Pre-Lie lgers, rooted trees nd relted lgeri strutures Mrh 23, 2004
Definition 1 A pre-lie lger is vetor spe W with mp : W W W suh tht (x y) z x (y z) = (x z) y x (z y). (1) Exmple 2 All ssoitive lgers re lso pre-lie lgers. Exmple 3 The vetor spe of polynomil vetor fields on ffine spe A n. i P (x) xi j = i Q(x) xj j Q(x) ( xj P (x) ) xi. (2) 1
Theorem 4 (CL) The free pre-lie lger on single genertor hs sis indexed y rooted trees. The pre-lie produt is given y the sum over ll possile grftings. 2
Corollry 5 For given polynomil vetor field P, there exists unique morphism from the free pre-lie lger on one genertor O to the pre-lie lger of polynomil vetor fields whih mps O to P. To ny rooted tree T, one n ssoite in this wy vetor field T P. Exmple 6 Consider the following vetor field (not polynomil, ut nlyti): V = exp(x) x. (3) Then for ny rooted tree T, one hs T V = T exp(x) x, (4) where T is the numer of verties of T. Exmple 7 Find wht is T V for V = x x. 3
Definition 8 The PreLie operd is the operd desriing pre-lie lgers. Rell tht n operd P is given y olletion of modules P(n) over the symmetri groups S n with omposition mps i : P(m) P(n) P(m + n 1), (5) stisfying nturl xioms. The stndrd exmple is given y P(n) = hom(w n, W ) (6) for some fixed vetor spe W, together with omposition of multi-liner opertions t position i. 4
One n reformulte nd enhne the desription of the free pre-lie lgers s desription of the PreLie operd. Theorem 9 (CL) The vetor spe PreLie(n) hs sis indexed y the set of rooted trees with verties in ijetion with {1,..., n} (lelled rooted trees). The tion of S n is y hnging the deortion. The omposition T i T of tree T t ple i of tree T is sum over the set of mps from inoming edges t i to verties of T. 5 4 3 5 3 1 2 2 1 2 1 2 3 2 1 4 4 5
For ny olletion P(n) of S n -modules, one defines n nlyti funtor whih mps vetor spe W to P(W ) = n 1 W n Sn P(n). (7) Rell tht one n define the derived funtor P of n nlyti funtor P y the formul P (W ) = P(W K{ }). (8) Equivlently, the S n -module P (n) is the restrition of the tion of S n+1 on P(n + 1) to the sugroup fixing n + 1. Theorem 10 If P is n operd, then, for ny vetor spe W, P (W ) hs nturl struture of ssoitive lger. The produt is given y omposition t the distinguished ple. 6
Exmple 11 Let us onsider the se of the PreLie operd nd vetor spe W = K{,, }. Then PreLie (W ) hs sis indexed y rooted trees with mp from verties to the set {,,, } suh tht is the imge of extly one vertex, lled the distinguished vertex (mrked deorted rooted trees). 7
Theorem 12 The suspe I(W ) spnned y trees where is not the root is twosided idel of PreLie (W ). The quotient lger hs nother desription. Rell tht the rket [x, y] = x y y x, (9) in pre-lie lger defines Lie lger. Theorem 13 The quotient of PreLie (W ) y I(W ) is isomorphi s n ssoitive lger to the universl enveloping lger of the Lie lger ssoited to the free pre-lie lger on W. PreLie (W )/I(W ) U(PreLie(W ) Lie ). (10) N.B. This enveloping lger is the dul of the Hopf lger of rooted trees, see Buther, Dür, Grossmn & Lrsson, Connes & Kreimer, Hoffmn, Foissy et.. 8
Definition 14 A lef is vertex without inoming edges. Theorem 15 The suspe Q(W ) spnned y trees where is lef is su-lger of PreLie (W ). Definition 16 A verterte is rooted tree with distinguished vertex lled the til. N.B. The til n e the sme s the root. A rooted tree where is lef n e onsidered s verterte: the distinguished vertex is removed nd the til is the vertex to whih is ws grfted. 9
Theorem 17 The lger Q(W ) is the free ssoitive lger on the suspe spnned y rooted trees where is lef tthed to the root. When onsidered s vertertes, the genertors re the vertertes where the til is the root. They n e identified with rooted trees. 10
Theorem 18 (Foissy) The universl enveloping lger U(PreLie(W ) Lie ) is free ssoitive lger. No expliit desription of suspe M(W ) of genertors is known. Theorem 19 The lger PreLie (W ) is free ssoitive lger on the spe of genertors M(W ) W, where the genertors in the prt W re the trees with two verties whose vertex mrked y is not the root. 11
Generting funtions To eh olletion P(n) of S n -modules, one ssoites generting funtion: P = n 0 dim P(n) x n /n!. (11) For PreLie, one gets PL = n 1 n n 1 x n /n!, (12) losely relted to the Lmert W-funtion. Mny lgeri theorems on free pre-lie lgers imply nlyti properties of the funtion PL. For exmple: Theorem 20 (CL) The free pre-lie lger PreLie(W ) is free module over the universl enveloping lger U(PreLie(W ) Lie ) over the genertors W. Hene one hs PL = x exp(pl). (13) 12
From now on, reple the tegory of vetor spes y the tegory of hins omplexes with vnishing differentils. Definition 21 A Λ-lger is hin omplex W with mp : W W W of degree 0 nd mp, : W W W of degree 1 suh tht is pre-lie produt,, is Lie rket of degree 1 nd the following reltion holds ± x y, z ± z y, x = z, x y, (14) where pproprite signs hve to e inserted ording to the Koszul sign rules. Mny good properties of pre-lie lgers should generlize to Λ-lgers. 13
Conjeture 22 The generting series for the nlyti funtor Λ is given y Λ = n 1 n 1 k=1 (n k t) x n /n!, (15) This would follow from the Koszul property for the operd Λ. These dimensions re known to e upper ounds. Note tht one reovers PL when t = 0. Conjeture 23 The free Λ-lger Λ(W ) is free s Lie lger nd its enveloping lger is free s n ssoitive lger. Conjeture 24 The ssoitive lger Λ (W ) is free ssoitive lger. Conjeture 25 There exists quotient mp of ssoitive lgers Λ (W ) U(Λ(W ) Lie ). (16) 14