PRE-LIE ALGEBRAS AND INCIDENCE CATEGORIES OF COLORED ROOTED TREES

Transcription

1 PRE-LIE ALGEBRAS AND INCIDENCE CATEGORIES OF COLORED ROOTED TREES MATT SZCZESNY Abstract. The incidence category C F of a family F of colored posets closed under disjoint unions and the operation of taking convex sub-posets was introduced by the author in [], where the Ringel-Hall algebra H F of C F was also defined. We show that if the Hasse diagrams underlying F are rooted trees, then the subspace n F of primitive elements of H F carries a pre-lie structure, defined over Z, and with positive structure constants. We give several examples of n F, including the nilpotent subalgebras of sl n, Lgl n, and several others.. Introduction A left pre-lie algebra is a k vector space A endowed with a binary bilinear operation satisfying the identity (.) (a b) c a (b c) = (b a) c b (a c) It follows easily from. that anti-symmetrizing yields a Lie bracket [a, b] = a b b a on A. However, not every Lie algebra arises from a pre-lie algebra. Pre-Lie algebras first appeared in the works of E.B. Vinberg [3] and M. Gerstenhaber [3], and have since found applications in several areas. One prominent example is perturbative quantum field theory [4], where insertion of Feynman graphs into each other equips them with a pre-lie structure which controls the combinatorics of the renormalization procedure. In this paper, we show that pre-lie algebras arise naturally from incidence categories introduced by that author in []. An incidence category is built from a collection F of colored posets, which is closed under the operations of disjoint union and convex subposet - we will denote it by C F. The objects of C F are the posets in F, and for P, P F Hom(P, P ) := {(I, I, f) I j is an order ideal in P j, f : P \I I an isomorphism } Here, the poset I should be viewed as the kernel of the morphism, and I as the image. All morphisms in C F have kernels and cokernels, and so the notion of exact sequence makes sense. In [], the Ringel-Hall algebra H CF of C F was defined. The author is supported by an NSA grant.

2 MATT SZCZESNY H CF is the Q vector space of finitely supported functions on isomorphism classes of C F : H CF := {f : Iso(C F ) Q supp(f) < } with product given by convolution: (.) f g(m) = A M f(a)g(m/a). H CF possesses a co-commutative co-product given by (.3) (f)(m, N) = f(m N) (where M N denotes the disjoint union of M and N) as well as an antipode, making it a Hopf algebra. H CF is graded, connected, and co-commutative, and so by the Milnor-Moore theorem isomorphic to U(n F ), where n F is the Lie algebra of its primitive elements. It follows from.3 that n F = span{δ P P F, P connected } We show that if F consists of posets whose Hasse diagrams are rooted trees, then n F carries a pre-lie structure, with δ P δ P := δ P δ P δ P P. A more concrete description of is the following: δ P δ P is a sum of deltafunctions supported on connected posets P F whose Hasse diagram is obtained by grafting the root of P onto a vertex of P. It follows from the definition of that the structure constants are non-negative integers. The paper is organized as follows. Section recalls the definition of pre-lie algebra and introduces the universal example, namely the pre-lie algebra of colored rooted trees. In section 3 we recall the construction of the incidence category C F as well as its main properties. The Ringel-Hall algebra of C F is introduced in section 4. In section 5 we define the pre-lie structure on n F and verify that it satisfies the identity.. Finally, section 6 is devoted to examples - among these are pre-lie structures on nilpotent Lie subalgebras of sl n and Lgl n. Acknowledgements: The author is very grateful to Pavel Etingof for valuable discussions and suggestions.. Pre-Lie algebras In this section, we recall the definition and some examples of (left) pre-lie algebras. Let k be a field. Definition. A left pre-lie algebra is a k vector space A endowed with a binary bilinear operation satisfying the left pre-lie identity (.) (a b) c a (b c) = (b a) c b (a c) for a, b, c A.

