ENVELOPING ALGEBRAS OF PRELIE ALGEBRAS, SOLOMON IDEMPOTENTS AND THE MAGNUS FORMULA FRE´DE´RICCHAPOTON AND FRE´DE´RICPATRAS 2 Abstract. We study the internal structure of enveloping algebras of preLie algebras. We show in 1 0 particularthatthecanonicalprojectionsarisingfromthePoincar´e-Birkhoff-Witttheoremcanbecom- 2 puted explicitely. They happen to be closely related to the Magnus formula for matrix differential n equations. Indeed,weshowthattheMagnusformulaprovidesawaytocomputethecanonical projec- a tion on the preLie algebra. Conversely, our results provide new insights on classical problems in the J theory of differential equations and on recent advances in theircombinatorial understanding. 0 1 ] A Q Contents . h 1. Introduction 1 t a 2. PreLie and enveloping algebras 2 m 3. The preLie PBW theorem 4 [ 4. The preLie PBW decomposition and the Magnus element 6 1 v Acknowledgements 7 9 References 7 5 1 2 . 1 0 1. Introduction 2 1 PreLiealgebrasplayanincreasinglyimportantroleinvariousfieldsofpureandappliedmathematics, : v as well as in mathematical physics. They appear naturally in the study of differential operators: the i X preLie product underlying the theory can be traced back to the work of Cayley in relation to tree r a expansions of compositions of differential operators. PreLie structures were defined formally much later in the work of Vinberg (they are sometimes refered to in the litterature as Vinberg algebras). TheywererediscoveredindeformationtheorybyGerstenhaber[6],whointroducedtheircurrentname. They appear currently in differential geometry, control theory, operads, perturbative quantum field theory... see e.g. [1, 2, 4] for some applications and further references on the subject. In most applications, preLie algebras structures appear through their representations, and their action can therefore be canonically lifted to the enveloping algebra of the underlying Lie algebra. The structure of enveloping algebras of preLie algebras, which are the subject of the present article, is richer than the structure of arbitrary enveloping algebras of Lie algebras. In the particular case of preLie algebras L, the usual Poincar´e-Birkhoff-Witt (PBW) theorem can indeed be refined: there exists on S(L), the symmetric algebra over L another associative, simply defined, product, making S(L) the enveloping algebra of L. Recall that, for Lie algebras, the PBW theorems asserts only that thecanonical map fromS(L)toU(L), theenveloping algebra constructed as thequotient of thetensor algebra by the ideal generated by Lie brackets, is a linear isomorphism; see e.g. [17] for details on the subject. 1 2 FRE´DE´RICCHAPOTONANDFRE´DE´RICPATRAS Inthecontextofdifferentialoperatorsandtrees,thedefinitionoftheproductonS(L)firstappeared in the work of Grossman-Larson [7]. The dual structure was discovered by Connes-Kreimer in the framework of perturbative quantum field theory [9, 3]. The work of Chapoton-Livernet [2] showed that free preLie algebras can be realized as Connes-Kreimer preLie algebras of trees, from which it follows that the enveloping algebra of a free preLie algebra L can be realized as a Grossman-Larson algebra, that is, as the symmetric algebra S(L) provided with a new productassociated to the process of insertion of trees. The general construction of the enveloping algebra of a preLie algebra L as S(L) is recent and due to Guin-Oudom, to which we refer for details [8]. In the present paper, we study the internal structure of S(L) in relation to the PBW theorem. The classical combinatorial study