Duality and Lie algebra of infinitesimal derivations

In the previous subsection we showed that H_rt is endowed with a structure of a graded Hopf algebra, connected and of finite type. Now we want to study the (restricted) dual Hopf algebra, H^?_rt =L

i≥0H_i^∗. We already know thatH^?_rtis aZ₊-graded, and of finite type Hopf algebra (see proposition 13).

SinceH_rt is commutative and (freely) generated by the set of all trees, i.e by the indecomposable elements, we have the following theorem (see theorem 8):

Theorem 15 H^?_rtis a cocommutative, primitively generated Hopf algebra, so that it is isomorphic, as a Hopf algebra, to the universal enveloping algebra of the Lie algebra of its primitive elements P(H^?_rt).

The set of primitive elements P(H^?_rt)⊂ H^?_rt is in one to one correspondence with the set of indecomposable elements I(H_rt) ⊂ H_rt; for each tree t ∈ I(H_rt), we get a linear form Z_t∈P(H^?_rt). Since H_rt is graded, of finite type and connected, we have the following proposition, (see theorem 10):

Proposition 17 Each Zt extends by zero to an algebra derivation.

Proof In fact: hZ_t, m(t₁⊗t₂)i=hZ_t, t₁iε(t₂) +hZ_t, t₂iε(t₁). But kerε ' L

i>0H_i. ♠

Definition 37 The derivationZ_tdefined in the previous proposition, is called infinitesimal character of the Hopf algebra H_rt.

For any t₁, t₂ ∈ H_rt let Z_t1, Z_t2 be the corresponding infinitesimal char-acters. The product between those is the one induced by the coproduct ∆ defined in H_rt: hm_∗(Z_t1 ⊗Z_t2), ti=hZ_t1 ⊗Z_t2,∆(t)i.

In particular, we have the following proposition:

Proposition 18 For Zt1, Zt2 and t as above, we have:

hm_∗(Z_t1 ⊗Z_t2), ti=X

˜ c

¡hZ_t1, P_c˜(t)ihZ_t2, R_c˜(t)i¢

, (3.1)

where ˜cruns over the set of all possible admissible cut of cardinality equal to one (i.e over the set of elementary cuts).

3.2. DUALITY AND LIE ALGEBRA OF INFINITESIMAL DERIVATIONS39 Proof In fact, for any give admissible cut c, the number of factors in P_c(t) is equal to the cardinality of c. Since the linear form Z_t1 extends by zero to a derivation of the algebra H_rt, the only terms which are not iden-tically equal to zero in (3.1) are the ones labelled by admissible cuts with cardinality equal to one.♠

The Lie algebra structure defined on P(H^?_rt) is given by the bracket:

[Z_t1, Z_t2] =m_∗(Z_t1 ⊗Z_t2 −Z_t2 ⊗Z_t1).

The associativity of the multiplication m_∗ (which follows from the coasso-ciativity of the coproduct ∆), ensures that such a bracket fulfills the Jacobi identity.

Remark 12 The product m_∗ is also called convolution product.

We will now analyze in greater details the Lie algebra P(H_rt).

We start with the following proposition:

Proposition 19 The set of the infinitesimal characters is not closed with respect to the convolution product.

Proof LetZ_t1, Z_t2 be two infinitesimal derivations and let T₁, T₂ be two trees. Then: hm_∗(Z_t1 ⊗Z_t2), m(T₁ ⊗T₂)i = hZ_t1 ⊗Z_t2, m(∆(T₁)⊗∆(T₂)).

On the other hand, by the definition of the coproduct, we have: ∆(T_i) = T_i⊗1+1⊗T_i+P

iT_i⁰⊗T_i⁰⁰, i= 1,2, where T_i⁰ and T_i⁰⁰ have degree greater than zero. So that: m(∆(T₁)⊗∆(T₂)) = T₁⊗T₂+T₂⊗T₁+ Σ⁰⊗Σ⁰⁰ where either Σ⁰ or Σ⁰⁰ are decomposable (product of indecomposable, i.e product of trees), so that they belong to the kernel of the linear forms Z_t1, Z_t2. From this, we can conclude:

hZ_t1 ⊗Z_t2, m(∆(T₁)⊗∆(T₂)) =hZ_t1, T₁ihZ_t2, T₂i+hZ_t2, T₁ihZ_t1, T₂i, which is, in general, different by zero. ♠

Remark 13 It is clear that the previous results hold for any Hopf algebra connected, graded of finite type, since the key for those results is the charac-terization of the coproduct, given in proposition 12.

40 CHAPTER 3. THE HOPF ALGEBRA OF ROOTED TREES Example 24 Let us calculate the convolution product betweenZ with itself.

From the definition:

m∗

¡Z ⊗Z ¢

=¡

Z ⊗Z ¢

◦∆. (3.2)

This is an element in H^?_rt, so that can be written as a linear combination of elements like ZF, where F is a forest. The generators in degree less or equal to 3, of the homogeneous components of the restricted dual are the following:

H₁^∗ = span_khZ i;H₂^∗ = span_khZ , Z i, H₃^∗ = span_khZ , Z , Z , Z i.