3 PRE-LIE ALGEBRAS AND INCIDENCE CATEGORIES OF COLORED ROOTED TREES 3 One checks easily that antisymmetrizing the operation gives A the structure of a Lie algebra. [a, b] = a b b a Example. Any associative k algebra A is a pre-lie algebra with the pre-lie structure given by a b := ab, where the right hand side refers to the associative multiplication in A. Example. One of the most important examples of pre-lie algebras is given by colored rooted trees. Recall that a tree is a graph with no cycles. We denote by E(t), V (t) the edge and vertex sets of t respectively. Let S be a finite set. By a rooted tree colored by S we mean a tree with a distinguished vertex r(t) V (t) called the root, and an assignment of an element of S to each v V (t). We adopt the convention that rooted trees are always drawn with the root on top. For example, if S = {a, b}, then the following are rooted trees colored by S: a b a a b b b b a a b b a Let T S denote the set of rooted trees whose vertices are colored by S. Given t T S, and e E(t), removing e disconnects t into two colored rooted trees: R e (t) containing r(t) and P e (t), whose root is the end of e. Let T S be the k vector space spanned by T S. We have T S = n=0 T S[n] where T S [n] is the subspace of T S spanned by trees with n vertices. For colored rooted trees t, t T S, let t t := s T S n(t, t, s)s where For example, we have n(t, t, s) = #{e E(s) P e (s) = t, R e (s) = t }. b a b = a b + ab b b It is well-known (see for instance []) that defines a pre-lie structure on T S. The following theorem is proven in [] Theorem. T S is the free pre-lie algebra on S generators. Remark. In what follows, unless stated otherwise, k = Q.

4 4 MATT SZCZESNY 3. Incidence categories 3.. Recollections on posets. We begin by recalling some basic notions and terminology pertaining to posets ( partially ordered sets) following [0, ]. () An interval is a poset having unique minimal and maximal elements. For x, y in a poset P, we denote by [x, y] the interval [x, y] := {z P : x z y} If P is an interval, we will often denote by 0 P and P the minimal and maximal elements. () An order ideal in a poset P is a subset L P such that whenever y L and x y in P, then x L. (3) A sub-poset Q of P is convex if, whenever x y in Q and z P satisfies x z y, then z Q. Equivalently, Q is convex if Q = L\I for order ideals I L in P. (4) Given two posets P, P, their disjoint union is naturally a poset, which we denote by P +P. In P +P, x y if both lie in either P or P, and x y there. (5) A poset which is not the union of two non-empty posets is said to be connected. (6) The cartesian product P P is a poset where (x, y) (x, y ) iff x x and y y. (7) A distributive lattice is a poset P equipped with two operations, that satisfy the following properties: (a), are commutative and associative (b), are idempotent - i.e. x x = x, x x = x (c) x (x y) = x = x (x y) (d) x y = x x y = y x y (e) x (y z) = (x y) (x z) (f) x (y z) = (x y) (x z) (8) For a poset P, denote by J P the poset of order ideals of P, ordered by inclusion. J P forms a distributive lattice with I I := I I and I I := I I for I, I J P. If P, P are posets, we have J P +P = J P J P, and if I, L J P, and I L, then [I, L] is naturally isomorphic to the lattice of order ideals J L\I. Remark. Suppose that the Hasse diagram of a poset P is a rooted tree - that is, P has a unique maximal element r(p), and the Hasse diagram contains no cycles. It is then easy to see that order ideals I P correspond to admissible cuts of P, where the latter is a collection of edges C E(P), having the property that at most one edge of C is encountered along any path from root to leaf. For

