Logarithmic derivatives and generalized Dynkin operators
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Abstract
Motivated by a recent surge of interest for Dynkin operators in mathematical physics and by problems in the combinatorial theory of dynamical systems, we propose here a systematic study of logarithmic derivatives in various contexts. In particular, we introduce and investigate generalizations of the Dynkin operator for which we obtain Magnustype formulas.
Keywords
Dynkin operator Free Lie algebras Logarithmic derivative Rota–Baxter algebra1 Introduction
 (1)
In the theory of free Lie algebras and noncommutative symmetric functions, it was shown that Dynkin operators generate the descent algebra (the direct sum of Solomon’s algebras of type A, see [9, 17]) and play a crucial role in the theory of Lie idempotents.
 (2)
Generalized Dynkin operators can be defined in the context of classical Hopf algebras [16]. The properties of these operators generalize the classical ones. Among others (see also [6, 7]), they can be used to derive fine properties of the renormalization process in perturbative quantum field theory (pQFT): Dynkin operators can be shown to give rise to the infinitesimal generator of the differential equation satisfied by renormalized Feynman rules [5]. This phenomenon has attracted attention to logarithmic derivatives in pQFT where generalized Dynkin operators are expected to lead to a renewed understanding of Dyson–Schwingertype equations, see e.g. [11].
The present article originated from different problems, namely problems in the combinatorial theory of dynamical systems (often referred to as Ecalle’s mould calculus, see e.g. [18], also for further references on the subject) and in particular in the theory of normal forms of vector fields. It appeared very soon to us that the same machinery that had been successfully used in the above mentioned fields and problems was also relevant for the study of dynamical systems. However, the particular form of logarithmic derivatives showing up in this field requires the generalization of the known results on logarithmic derivatives and Dynkin operators to a broader framework: we refer to Sect. 3.1 of the present article for more details of the subject.
The purpose of the present article is therefore to develop further the algebraic and combinatorial theory of logarithmic derivatives, following various directions and perspectives (free Lie algebras, Hopf algebras, Rota–Baxter algebras, …), all of them known to be relevant for the study of dynamical systems but also to various other fields, running from the numerical analysis of differential equations (see e.g. [12, 13]) to pQFT.
We limit the scope of the present article to the general theory and plan to use the results in forthcoming articles.
2 Twisted Dynkin operators on free Lie algebras
We write δ an arbitrary derivation of T(X) (in particular δ acts as the null application on the scalars, T _{0}(X)). The simplest and most common derivations are induced by maps f from X to its linear span: the associated derivation, written \(\tilde{f}\), is then defined by \({\tilde{f}}(y_{1}\ldots y_{n})=\sum_{i=0}^{n}y_{1}\ldots y_{i1}f(y_{i})y_{i+1}\ldots y_{n}\). Since for \(a,b\in T(X),\ \tilde{f}([a,b])=[f(a),b]+[a,f(b)]\), where [a,b]:=ab−ba, these particular derivations are also derivations of the free Lie algebra over X. These are the derivations we will be most interested in practice (the ones that map the free Lie algebra over X to itself), we call them Lie derivations.
In the particular case f=Id, we also write Y for \(\tilde {\mathrm{Id}}\): Y is the graduation operator, Y(y _{1}…y _{ n })=n⋅y _{1}…y _{ n }. When \(f=\delta_{x_{i}}\) (f(x _{ i })=x _{ i },f(x _{ j })=0 for j≠i), \(\tilde{f}\) counts the multiplicity of the letter x _{ i } in words and is the noncommutative analog of the derivative with respect to x _{ i } of a monomial in the letters in X.
Proposition 1
The first and third term of the summation sum up to [S∗δ(y _{1}…y _{ n−1}),y _{ n }] which is, by induction, equal to […[[δ(y _{1}),y _{2}],y _{3}]…,y _{ n }]. The second computes S∗Id(y _{1}…y _{ n−1})δ(y _{ n }), which is equal to 0 for n>1. The Proposition follows.
Corollary 2
We recover in particular the theorem of Dynkin [3], Specht [19], Wever [20] (case f=Id) and obtain an extension thereof to the case \(f=\delta_{x_{i}}\). We let the reader derive similar results for other families of Lie derivations.
Corollary 3
The definition D=S∗Y of the Dynkin operator seems to be due to von Waldenfels, see [17].
Let us write \(T_{n}^{i}(X)\) for the linear span of words over X such that the letter x _{ i } appears exactly n times. The derivation \(\tilde{\delta}_{x_{i}}\) acts as the multiplication by n on \(T_{n}^{i}(X)\).
