all a,b ∈ A, ε(a•b) = 0. Proof. 1. Indeed, 1 •a =(1 .1 )•a = (1 •a).1 +1 .(1 •a)= 2(1 •a), so 1 •a = 0. A A A A A A A A A 2. For all a,b ∈ A: ε(a•b) = (ε⊗ε)◦∆(a•b) = ε(a(1))ε(a(2) •b)+ε(a(1) •b(1))ε(a(2) b(2)) = ε(a(1))ε(a(2) •b)+ε(a(1) •b(1))ε(a(2))ε(b(2)) = ε(a•b)+ε(a•b), so ε(a•b) = 0. (cid:3) Remark. Consequently, if a is primitive: ∆(a•b)= 1 ⊗a•b+a•b(1)⊗b(2). A 5 So the map b −→ a•b is a 1-cocycle for the Cartier-Quillen cohomology [1]. If A is a Com-PreLie bialgebra, we denote by Prim(A) the space of its primitive elements: Prim(A) = {a ∈ A| ∆(a)= a⊗1+1⊗a}. We define an endomorphism of Prim(A) in the following way: Proposition 4 Let A be a Com-PreLie bialgebra. 1. If x ∈ Prim(A), then x•1 ∈Prim(A). We denote by f the map: A A Prim(A) −→ Prim(A) f : A a −→ a•1 . A (cid:26) 2. If f = 0, then Prim(A) is a preLie subalgebra of A. A Proof. 1. Indeed, if a is primitive: ∆(a•1 )= a⊗1 •1 +1 ⊗a•1 +a•1 ⊗1 1 +1 •1 ⊗a 1 A A A A A A A A A A A = 0+1 ⊗1 •a+a•1 ⊗1 +0, A A A A so a•1 is primitive. A 2. Let a,b ∈ Prim(A). ∆(a•b)= a⊗1 •b+1 ⊗a•b+1 •1 ⊗a b+a•1 ⊗b+1 •b⊗a+a•b⊗1 A A A A A A A = 1 ⊗a•b+a•b⊗1 . A A So a•b ∈ Prim(A). (cid:3) 2 Examples on shuffle algebras Let V be a vector space and let f : V −→ V be any linear map. The tensor algebra T(V) is given the shuffle product , the half-shuffle ≺ and the deconcatenation coproduct ∆, making it a bialgebra. Recall that these products can be inductively defined in the following way: if x,y ∈ V, u,v ∈ T(V): 1 ≺ yv = 0, 1 v = 0, xu ≺v = x(u≺ v+v ≺ u), xu yv = x(u yv)+y(xu v). (cid:26) (cid:26) For any x ,...,x ∈ V: 1 n n ∆(x ...x ) = x ...x ⊗x ...x . 1 n 1 i i+1 n i=0 X For all linear map F :V −→ W, we define the map: T(V) −→ T(W) T(F) : x ...x −→ F(x )...F(x ). 1 n 1 n (cid:26) This a Hopf algebra morphism from T(V) to T(W). The subalgebra of (T(V), ) generated by V is denoted by coS(V). It is the largest cocom- mutative Hopf subalgebra of (T(V), ,∆); it is generated by the symmetric tensors of elements of V. 6 2.1 Com-PreLie algebra attached to a linear endomorphism We described in [4] a first family of Zinbiel-PreLie bialgebras; coming from a problem of com- position of Fliess operators in Control Theory. Let f be an endomorphism of a vector space V. We define a bilinear product • on T(V) inductively on the length of words in the following way: if x ∈ V, v,w ∈ T(V), 1•w = 0, xv•w = x(v•w)+f(x)(v w). Then (T(V),≺,•,∆) is a Zinbiel-PreLie bialgebra, denoted by T(V,f). Moreover, f = f. T(V,f) Examples. If x ,x ,x ∈ V, w ∈ T(V): 1 2 3 x •w = f(x )w, 1 1 x x •w = x f(x )w+f(x )(x w), 1 2 1 2 1 2 x x x •w = x x f(x )w+x f(x )(x w)+f(x )(x x w). 