5 PRE-LIE ALGEBRAS AND INCIDENCE CATEGORIES OF COLORED ROOTED TREES 5 instance, the dotted edges of the poset T below yield an admissible cut: b b a b Each admissible cut C E(P) divides the tree into a rooted connected tree R C (P) containing r(p), and a rooted forest (a disjoint union of rooted trees) P C (P). The notation is clearly an extension of that used in example. In the last example, we have R C (T) = b P C (T) = b b a 3.. From posets to categories. Let F be a family of colored posets which is closed under the formation of disjoint unions and the operation of taking convex subposets, and let P(F) = {J P : P F} be the corresponding family of distributive lattices of order ideals. For each pair P, P F, let M(P, P ) denote the set of colored poset isomorphisms P P. It follows that M(P, P) forms a group, which we denote Aut M (P) The category C F. We proceed to define a category C F, called the incidence category of F as follows. Let Ob(C F ) := F = {P F} and Hom(P, P ) := {(I, I, f) : I i J Pi, f M(P \I, I )} i =, We need to define the composition of morphisms Hom(P, P ) Hom(P, P 3 ) Hom(P, P 3 ) Suppose that (I, I, f) Hom(P, P ) and (I, I 3, g) Hom(P, P 3 ). Their composition is the morphism (K, K 3, h) defined as follows. We have I I I, and since f : P \I I is an isomorphism, f (I I ) is an order ideal of P \I. Since in J P, [I, P] J P \I, we have that f (I I ) corresponds to an order ideal K J P such that I K. We have I I I, and since [I, P ] J P \I, I I corresponds to an order ideal L J P \I. Since g : P \I I 3 is an isomorphism, g(l ) J I 3, and since J I 3 J P3, g(l ) corresponds to an order ideal K 3 J P3 contained in I 3. The isomorphism f : P \I I restricts to an isomorphism f : P \K I \I I = I \I, and the isomorphism g : P \I restricts to an isomorphism ḡ : I I \I = I \I K 3. Thus, g f : P \K K 3 is an isomorphism and g f M(P \K, K 3 ) by the property (4) above.

6 6 MATT SZCZESNY As shown in [], the composition of morphisms is associative. Remark 3. We refer to I as the image of the morphism (I, I, f) : P P. We denote by Iso(C F ) the collection of isomorphism classes of objects in C F, and by [P] the isomorphism class of P C F Properties of the categories C F. We now enumerate some of the properties of the categories C F. () The empty poset is an initial, terminal, and therefore null object. We will sometimes denote it by. () We can equip C F with a symmetric monoidal structure by defining P P := P + P. (3) The indecomposable objects of C F are the P with P a connected poset in F. (4) The simple objects of C F are the P where P is a one-element poset. (5) Every morphism (3.) (I, I, f) : P P has a kernel (, I, id) : I P (6) Similarly, every morphism 3. possesses a cokernel (I, P \I, id) : P P \I We will use the notation P /P for coker((i, I, f)). Note: Properties 5 and 6 imply that the notion of exact sequence makes sense in C F. (7) All monomorphisms are of the form (, I, f) : Q P where I J P, and f : Q I M(Q, I). Monomorphisms Q P with a fixed image I form a torsor over Aut M (I). All epimorphisms are of the form (I,, g) : P Q

7 PRE-LIE ALGEBRAS AND INCIDENCE CATEGORIES OF COLORED ROOTED TREES 7 where I J P and g : P \I Q M(P \I, Q). Epimorphisms with fixed kernel I form a torsor over Aut M (P \I) (8) Sequences of the form (3.) (,,id) I (,I,id) P (I,,id) (P \I,,id) P \I with I J P are short exact, and all other short exact sequences with P in the middle arise by composing with isomorphisms I I and P \I Q on the left and right. (9) Given an object P and a subobject I, I J P, the isomorphism J P \I [I, P] translates into the statement that there is a bijection between subobjects of P/I and order ideals J J P such that I J P. The bijection is compatible with quotients, in the sense that (P/I)/(J/I) J/I. (0) Since the posets in F are finite, Hom(P, P ) is a finite set. () We may define Yoneda Ext n (P, P ) as the equivalence class of n step exact sequences with P, P on the right and left respectively. Ext n (P, P ) is a finite set. Concatenation of exact sequences makes into a monoid. Ext := A,B I(CF ),n Ext n (A, B) () We may define the Grothendieck group of C F, K 0 (C F ), as K(C F ) = A C F Z[A]/ where is generated by A + B C for short exact sequences A C B We denote by k(a) the class of an object in K 0 (C F ). 4. Ringel-Hall algebras For an introduction to Ringel-Hall algebras in the context of abelian categories, see [8]. We define the Ringel-Hall algebra of C F, denoted H CF, to be the Q vector space of finitely supported functions on isomorphism classes of C F. I.e. H CF := {f : Iso(C F ) Q supp(f) < } As a Q vector space it is spanned by the delta functions δ A, A Iso(C F ). The algebra structure on H CF is given by the convolution product: (4.) f g(m) = A M f(a)g(m/a)