of enveloping algebras, due originally to Solomon [18] and developped further in the work of Reutenauer and coauthors [16, 17], shows that the canonical projections on the components of the direct sum decomposition of U(L) induced by the PBW isomorphisms and the natural decomposition of the symmetric algebra into homogeneous components: S(L) = Sn(L), n∈N where L is the free Lie algebra over an arbitrary set of generators, give rise to a family of oLrthogonal projections (i.e. idempotents) in the symmetric group group algebras Q[S ]. These idempotents, n the Solomon idempotents, play a key role in many fields of mathematics, partially due to their close connexion to enveloping algebra structures: free Lie algebras, control theory, group theory (both in the study of finiteCoxeter groups and the general theory of transformation groups), iterated integrals, Baker-Campbell-Hausdorff-type formulas (they were discovered independentlyof Solomon by Mielnik- Plebanskiin this setting [11]), noncommutative symmetric functions, homological algebra, andsoon... See e.g. [17, 5] for further details on the subject. These idempotents have been given many names in the litterature: canonical idempotents (as canonical projections in the enveloping algebras), Barr idempotent (for the first idempotent of the series), eulerian idempotents (because, according to the seminal work of Solomon, their coefficients in the symmetric group algebras are related to the eulerian numbers). Theoneof “Solomon idempotents” fits thebestwiththe practice for other Lie idempotents (e.g. the Dynkin, Garsia, Klyachko or Garsia-Reutenauer idempotents). In the particular case of enveloping algebras of free preLie algebras, new structures and formulas arise in relation to the Solomon idempotents, due to the possibility of realizing these enveloping algebras as S(L) equippedwith a new product. This is the subjectof the present article, which relates in particular the first Solomon idempotent to the Magnus series familiar in the theory of differential equations. 2. PreLie and enveloping algebras In this article, all vector spaces are over an arbitrary ground field k of characteristic zero. Definition 1. A preLie algebra is a vector space L equipped with a bilinear map x such that, for all x,y,z in L: (x x y) x z−x x (y x z) = (x x z) x y−x x (z x y). The vector space L is then equipped with a Lie bracket [x,y] := x x y−y x x. We write L for Lie the associated Lie algebra, when we want to emphasize that we view L as a Lie algebra. Most of the time this will be clear from the context, and we will then simply write L although making use of the Lie algebra structure. For example, we will write U(L) and not U(L ) for the enveloping algebra of Lie L . We will also denote by x the right action of the universal enveloping algebra of L on L that Lie Lie ON ENVELOPING ALGEBRAS OF PRELIE ALGEBRAS 3 extends the pre-Lie product: ∀a,b ∈ L, (b)a := b x a. Notice that this action is well defined since the product x makes L a module over the Lie algebra L : Lie ∀a,b,c ∈ L, ((c)b)a−((c)a)b = (c x b) x a−(c x a)x b = c x (b x a−a x b) = (c)[b,a]. Recall that one can consider S(L), the symmetric algebra over L, equipped with a product ∗ induced by x, as the enveloping algebra of L . Here, we realize S(L) = Sn(L) as the subspace Lie n∈N of symmetric tensors in T(L) := L⊗n, so that Sn(L) identifies withLthe S -invariant tensors n n∈N in L⊗n: Sn(L) = (L⊗n)Sn, where SL stands for the symmetric group of order n -the permutation n group of [n] := {1,...,n}. For a ∈ L, we write an for its n-fold symmetric tensor power a⊗n in S(L). More generally, for l ,...,l ∈ L, we write l ...l for the n-fold symmetric tensor product: 1 n 1 n 1 l ⊗...