Since m_∗ is grading preserving: m_∗¡

Z ⊗Z ¢

∈H₂^∗, so that we can write:

m_∗¡

Z ⊗Z ¢

=a₁Z +a₂Z

for some a₁, a₂. It is easy to find such coefficients using the equality (3.2);

m∗

¡Z ⊗Z ¢

=Z + 2Z .

Remark 14 Using the previous argument, it is possible, at least in principle, to write each generator ofH^?_rtas linear combinations of products (with respect to m∗) of infinitesimal characters. In particular, from the previous example we have that:

Z = 1

2

³ m_∗¡

Z ⊗Z ¢

−Z

´ .

In the case under examination, i.e for the Hopf algebraH_rt, it is possible to introduce a new product, that we will indicate with ∗, with respect to which P(H_rt^?) turn out to be closed. Such a remarkable product will induce a Lie algebra structure on the vector spaceP(H^?_rt) that will coincide with the one defined via the convolution product.

Definition 38 For any given Zt1, Zt2 ∈P(H^?_rt), define:

Zt1 ∗Zt2 =X

˜ c

n(t;t1, t2)Zt, (3.3)

3.2. DUALITY AND LIE ALGEBRA OF INFINITESIMAL DERIVATIONS41 where ˜cruns over the set of all admissible cut of cardinality equal to one. The numerical coefficients n(t;t₁, t₂), express the number of ways in which t can be decomposed by an elementary cut c˜(i.e an admissible cut with cardinality equal to one), in such a way P_˜c(t) = t₁ and R_˜c(t) = t₂.

Example 25 Let us calculate the ∗-product in some simple case:

1) Z ∗Z =Z ;

2)Z ∗Z =Z + 2 Z ,

3) Z ∗Z =Z ;

As it is evident in the previous examples, the ∗-product is not commutative.

Moreover, it is not associative as the next example shows:

Example 26

¡Z ∗Z ¢

∗Z =Z ∗Z =Z ; on the other hand:

Z ∗¡

Z ∗Z ¢

=Z ∗Z =Z + 2 Z .

Nevertheless, the ∗-product defined in (3.3) fulfills the following property:

Theorem 16 For each triple of rooted trees t₁, t₂ and t₃ we have:

Z_t1∗¡

Z_t2∗Z_t3¢

−(Z_t1∗Z_t2)∗Z_t3 =Z_t2∗¡

Z_t1∗Z_t3¢

−¡

Z_t2∗Z_t1¢

∗Z_t3. (3.4) Proof The proof is a consequence of the following lemma. Let us define the following triple product:

A(t₁, t₂, t₃) = Z_t1 ∗¡

Z_t2 ∗Z_t3¢

−¡

Z_t1 ∗Z_t2¢

∗Z_t3. (3.5) Lemma 2

A(t1, t2, t3) =X

˜ c

n(t1, t2, t3;t)Zt,

wheren(t1, t2, t3;t)is the number of admissible cuts˜cof tsuch thatR˜c(t) = t3

and P_˜c(t) = t₁t₂ and card ˜c= 2.

42 CHAPTER 3. THE HOPF ALGEBRA OF ROOTED TREES Proof(of the Lemma) By the definition of the product∗, we can write:

A(t1, t2, t3) =X

t

¡ X

t⁰

n(t2, t3;t⁰)n(t1, t⁰;t)−n(t1, t2;t⁰)n(t⁰, t3;t)¢ Zt.

The first sum corresponds to the couples of elementary cuts (˜c,c˜⁰) where ˜cis a cut of the treeR_˜c(t). Such a set of couples is the (disjoint) union of the set of couples (˜c,˜c⁰), with the property that ˜c∪˜c⁰ is still admissible for t, with the set of couples which do not have such a property. On the other hand, the second sum corresponds to the set of couples of admissible cuts (˜c,˜c⁰), such that R_˜c(t) = t₃ and ˜c⁰ is cut of P_˜c⁰(t). This means that for such couples we never have the case ˜c∪c˜⁰ is admissible for t. From this follows the lemma, since the difference representing A(t₁, t₂, t₃) will count only the couple (˜c,˜c⁰) of admissible cuts such that ˜c∪˜c⁰ is still admissible. ♠

To end the proof of the theorem, it suffices to observe that:

A(t1, t2, t3) =A(t2, t1, t3),

which follows from the definition given in the equation (4.1). ♠

Definition 39 A vector space V endowed with a product ∗ that fulfills the condition expressed in (3.4) is called a pre-Lie algebra.

Remark 15 The trilinear form A defined in (4.1) can be thought as a mea-sure of the non-associativity of the product ∗ and it is called associator.

Pre-Lie algebras are also known in the literature as left-symmetric or right-symmetric. The one we introduced is left symmetric since:

A(t₁, t₂, t₃) =A(t₂, t₁, t₃).

Right-symmetric ones would be defined using the following associator:

A(t₁, t₂, t₃) =A(t₁, t₃, t₂).