8 8 MATT SZCZESNY for M Iso(C F ). In what follows, it will be conceptually useful to choose a representative in each isomorphism class. For M, N, Q Iso(C F ), let F Q M,N be the number of exact sequences M i Q π N where (i, π) and (i, π ) are considered equivalent iff i = i and π = π (this makes sense, since we have fixed a representative in each isomorphism class). It follows from the definition 4. that δ M δ N = Q Iso(C F ) F Q M,N Aut(M) Aut(N) δ Q, from which it is apparent that H CF encodes the structure of extensions in C F. H CF possesses a co-commutative co-product given by (4.) (f)(m, N) = f(m N) as well as a natural K + 0 (C F ) grading in which δ A has degree k(a) K + 0 (C F ). If F is colored by the set S, it is easy to see that K + 0 (C F) N S. The subobjects of P C F are exactly I J P, and the product 4. becomes f g([p]) = I J P f([i])g([p \I]). It is shown in [8] that the product is associative, the co-product co-associative and co-commutative, and that the two are compatible, making H CF into a cocommutative bialgebra. Recall that a bialgebra A over a field k is connected if it possesses a Z 0 grading such that A 0 = k. In addition to the K 0 + (C F ) grading, H CF possesses a grading by the order of the poset - i.e. we may assign deg(δ P ) = P. This gives it the structure of graded connected bialgebra, and hence Hopf algebra. The Milnor-Moore theorem implies that H CF is the enveloping algebra of the Lie algebra of its primitive elements, which we denote by n F - i.e. H CF U(n F ). It follows from 4. that f n F is primitive if it is supported on the isomorphism classes of connected posets. Thus, we have that n F = span{δ P P F, P connected } We will use the notation F conn F to denote the sub-collection of F consisting of connected posets. We have thus established the following: Theorem. The Ringel-Hall algebra of the category C F is a co-commutative graded connected Hopf algebra, isomorphic to U(n F ), where n F denotes the graded Lie algebra of its primitive elements. n F = span{δ P P F conn }. Remark 4. H CF is a special case of an incidence Hopf algebra introduced by Schmitt in [0, 9].

9 PRE-LIE ALGEBRAS AND INCIDENCE CATEGORIES OF COLORED ROOTED TREES 9 5. A Pre-Lie structure on n F We assume now that the collection F consists of colored posets whose underlying Hasse diagrams are rooted trees. Recall that F was assumed to be: closed under the operation of taking convex sub-posets closed under disjoint unions It is immediate that to produce an F satisfying these two requirements, one may start with an arbitrary collection F of colored posets, and close it with respect to each operation - i.e. adjoin to F all convex sub-posets and all disjoint unions of these. If F arises in this way as the closure of F, we will write F = F. Example 3. Suppose that F consists of a single poset, whose Hasse diagram is an n vertex ladder colored by the set S = {,, n}. 3 n n Let us adopt the notation L(a, a,..., a k ) for a k vertex ladder Hasse diagram labeled by a, a,..., a k root-to-leaf (F thus consisting of L(,,, n)). To close F with respect to convex subsets, we must adjoin to it L(r, r+, r+,, r+m), where r r + m n.... n,... n,... n n 3 n n 3 n n Finally, closing with respect to disjoint unions, we can identify elements of F = F with Young diagrams having at most n rows, each of whose columns is labeled by k, k +,, k + m. For instance