1 2 3 1 2 3 1 2 3 1 2 3 More generally, if x ,...,x ∈ V and w ∈ T(V): 1 n n x ...x •w = x ...x f(x )(x ...x w). 1 n 1 i−1 i i+1 n i=1 X This construction is functorial: let V and W be two vector spaces, f an endomorphism of V and g an endomorphism of W; let F : V −→ W, such that g ◦F = F ◦f. Then T(F) is a morphism of Zinbiel-PreLie bialgebras from T(V,f) to T(W,g). Proposition 5 Let (cid:7) be a preLie product on (T(V), ,∆), making it a Com-PreLie bialge- bra, such that for all k,l ∈ N, V⊗k blacklozengeV⊗l ⊆ V⊗(k+l). There exists a f ∈ End(V), such that (T(V), ,(cid:7),∆) = T(V,f). Proof. Let f = f . We denote by • the preLie product of T(V,f). Let us prove that for T(V) any x = x ...x ,y = y ...y ∈ T(V), x•y = x(cid:7)y. If k = 0, we obtain 1•y = 1(cid:7)y = 0. We 1 k 1 l now treat the case l = 0. We proceed by induction on k. It is already done for k = 0. If k = 1, then x ∈ V and x•1 = f(x) = x(cid:7)1. Let us assume the result at all ranks < k, with k ≥ 2. Then, as the length of x′ andx′′ is < k: ∆(x•1) = x(1) ⊗x(2)•1+x(1) •1⊗x(2) = 1⊗x•1+x′•1⊗1+x′⊗x′′•1+x⊗1⊗1 = 1⊗x•1+x′(cid:7)1⊗1+x′⊗x′′(cid:7)1+x⊗1⊗1 = ∆(x(cid:7)1)+(x•y−x(cid:7)y)⊗1+1⊗(x•y−x(cid:7)y). We deduce that x•1−x(cid:7)1 is primitive, so belongs to V. As it is homogeneous of length k ≥ 2, it is zero, and x•1 = x(cid:7)1. We can now assume that k,l ≥ 1. We proceed by induction on k+l. There is nothing left to do for k+l = 0 or 1. Let us assume that the result is true at all rank < k+l, with k+l ≥ 2. Then, using the induction hypothesis, as x′ and x′′ have lengths < k and y′ has a length < l: ∆(x•y)= 1⊗x•y+x′⊗x′′•y+x⊗1•y+x•1⊗y+x′•1⊗x′′ y+1•1⊗x y +x•y⊗1+x′•y⊗x′′+1•y⊗x+x•y′⊗y′′+x′•y′⊗x′′ y′′+1•y′⊗x y′′ = 1⊗x•y+x′⊗x′′(cid:7)y+x⊗1(cid:7)y+x(cid:7)1⊗y+x′(cid:7)1⊗x′′ y+1(cid:7)1⊗x y +x•y⊗1+x′(cid:7)y⊗x′′+1(cid:7)y⊗x+x(cid:7)y′⊗y′′+x′(cid:7)y′⊗x′′ y′′+1(cid:7)y′⊗x y′′ = ∆(x(cid:7)y)+(x•y−x(cid:7)y)⊗+1⊗(x•y−x(cid:7)y). We deduce that x • y − x(cid:7)y is primitive, hence belongs to V. As it belongs to V⊗(k+l) and k+l ≥ 2, it is zero. Finally, x•y = x(cid:7)y. (cid:3) 7 Proposition 6 The Com-PreLie bialgebras T(V,f) and T(W,g) are