⊗l , where σ runs over S . n! σ σ(1) σ(n) n Anenveloping algebra carries thestructureof acocommutative Hopf algebra forwhich theelements P of the Lie algebra identify with the primitive elements. The corresponding coproduct on S(L) is given by: for arbitrary a ,...,a ∈ L, 1 n ∆(a ...a ) = a ⊗a 1 n I J I X where, for a subset I of [n], a := a , where I runs over the (possibly empty) subsets of [n] and I i∈I i J := [n]−I. The product ∗ is associative but not commutative and is defined as follows: Q (a ...a )∗(b ...b ) = B (a x B )...(a x B ), 1 l 1 m 0 1 1 l l f X where the sum is over all functions f from {1,...,m} to {0,...,l} and B := f−1(i). The increasing i filtration of S(L) by the degree is respected by the product ∗, but the direct sum decomposition into graded components is not. A symbol means that we consider the completion of a graded vector space: for example, L stands for the product L , where L stands for the linear span of preLie products of length n of elements n n n of L. The sberies exp(a) := an belongs for example to S[(L), and so on. A graded vectbor space Q n! n V = Vn is connected if V0P= k. n∈N SinLce we are interested in universal properties of preLie algebras, we assume from now on that L is a free preLie algebra. It follows that both L and L are naturally graded by the lengths of preLie Lie products. For clarity, we will distinguish notationally between: • U(L), the enveloping algebra of L defined as the quotient of the tensor algebra T(L) over L by the ideal generated by the sums [x,y]−x⊗y +y ⊗x, x,y ∈ L. The product on U(L) is written simply · (so that e.g. for x,y ∈ L,x·y = x⊗y in U(L)). • S∗(L), the symmetric algebra over L equipped with the structure of an enveloping algebra of L by the product ∗. Lie • S(L), the symmetric algebra over L equipped with the structure of a polynomial algebra. Recall that the product of l ,...,l ∈L in S(L) is written l l ...l . 1 n 1 2 n WewriteS∗,n(L)andSn(L)forthesymmetrictensorsoflengthninS∗(L)andS(L); wewriteS≥n(L) for sums of symmetric tensors of length at least n. The coproduct on S∗(L) and S(L) for which L is theprimitivepartis written∆inbothcases. ItprovidesS∗(L)withtheHopfalgebrastructurearising 4 FRE´DE´RICCHAPOTONANDFRE´DE´RICPATRAS from its enveloping algebra structure, whereas S(L) inherits a bicommutative Hopf algebra structure. Both S∗(L) and S(L) are free cocommutative coalgebras over L. We refer to [12, 8] for details. Thenaturallinearisomorphism(thePBWisomorphism)fromS(L)toU(L)obtained byidentifying S(L) with the symmetric tensors in T(X) is written π : nat 1 π (l ...l ) = l ·l ·...·l . nat 1 n n! σ(1) σ(2) σ(n) σX∈Sn It is a coalgebra isomorphism since it maps L to L and since bothS(L) and U(L) are cofree cocommu- tative coalgebras over L. Similarly, the identity map Id from S∗(L) to S(L) is (obviously) a coalgebra isomorphism. The canonical isomorphism, written is, between U(L) and S∗(L) maps a product of elements of L x ...x in U(L) to their product x ∗...