10 0 MATT SZCZESNY is identified with the poset L(, 3, 4, 5) + L(,, 3) + L(3) + L(4). We proceed to equip n F with a pre-lie structure. For a, b F conn, we define (5.) δ a δ b = δ a δ b δ a b and extend the product to all of n F by linearity. The subtraction of the term δ a b in 5. has the effect of removing the delta-function supported on the one split extension of b by a, and so the right-hand side of 5. does indeed lie in n F. It follows easily that we may re-write the definition 5. as: (5.) δ a δ b = t F n(a, b, t)δ t where n(a, b, t) is defined as in example. Theorem 3. Let F be a collection of colored posets closed with respect to taking convex sub-posets and disjoint unions. If the Hasse diagrams of posets in F are rooted trees, then equips n F with the structure of a pre-lie algebra. Proof. A two-sided pre-lie ideal in a pre-lie algebra A is a subspace I A such that if x I, then a x I and x a I a A. One checks easily that the quotient A/I inherits a pre-lie structure. Let F be a collection of colored rooted forests colored by S, closed under the operations of disjoint union and convex sub-poset, and F conn F the connected ones (i.e. the rooted trees). F conn is closed under taking convex sub-posets. I claim that J = T S \F conn is a two-sided pre-lie ideal in T S. Let u T S and s J. We have δ u δ s = t T S n(u, s, t)δ t Suppose that n(u, s, t) 0 and t T S \J = F conn. t has an edge e such that P e (t) = u and R e (t) = s, and since both are convex sub-posets of the poset t F conn, u, s F conn, contradicting the fact that s J. It follows that δ u δ s J. The same argument shows that δ s δ u J. The quotient T S /J is canonically identified with n F with the bracket 5.. We give a second proof, very close to the one for F = T S given in []. Proof. We need to verify the identity.. It follows from 5. that for a, b, c F conn,

11 PRE-LIE ALGEBRAS AND INCIDENCE CATEGORIES OF COLORED ROOTED TREES (δ a δ b ) δ c = ( n(a, b, t)t) c t F conn = n(a, b, t)n(t, c, s)s s,t F conn and δ a (δ b δ c ) = a ( n(b, c, t)t) s,t F conn t F conn = n(b, c, t)n(a, t, s)s s,t F conn Because F is closed under taking convex sub-posets, P e (t) F conn and R e (t) F conn, t F conn. The sum t F conn n(a, b, t)n(t, c, s) may be identified with the number of pairs of edges π = {e, e } E(s), such that the resulting cut is NOT admissible (i.e. both edges lie along a single path from root to leaf in s), and the three connected components when π is removed, are, top-to-bottom, c, b and a. Similarly, the sum t F conn n(b, c, t)n(a, t, s) may identified with the number of pairs π = {e, e } E(s) such that the corresponding cut of s results in three components a, b, c, with r(s) c, and no element of a greater than an element of b. The coefficient of δ s in δ a (δ b δ c ) (δ a δ b ) δ c therefore counts the number of admissible two-edge cuts of s such that the connected component containing r(s) is isomorphic to c, and the remaining two to a, b respectively. Applying the same analysis to the right-hand-side of. proves the equality. Remark 5. It follows from 5. that n F is defined over Z, and that the structure constants are non-negative. 6. Examples In this section, we consider different examples of families F, and the resulting pre-lie algebras n F. Recall that since n F is graded by N, the Lie algebra n F is pro-nilpotent (nilpotent if n F is finite-dimensional). Example 4. Let S be a finite set, and F = T S, the set of rooted forests colored by S. We then obtain the pre-lie algebra structure on S labeled rooted trees described in example.