Corollary 4
3 Abstract logarithmic derivatives
Quite often, the logarithmic derivatives one is interested in arise from dynamical systems and geometry. We explain briefly why this is so in the context of a fundamental example, the classification of singular vector fields (Sect. 3.1). Although we settle our later computations in the general framework of Lie and enveloping algebras, the reader may want to keep this motivation in mind.
The second Sect. 3.2 shows briefly how to extend the results on generalized Dynkin operators obtained previously in the tensor algebra to the general setting of enveloping algebras.
We show finally (Sect. 3.3) how these results connect to the theory of Rota–Baxter algebras, which is known to be the right framework to investigate formal properties of derivations. Indeed, as we will recall below, Rota–Baxter algebra structures show up naturally when derivations have to be inverted. See also [1, 6, 7] for further details on the subject of Rota–Baxter algebras and their applications.
3.1 An example from the theory of dynamical systems
Derivations on graded complete Lie algebras appear naturally in the framework of dynamical systems, especially when dealing with the formal classification (up to formal change of coordinates) of singular vector fields.
The reader is referred to [10] for an overview and further details on the objects we consider (such as identitytangent diffeomorphisms or substitution automorphisms); let us also mention that the reader who is interested only in formal aspects of logarithmic derivatives may skip the current section.

The vector space L of vector fields without linear part (or without component of graduation degree 0) is a graded complete Lie algebra. The vector space of linear vector fields is written L _{0}.
 The exponential of a vector field in L gives a one to one correspondence between vector fields and substitution automorphisms on formal power series, that is, operators F such thatwhere (F(x _{1}),…,F(x _{ ν })) is a formal identitytangent diffeomorphism.$$F\bigl(A(x_1,\dots,x_\nu)\bigr)=A\bigl(F(x_1), \dots,F(x_{\nu})\bigr) $$

The previous equation also determines an isomorphism between the Lie group of L and the group of formal identitytangent diffeomorphism G _{1}.
In the framework of dynamical systems, the forthcoming Theorem 15 ensures that if the derivation \(\mathrm{ad}_{X_{0}}\) is invertible on L, any vector field X _{0}+Y can be linearized. This is the kind of problems that can be addressed using the general theory of logarithmic derivatives to be developed in the next sections.
3.2 Hopf and enveloping algebras
We use freely in this section the results in [16] to which we refer for further details and proofs. The purpose of this section is to extend the results in [16] on the Dynkin operator to more general logarithmic derivatives.
Let \(L=\bigoplus_{n\in\mathbf{N}^{\ast}}L_{n}\) be a graded Lie algebra, \(\hat{L}=\prod_{n\in\mathbf{N}^{\ast}}L_{n}\) its completion, \(U(L)=\bigoplus_{n\in\mathbf{N}^{\ast}}U(L)_{n}\) the (graded) enveloping algebra of L and \(\hat{U}(L)= \prod_{n\in\mathbf{N}^{\ast}}U(L)_{n}\) the completion of U(L) with respect to the graduation (N ^{∗} stands for the set of strictly positive integers).
The ground field is chosen to be Q (but the results in the article would hold for an arbitrary ground field of characteristic zero and, due to the Cartier–Milnor–Moore theorem [14, 15], for arbitrary graded connected cocommutative Hopf algebras).
The enveloping algebra U(L) is naturally provided with the structure of a Hopf algebra. We denote by ϵ:Q=U(L)_{0}→U(L) the unit of U(L), by η:U(L)→Q the counit, by Δ:U(L)→U(L)⊗U(L) the coproduct and by μ:U(L)⊗U(L)→U(L) the product. An element l of U(L) is primitive if Δ(l)=l⊗1+1⊗l; the set of primitive elements identifies canonically with L. Recall that the convolution product ∗ of linear endomorphisms of U(L) is defined by f∗g=μ∘(f⊗g)∘Δ; ν:=ϵ∘η is the neutral element of ∗. The antipode is written S, as usual.
Definition 5
An element f of End(U(L)) admits F∈End(U(L))⊗End(U(L)) as a pseudocoproduct if F∘Δ=Δ∘f. If f admits the pseudocoproduct f⊗ν+ν⊗f, we say that f is pseudoprimitive.
In general, an element of End(U(L)) may admit several pseudocoproducts. However, this concept is very flexible, as shows the following result [16, Thm. 2].
Proposition 6
If f,g admit the pseudocoproducts F,G and α∈Q, then f+g, αf, f∗g, f∘g admit, respectively, the pseudocoproducts F+G,αF,F∗G,F∘G, where the products ∗ and ∘ are naturally extended to End(U(L))⊗End(U(L)).