∗x in S∗(L). Take care that is is not the inverse of the PBW 1 n 1 n map π . nat All these maps are summarized by the following diagram: S∗(L) −I→d S(L) −πn→at U(L)−i→s S∗(L). Recall at last from [14, 15] the following two results: Lemma 1. The Solomon first idempotent sol , that is the map from U(L) (resp. S∗(L)) to L orthog- 1 onally to the image of S≥2(L) by π (resp. is◦π ) can be expressed as: nat nat sol = log⋆(Id), 1 where Id is the identity map of U(L), resp. S∗(L). More generally, the map to the image of Sn(L) orthogonally to the image of the other graded components Si(L), i 6= n, reads sol⋆n sol = 1 . n n! TheLemmageneralizestoarbitraryenvelopingalgebrasofgradedconnectedLiealgebrasReutenauer’s computation of theSolomon idempotentin [16]. Herethelogarithm is computed intheconvolution al- gebrasoflinearendomorphismsofU(L)andS∗(L). Thatis,forf,g ∈ End(U(L))(resp. End(S∗(L))): f ⋆g := m◦(f ⊗g)◦∆, where m stands for the product in U(L), resp. S∗(L). Proposition 2. The inverse maps inv, resp inv to the PBW isomorphisms π and is◦π are S nat nat given by: sol⊗n inv(x) := 1 ◦∆[n], n! n X where ∆[n] is the iterated coproduct map from U(L) to U(L)⊗n. The same formula holds for inv . S 3. The preLie PBW theorem LetusconsidernowthecombinatorialPBWproblemforpreLiealgebras,thatistheparticularization of Solomon’s PBW problem for Lie algebras [18] to the preLie case: compute explicit formulas for the canonical decomposition of the enveloping algebra S∗(L) induced by the isomorphism is◦π with nat S(L) = Sn(L). n We arLe interested in the general form of the decomposition, that is, the way an arbitrary element l = l ...l of S∗(L) decomposes into l = sol (l)+...+sol (l). We can therefore assume that the l 1 n 1 n i ON ENVELOPING ALGEBRAS OF PRELIE ALGEBRAS 5 are algebraically independent in L. Moreover, by the classical polarization argument, setting a := l + ... + l , we get that l is the l ,...,l -multilinear component of an; similarly, sol (l) identifies 1 n 1 n n! i to the l ,...,l -multilinear component of sol (an) and of sol (exp(a)). In the end, the computations 1 n i n! i can therefore be handled in the sub free preLie algebra of L over a and amount to computing the sol (exp(a)). i We get: exp(a) = Id(exp(a)) = (exp⋆◦log⋆(Id))(exp(a)), where we recall that log⋆(Id) = sol . 1 Since ∆(an) = n ai ⊗an−i, a := exp(a) is a group-like element in S∗(L) (i.e. ∆(a) = a⊗a). i i≤n However, the logarPith(cid:0)m(cid:1)of a group-like element x is a primitive element since ∆(log∗(x)) = log∗(x⊗x) = log∗(x⊗1)·(1⊗x)= log∗(x)⊗1+1⊗log∗(x) and therefore log⋆(Id)(a) = log∗(a) is a primitive element in S∗(L) (here log⋆, resp. log∗ means that we compute the logarithm for the ⋆ product, resp. the ∗ product, and so on). We get finally the first, polarized, form of the preLie PBW decomposition: π[n]◦sol⊗n◦∆[n] (log∗(a))∗n a = 1 (a) = , n! n! n n X X where π[n] stands for the iterated ∗ product from (S∗(L))⊗n to S∗(L), resp. ∆[n] for the iterated coproduct from S∗(L) to (S∗(L))⊗n. Now, the l ,...,l -multilinear part sol (l ...l ) of log∗(a) reads 1 n 1 1 n n (−1)i−1 sol (l ...l ) = l ∗...∗l 1 1 n i I1 Ii Xi=1 I1X,...,Ii where the I ,...,I run over all ordered partitions of {l ,...,l } (I ... I = {l ,...,l }) where 1 i 1 n 1 i 1 n ∀j ∈ [n], I 6= ∅ and, for any subset I of [1,n], l := l . We get the first part of the following j I i∈I i ` ` Proposition: Q Proposition 3. The PBW decomposition of a generic element l ...l ∈ S∗(L) reads: 1 n n (−1)i−1 sol (l ...l )= l ∗...∗l . 1 1 n i I1 Ii Xi=1 I1X,...,Ii More generally, for higher components of the PBW decomposition, we have: n s(j,i) sol (l ...l ) = l ∗...∗l , i 1 n j! I1 Ij Xj=i I1X,...,Ij where the s(j,i) are Stirling numbers of the first kind. The proof of the general case follows from a Ψ-ring argument as for the geometric computation of the sol in [13]. Notice first that i Ψk := Id⋆k = (exp⋆(log⋆(Id)))⋆k = exp⋆(klog⋆Id) = knsol . n n X 6 FRE´DE´RICCHAPOTONANDFRE´DE´RICPATRAS Besides, with the same notation as above for ordered partitions, we get from the definition of the ⋆ product and the group-like behaviour of a: n k Ψk(l ...l )= l ∗...