12 MATT SZCZESNY Example 5. Suppose S consists of a single element, and let F = F, where F is the collection of all ladders: (since there is only one color, we suppress the labeling). Denote by L n the n- vertex ladder. We have δ Ln δ Lm = δ Lm+n. so the Lie algebra n F is abelian. In the Ringel-Hall algebra H CF δ Ln δ Lm = δ Lm+n + δ Lm L n we have and (L m ) = L m + L m It is well-known (see eg. [6]) that the Hopf algebra H CF is isomorphic to the Hopf algebra of symmetric functions, with L m corresponding to the mth power sum. Example 6. Let S = {,,, n}, and let F = F, where F consists of singleton vertices colored by S. F is thus the collection of all finite sets colored by S, with trivial partial order. Denote by X(m, m,, m n ) the set of m + m + + m n elements, with m i colored i, i n. n F is therefore spanned by the δ X(0,,i,,0). The operation is identically 0, so the Lie algebra n F is abelian. In H CF we have ( n ( ) ) mi + m i δ X(m,,m n) δ X(m,,m n) = δ (m+m,,mn+m n) i Example 7. Let S = {,,, n}, and let F = F, where F consists of all S colored ladder trees 3 m i

13 PRE-LIE ALGEBRAS AND INCIDENCE CATEGORIES OF COLORED ROOTED TREES 3 Denote by L(a,, a k ) the k-vertex ladder whose ith vertex counting from the leaf is colored a i. We have (6.) δ L(a,,a n) δ L(b,,b m) = δ L(a,,a n,b,,b m) Let Q < X,, X s > denote the free associative algebra on S viewed as a Lie algebra. There is a linear isomorphism ρ : n F Q < X,, X s > ρ(l(a,, a k )) = X a X a X ak It follows from 6. that ρ is a Lie algebra isomorphism. Example 8. Consider the collection F from example 3, where F = L(,,, n). Here n F = span{δ L(k,,k+m) }, k k + m n. We have { δ L(p,,p+r) δ L(k,,k+m) = δl(k,,p+r) if k + m + = p 0 otherwise so that in the Lie algebra n F, { (6.) [δ L(p,,p+r), δ L(k,,k+m) ] = δl(k,,p+r) if k + m + = p 0 otherwise Let E i,j denote the (n + ) (n + ) matrix with a in entry (i, j) and zeros everywhere else. Then the commutation relations 6. imply that the map φ : n F Mat n+ φ(δ L(k,,k+m) ) = E k,k+m+ is an isomorphism of n F onto the Lie algebra of upper-triangular (n+) (n+) matrices. Example 9. Let S = {, }, and let F = F, where F consists of all S colored ladders where the colors alternate., Let us denote by L(i, n), i S, n the alternating ladder with n vertices, whose root is colored i. Then n F = span{l(i, n)}, i S, n. We have { δl(i,n+m) if m 0 mod, i S δ L(i,n) δ L(i,m) = 0 otherwise { δl(j,n+m) if m mod, i j S δ L(i,n) δ L(j,m) = 0 otherwise

14 4 MATT SZCZESNY It follows that (6.3) [δ L(i,k), δ L(j,l) ] = 0 { δl(j,(k+l)+) if i = j [δ L(i,k), δ L(j,l+) ] = if i j δ L(j,(k+l)+) [δ L(i,k+), δ L(j,l+) ] = δ L(j,(k+l+)) δ L(i,(k+l+)) Recall that gl = Mat = n h n +, where n = span{f}, n + = span{e}, h = span{h, h } and f = ( ) h = ( ) h = ( ) e = ( ) Let Lgl = gl Q[t, t ] be the loop algebra of gl, with bracket [X t m, Y t n ] = [X, Y ] t n+m Lgl also has a triangular decomposition Lgl = Lgl + h Lgl, where Lgl + = n + gl tq[t] Lgl = n gl t Q[t ] Let φ : n F Lgl + φ(δ L(,k+) ) = e t k φ(δ L(,k+) ) = f t k+ φ(δ L(,k) ) = h t k φ(δ L(,k) ) = h t k It follows from 6.3 that φ is an isomorphism. It follows that U(Lgl + ) has an integral basis which may be identified with Young diagrams whose columns are colored by alternating strings of s and s. Example 0. A straightforward generalization of the previous example, with S = {,, n} and F consisting of ladders periodically colored by,, n yields n F Lgl + n.