Proposition 20 If(V,∗)is a pre-Lie algebra then L(V)is a Lie algebra; i.e the bracket[·, ·] :V ⊗V −→V, [x, y] =x∗y−y∗x, for each x, y ∈V fulfills the Jacobi identity.

3.2. DUALITY AND LIE ALGEBRA OF INFINITESIMAL DERIVATIONS43 Proof We need to show that for each x, y and z in V, [x,[y, z]] + [z,[x, y]] + [y,[z, x]] = 0. This follows after applying the definition of the bracket [x, y] = x∗y−y∗x, and from the property (3.4). ♠

OnP(H^?_rt) there are defined two products: the convolution productm_∗ = ∆^t, and the pre-Lie product (3.3). The first one does not close to a product for the vector space of the primitive elements P(H^?_rt) (see proposition 19) while, by its very definition, the pre-Lie product ∗ does close to such a product.

Nevertheless, we have the following result:

Proposition 21 The convolution product between Z_t1 andZ_t2, (Z_t1⊗Z_t2)◦

∆, is a linear form onHrt, whose restriction to the indecomposable elements I(H_rt) coincides with the linear form Z_t1 ∗Z_t2.

Proof Let us consider the t₁, t₂, t ∈ I(H_rt). The coproduct gives us

∆(t) = t⊗1+1⊗t+P

˜

cPc˜(t)⊗R˜c(t), where the sum is taken over the set of admissible cuts. The convolution between Z_t1 and Z_t2 will give us:

¡(Z_t1 ⊗Z_t2)◦∆¢

(t) = X

˜ c

hZ_t1, P_c˜(t)ihZ_t2, R_c˜(t)i. (3.6) Since Z_t1, Z_t2 are infinitesimal characters, the the sum in (3.6) reduces to a sum over the elementary cuts, i.e:

¡(Z_t1 ⊗Z_t2)◦∆¢

(t) = X

˜ c

hZ_t1, P_c˜(t)ihZ_t2, R_c˜(t)i.

This sum represents the number of ways in which t can be decomposed, via an elementary cut ˜c, asP_˜c=t₁ andR_˜c(t) = t₂; such a number is by definition n(t;t₁, t₂)♠

Moreover, we have that:

Theorem 17 For each t₁ and t₂ in I(H_rt),

m_∗(Z_t1 ⊗Z_t2 −Z_t2 ⊗Z_t1) =Z_t1 ∗Z_t2 −Z_t2 ∗Z_t1,

so that the Lie algebras structures induced on P(H^?_rt) by these two products coincide.

44 CHAPTER 3. THE HOPF ALGEBRA OF ROOTED TREES Proof We already know that the restriction of m_∗(Z_t1 ⊗Z_t2) to the in-decomposable elements in H_rt coincides with Z_t1 ∗Z_t2, so that m_∗(Z_t1 ⊗ Z_t2 −Z_t2 ⊗Z_t1)(t) = (Z_t1 ∗Z_t2 −Z_t2 ∗Z_t1)(t) for any t ∈ I(H_rt). There-fore, we are left to prove that m_∗(Z_t1 ⊗ Z_t2 −Z_t2 ⊗Z_t1) restricts to zero to any decomposable element in H_rt. Let T = T₁T₂ such an element, then

∆(T) =T₁⊗T₂+T₂⊗T₁ +P

σ⊗σ⁰ where σ and or σ⁰ are decomposable.

Applying Z_t1 ⊗Z_t2 −Z_t2 ⊗Z_t1 to right hand side of the previous equality we get zero: in fact (Z_t1 ⊗Z_t2 −Z_t2 ⊗Z_t1) is zero for each term of the sum Pσ⊗σ⁰ sinceZ_t1, Z_t2 are infinitesimal derivations, and (Z_t1⊗Z_t2−Z_t2⊗Z_t1) is zero on T₁ ⊗T₂+T₂⊗T₁ since the last one is symmetric in T₁, T₂ while the first one in antisymmetric int₁, t₂. ♠

Remark 16 The convolution product m_∗ is associative, while, as already remarked, the product ∗ is pre-Lie. Nevertheless,

Z_t1 ∗Z_t2|_I(Hrt)=m_∗(Z_t1 ⊗Z_t2)|_I(Hrt).

Let us clarify this apparent anomaly discussing one example. Let us consider the triple product of the infinitesimal character Z with itself.

Let us calculate first the triple products using the ∗-product:

Z ∗¡

Z ∗Z ¢

= 2Z +Z , (3.7)

and

¡Z ∗Z ¢

∗Z =Z . (3.8)

Using the convolution product, we get instead:

m_∗

³

Z ⊗m_∗¡

Z ⊗Z ¢´

=m_∗³¡

2Z +Z ¢

⊗Z

´

= 6Z + 2Z + (3.9)

+3Z +Z

and similarly:

m_∗

³ m_∗¡

Z ⊗Z ¢

⊗Z

´

=m_∗

³ Z ⊗¡

2Z +Z ¢´

= 6Z +2Z + (3.10)

Im Dokument Hopf algebras (Seite 38-45)