∗l . 1 n j I1 Ij Xj=1I1X,...,Ij(cid:18) (cid:19) From the definition of the Stirling numbers s(n,k) of the first kind, x(x−1)...(x−j+1) = s(j,i)xj, 1≤i≤j X we get: n n s(j,i) Ψk(l ...l ) = l ∗...∗l ki 1 n  j! I1 Ij Xi=1 Xj=i I1X,...,Ij and finally, by identification of the coefficient of ki:  n s(j,i) sol (l ...l ) = l ∗...∗l , i 1 n j! I1 Ij Xj=i I1X,...,Ij which concludes the proof. 4. The preLie PBW decomposition and the Magnus element Definition 2. The Magnus element in the free preLie algebra L over a single generator a is the (necessarily unique) solution Ω to the equation: b Ω a x = Ω. exp(Ω)−1 (cid:18) (cid:19) Theterminology is motivated by the so-called Magnus solution to an arbitrary matrix (or operator) differential equation X′(t) = A(t)X(t), X(0) = 1: X(t) = exp(Ω(t)), where ad B Ω′(t) = Ω(t) A(t) =A(t)+ nadn (A(t)), expadΩ(t)−1 n! Ω(t) n>0 X where ad stands for the adjoint representation and the B for the Bernoulli numbers. The link n with preLie algebras follows from the observation that (under the hypothesis that the integrals and derivatives are well-defined), for arbitrary time-dependent operators, the preLie product t M(t) x N(t) := [N(u),M′(u)]du Z0 satisfies (M(t) x N(t))′ = ad M′(t). The Magnus formula rewrites therefore: N(t) ′ Ω Ω′(t) = A(t) x exp(Ω)−1 (cid:18) (cid:18) (cid:19)(cid:19) where Ω iscomputedintheenvelopingalgebraofthepreLiealgebraoftime-dependentoperators. exp(Ω)−1 See e.g. the recent works by K. Ebrahimi-Fard and D. Manchon for further insights and an up-to-date point of view on the Magnus formula, in particular [4]. Theorem 4. The Magnus element identifies with sol (exp(a)): 1 Ω = sol (exp(a)) = log∗(exp(a)). 1 ON ENVELOPING ALGEBRAS OF PRELIE ALGEBRAS 7 Indeed, we have: (a−1)∗n sol (exp(a)) = log∗(a)= (−1)n−1 . 1 n n>0 X Let us write a = b+o(1) to mean that a and b in S∗(L) are equal up to an element of S∗,≥2(L). We then have, by definition of the ∗ product: (a−1)∗n−1 sol (exp(a)) = (a+o(1))∗ (−1)n−1 1 n ! n X (a−1)∗n−1 = a∗ (−1)n−1 +o(1). n ! n X Moreover, for an arbitrary element b of S∗,n(L) the product a∗b reads a x b+o(1). Finally, since sol (exp(a)) ∈ L, we get 1 (a−1)∗n−1 log∗(a) sol (exp(a)) = a x (−1)n−1 = a x , 1 n a−1 ! n (cid:18) (cid:19) X from which the theorem follows. Acknowledgements. This work originated with discussions at the IESC conference Institute of Carg`ese with K. Ebrahimi-Fard, F. Hivert, F. Menous, J.-Y. Thibon and the other participants to the CNRS PEPS program “Mould calculus”. We thank them warmly for the stimulating exchanges, as well as the IESC for its hospitality. References [1] A.Agrachev,R.Gamkrelidze,Chronological algebras and nonstationary vector fields,J.Sov.Math.17No.1(1981), 1650-1675. [2] F.Chapoton andM. Livernet,Pre-Lie algebras and the rooted trees operad Internat.Math. Res.Notices, 8, (2001), 395–408. [3] A.ConnesandD.Kreimer,Renormalization in quantum field theory and the Riemann–Hilbert problem I: The Hopf algebra structure of graphs and the main theorem, Commun. Math. Phys.210 (2000) 249–273. [4] K. Ebrahimi-Fard, D. 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Theory 4 (1968) 363375 Institut Camille Jordan Universit´e Claude Bernard Lyon 1 Baˆtiment Braconnier 21 Avenue Claude Bernard F-69622 VILLEURBANNE Cedex FRANCE Laboratoire J.-A. Dieudonn´e UMR 6621, CNRS, Parc Valrose, 06108 Nice Cedex 02, France