isomorphic if, and only if, there exists a linear isomorphism F : V −→ W, such that g◦F = F ◦f. Proof. Ifsuch anF exists,by functoriality T(F)isan isomorphism from T(V,f)toT(W,g). Let us assume that φ : T(V,f) −→ T(V,g) is an isomorphism of Com-PreLie bialgebras. Then φ(1) = 1, and φ induces an isomorphism from V = Prim(T(V)) to W = Prim(T(W)), denoted by F. For all x ∈ V: φ(x•1) =φ(f(x)) = F ◦f(x) = F(x)•1 = g◦F(x). So such an F exists. (cid:3) 2.2 Com-PreLie algebra attached to a linear form Let V be a a vector space, f :V −→ K be a linear form, and λ ∈ K. Theorem 7 Let • be the product on T(V) such that for all x ,...,x ,y ,...,y ∈ V: 1 m 1 n n−1 x ...x •y ...y = λif(x )f(y )...f(y )x ...x y ...y . 1 m 1 n 1 1 i 2 m i+1 n i=0 X Then (T(V), ,•) is a Com-PreLie algebra. It is denoted by T(V,f,λ). Examples. If x ,x ,x ∈ V, w ∈ T(V): 1 2 3 x •w = f(x )w, 1 1 x x •w = x f(x )w+f(x )(x w), 1 2 1 2 1 2 x x x •w = x x f(x )w+x f(x )(x w)+f(x )(x x w). 1 2 3 1 2 3 1 2 3 1 2 3 In particular if x =... = x = y = ... = y = x: 1 n 1 n Lemma 8 Let x ∈ V. We put f(x) = ν and µ = λf(x). Then, for all m,n ≥ 0, in T(V,f,λ): m+n−1 j xm•xn = ν µm+n−j−1 xj. m−1 j=m (cid:18) (cid:19) X The proof of theorem 7 will use definition 9 and lemma 10: Definition 9 Let ∂ and φ be the linear maps defined by: T(V) −→ T(V) T(V) −→ T(V) 1 −→ 0, ∂ : 1 −→ 0, φ: n−1   x ...x −→ f(x )x ...x ,  x ...x −→ λif(x )...f(x )x ...x .  1 n 1 2 n  1 n 1 i i+1 n i=0 X     Lemma 10 1. For all u,v ∈ T(V):  (a) ∂(u v) = ∂(u) v+u ∂(v). (b) ∂ ◦φ(u) φ(v)−φ(∂(u) φ(v)) = ∂ ◦φ(v) φ(u)−φ(∂(v) φ(u)). 2. For all u∈ T(V,f,λ): ∆◦∂(u) = (∂ ⊗Id)◦∆(u), ∆◦φ(u) = (φ⊗Id)◦∆(u)+1⊗φ(u). 8 Proof. 1. (a) This is obvious if u = 1 or v = 1, as ∂(1) = 0. Let us assume that u,v are nonempty words. We put v = xu′,v = yv′, with x,y ∈ V. Then: ∂(u v)= ∂(x(u′ v)+y(u v′)) = f(x)u′ v+f(y)u v′ = (f(x)u′) v+u (f(y)v′) = ∂(u) v+u ∂(v). 1. (b) Let us take u= x ...x and y = y ...y be two words of T(V) of respective lengths 1 m 1 n m and n. First, observe that φ(∂u φ(v)) is a linear span of terms: λi+j−1f(x )...f(x )f(y )...f(y )x ...x y ...y , 1 i 1 j i+1 m j+1 m with 1 ≤ i ≤m, 0≤ j ≤ n, (i,j) 6= (0,0). Let us compute the coefficient of such a term: j i+j−1 i−1+j −p p i+j • If j < n, it is = = . i−1 i−1 i p=0(cid:18) (cid:19) p=i−1(cid:18) (cid:19) (cid:18) (cid:19) X X n−1 i+j−1 i+j−1 i−1+j −p p p i+j • If j = n, its