An element f∈End(U(L)) takes values in Prim(U(L)) if and only if it is pseudoprimitive.
Proposition 7
The logarithmic derivative D _{ δ } is a pseudoprimitive: it maps U(L) to L.
Proposition 8
For l∈L, we have δ(l)=D _{ δ }(l). In particular, when δ is invertible on L, D _{ δ }(l) is a projection from U(L) onto L.
The proof is similar to the one in the free Lie algebra. We have D _{ δ }(l)=(S∗δ)(l)=π∘(S⊗δ)∘Δ(l)=π∘(S⊗δ)(l⊗1+1⊗l) =π(S(l)⊗δ(1)+S(1)⊗δ(l))=δ(l), since δ(1)=0 and S(1)=1.
3.3 Integrodifferential calculus
The notations are the same as in the previous section, but we assume now that the derivation δ is invertible on L and extends to an invertible derivation on U(L)^{+}:=⨁_{ n≥1} U(L)_{ n }. The simplest example is provided by the graduation operator Y(l)=n⋅l for l∈L _{ n } (resp. Y(x)=n⋅x for x∈U(L)_{ n }). This includes the particular case, generic for various applications to the study of dynamical systems, where L is the graded Lie algebra of polynomial vector fields spanned linearly by the x ^{ n } ∂ _{ x } and δ:=x∂ _{ x } acting on P(x)∂ _{ x } as δ(P(x)∂ _{ x }):=xP′(x)∂ _{ x }.
Lemma 9
The solution to the Atkinson recursion is a grouplike element in 1+U(L)^{+}. In particular, S(ϕ)=ϕ ^{−1}.
We are interested now in the situation where another Lie derivation d acts on U(L) and commutes with R (or equivalently with δ when R is the inverse of a derivation). A typical situation is given by Schwarz commuting rules between two different differential operators associated to two independent variables.
Theorem 10
The first identity D _{ d }(ϕ)=ϕ ^{−1}⋅d(ϕ) follows from the definition of the logarithmic derivative D _{ d }:=S∗d and from the previous Lemma.
The general case follows by induction. Let us assume that the identity holds for the components in degree n<p of D _{ d }(ϕ). We summarize in a technical Lemma the main ingredient of the proof. Notice that the Lemma follows directly from the Rota–Baxter relation and the definition of \(R_{d}^{[n+1]}(x)\).
Lemma 11
4 Magnustype formulas
The Magnus formula is a useful tool for numerical applications (computing the logarithm of the solution improves the convergence at a given order of approximation). It has recently been investigated and generalized from various points of view, see e.g. [4], where the formula has been extended to general dendriform algebras (i.e. noncommutative shuffle algebras such as the algebras of iterated integrals of operators), [2] where the algebraic structure of the equation was investigated from a purely preLie algebras point of view, or [1] where a generalization of the formula has been introduced to model the commutation of timeordered products with timederivations.
Here, we would like to go one step further and extend the formula to general logarithmic derivatives in the suitable framework in view of applications to dynamical systems and geometry, that is, the framework of enveloping algebras of graded Lie algebras and Lie derivations actions. Notations are as before, that is, L is a graded Lie algebra and d a graded Lie derivation (notice that we do not assume its invertibility on L or U(L)).
Lemma 12
Theorem 13
Let us show as a direct application of Theorem 13 how to recover the classical Magnus theorem (other applications to mould calculus and to the formal and analytic classification of vector fields are postponed to later works).
Example 14
Let us consider once again an operatorvalued linear differential equation X′(t)=X(t)λA(t), X(0)=1. Notice that the generator A(t) is written to the right, for consistency with our definition of logarithmic derivatives D _{ d }=S∗d. All our results can of course be easily adapted to the case d∗S (in the case of linear differential equations this amounts to consider instead X′(t)=A(t)X(t)), this easy task is left to the reader. Notice also that we introduce an extra parameter λ, so that the perturbative expansion of X(t) is a formal power series in λ.
Consider then the Lie algebra O of operators M(t) (equipped with the bracket of operators) and the graded Lie algebra \(L=\bigoplus_{n\in \mathbf{N}^{\ast}}\lambda^{n} O=O\otimes\lambda\mathbf{C}[\lambda]\). Applying Theorem 13, we recover the classical Magnus formula.
Although a direct consequence of the previous theorem (recall that any grouplike element in the enveloping algebra U(L) can be written as an exponential), the following proposition is important and we state it also as a theorem.
Theorem 15
When d is invertible on L, the logarithmic derivative D _{ d } is a bijection between the set of grouplike elements in U(L) and L.
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