15 PRE-LIE ALGEBRAS AND INCIDENCE CATEGORIES OF COLORED ROOTED TREES 5 Example. Let S = {, }, and let F = F, where F is the set of all ladders colored by a sequence of s followed by a sequence of s. Denote by L(i, j) the ladder with i s followed by j s. We have δ L(i,j) δ L(m,n) = 0 if ij > 0 and mn > 0 { δl(i+m,0) if n = 0 δ L(i,0) δ L(m,n) = 0 otherwise δ L(0,j) δ L(m,n) = δ L(m,n+j) δ L(i,j) δ L(m,0) = δ L(i+m,j) { δl(0,j+n) if i = 0 δ L(i,j) δ L(0,n) = 0 otherwise so that we obtain the following non-zero commutation relations (i.e. all other commutators are 0): [δ L(i,0), δ L(0,n) ] = δ L(i,n) [δ L(i,0), δ L(m,n) ] = δ L(m+i,n) if n > 0 [δ L(0,j), δ L(m,n) ] = δ L(m,n+j) if m > 0 Example. Let S = {,,, n}, and let F = F, where F consists of all S colored corollas (rooted trees where all leaves are connected directly to the root) 3 Closing F with respect to convex sub-posets means adjoining singleton colored trees. Denote by X(i) the singleton tree colored by i n, and by Y (i, a,, a n ) the corolla whose root is colored i and which has a + a + a n

16 6 MATT SZCZESNY leaves, with a colored, a colored etc. In n F we have δ X(i) δ X(j) = δ Y (j,0,,i,,0) δ X(i) δ Y (j,a,,a n) = δ Y (j,a,,a i +,,a n) δ Y (j,a,,a n) δ X(i) = 0 δ Y (j,a,,a n) δ Y (j,b,,b n) = 0 which leads to the following commutation relations: [δ X(i), δ X(j) ] = δ Y (j,0,,i,,0) δ Y (i,0,,j,,0) [δ X(i), δ Y (j,a,,a n)] = δ Y (j,a,,a i +,,a n) [δ Y (j,a,,a n), δ Y (j,b,,b n)] = 0

17 PRE-LIE ALGEBRAS AND INCIDENCE CATEGORIES OF COLORED ROOTED TREES 7 References [] Cartier, P. A primer of Hopf algebras. Frontiers in number theory, physics, and geometry. II, , Springer, Berlin, 007. [] Chapoton, F.; Livernet, M. Pre-Lie algebras and the rooted trees operad. Internat. Math. Res. Notices 00, no. 8, [3] Gerstenhaber, M. The cohomology structure of an associative ring, Ann. of Math. 78 (963), [4] Kreimer, D. On the Hopf algebra structure of perturbative quantum field theory. Adv. Theor. Math. Phys (998). [5] Kremnizer K. and Szczesny M. Feynman graphs, rooted trees, and Ringel-Hall algebras. Comm. Math. Phys. 89 (009), no [6] Macdonald, I. G. Symmetric functions and Hall polynomials. Second edition. With contributions by A. Zelevinsky. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 995. x+475 pp. [7] Rota, G-C. On the Foundations of Combinatorial Theory I: Theory of Mbius Functions, Zeitschrift fr Wahrscheinlichkeitstheorie und Verwandte Gebiete : , (964). [8] Schiffmann, O. Lectures on Hall algebras. Preprint math.rt/0667. [9] Schmitt, W. R. Antipodes and Incidence Coalgebras. Journal of Comb. Theory. A 46 (987), [0] Schmitt, W. R. Incidence Hopf algebras. J. Pure Appl. Algebra 96 (994), no. 3, [] Stanley, R. Enumerative combinatorics Vol.. Cambridge Studies in Advanced Mathematics, 49. Cambridge University Press, Cambridge, 997. [] Szczesny, M. Inicidence categories. J. Pure and Appl. Algebra, in press. [3] Vinberg, E.B. The theory of homogeneous convex cones, Transl. Moscow Math. Soc. (963), Department of Mathematics and Statistics, Boston University, Boston MA, USA address: szczesny@math.bu.edu