is = = −1= −1. i−1 i−1 i−1 i p=0(cid:18) (cid:19) p=i (cid:18) (cid:19) p=i−1(cid:18) (cid:19) (cid:18) (cid:19) X X X We obtain: m n i+j φ(∂u φ(v)) = λi+j−1 f(x )...f(x )f(y )...f(y )x ...x y ...y 1 i 1 j i+1 m j+1 n i i=1 j=0 (cid:18) (cid:19) XX m−1 − λi+n−1f(x )...f(x )f(y )...f(y )x ...x 1 i 1 n i+1 m i=1 X m+n −λm+n−1 f(x )...f(x )f(y )...f(y ) 1 m 1 n m (cid:18) (cid:19) m n i+j = λi+j−1 f(x )...f(x )f(y )...f(y )x ...x y ...y 1 i 1 j i+1 m j+1 n i i=1 j=1 (cid:18) (cid:19) XX m + λi−1f(x )...f(x )x ...x y ...y 1 i i+1 m 1 n i=1 X m−1 − λi+n−1f(x )...f(x )f(y )...f(y )x ...x 1 i 1 n i+1 m i=1 X m+n −λm+n−1 f(x )...f(x )f(y )...f(y ). 1 m 1 n m (cid:18) (cid:19) Moreover: m n−1 i+j ∂ ◦φ(u) φ(v) = λi+j−1 f(x )...f(x )f(y )...f(y )x ...x y ...y 1 i 1 j i+1 m j+1 n i i=1 j=0 (cid:18) (cid:19) XX m−1n−1 i+j = λi+j−1 f(x )...f(x )f(y )...f(y )x ...x y ...y 1 i 1 j i+1 m j+1 n i i=1 j=1 (cid:18) (cid:19) X X n−1 + λj+m−1f(x )...f(x )f(y )...f(y )y ...y 1 m 1 j j+1 n j=1 X m + λi−1f(x )...f(x )x ...x y ...y . 1 i i+1 m 1 n i=1 X 9 Hence: ∂ ◦φ(u) φ(v)−φ(∂u φ(v)) m−1n−1 i+j = λi+j−1 f(x )...f(x )f(y )...f(y )x ...x y ...y 1 i 1 j i+1 m j+1 n i i=1 j=1 (cid:18) (cid:19) X X m n i+j − λi+j−1 f(x )...f(x )f(y )...f(y )x ...x y ...y 1 i 1 j i+1 m j+1 n i i=1 j=1 (cid:18) (cid:19) XX m+n +λm+n−1 f(x )...f(x )f(y )...f(y ) 1 m 1 n m (cid:18) (cid:19) n−1 + λj+m−1f(x )...f(x )f(y )...f(y )y ...y 1 m 1 j j+1 n j=1 X m−1 + λi+n−1f(x )...f(x )f(y )...f(y )x ...x . 1 i 1 n i+1 m i=1 X The three first rows are symmetric in u and v, whereas the sum of the fourth and fifth rows is symmetric in u and v. So ∂◦φ(u) φ(v)−φ(∂u φ(v)) is symmetric in u and v. 2. Let us take u = x ...x , with x ,...,x ∈ V. Then: 1 n 1 n n ∆◦∂(u) =f(x ) x ...x ⊗x ...x 1 2 i i+1 n i=1 X n = ∂(x ...x )⊗x ...x +∂(1)⊗x ...x 1 i i+1 n 1 n i=1 X n = ∂(x ...x )⊗x ...x 1 i i+1 n i=0 X =(∂ ⊗Id)◦∆(u). Moreover: n−1 ∆◦φ(u) = λif(x )...f(x )∆(x ...x ) 1 i i+1 n i=0 X n−1 n = λif(x )...f(x )x ...x ⊗x ...x 1 i i+1 j j+1 n i=0 j=i XX n j = λif(x )...f(x )x ...x ⊗x ...x −λnf(x )...f(x )⊗1 1 i i+1 j j+1 n 1 n j=0 i=0 XX n n−1 = φ(x ...x )⊗x ...x + λjf(x )...f(x )⊗x ...x 1 j j+1 n 1 j j+1 n j=0 j=0 X X n n−1 = φ(x ...x )⊗x ...x +1⊗ λjf(x )...f(x )x ...x 1 j j+1 n 1 j j+1 n   j=0 j=0 X X   = (φ⊗Id)◦∆(u)+1⊗φ(u). (cid:3) Proof. (Theorem 7). By definition, for all u,v ∈ T(V): u•v = ∂(u) φ(v). 10