differential algebra structures on families of trees › download › pdf › 81129346.pdf · 100...

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Advances in Applied Mathematics 35 (2005) 97–119

www.elsevier.com/locate/yaam

Differential algebra structures on families of tree

Robert L. Grossman∗, Richard G. Larson

Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago,851 S. Morgan Street, Chicago, IL 60607-7045, USA

Received 7 July 2002; accepted 24 January 2005

Available online 28 April 2005

Abstract

It is known that the vector space spanned by labeled rooted trees forms a Hopf algebra. Lk bea field and letR be a commutativek-algebra. LetH denote the Hopf algebra of rooted trees labeusing derivations in Der(R). In this paper, we introduce a construction that givesR a H -modulealgebra structure and show this induces a differential algebra structure ofH acting onR. The workhere extends the notion of aR/k-bialgebra introduced by Nichols and Weisfeiler. 2005 Elsevier Inc. All rights reserved.

1. Introduction

Let k be a field,R be a commutativek-algebra, and Der(R) the Lie algebra of derivations ofR. It is known that the vector space spanned by labeled rooted trees forms aalgebra [4]. LetH denote the Hopf algebra of rooted trees whose non-root nodes abeled using derivationsD ∈ Der(R) [4]. For such a Hopf algebra, we introduce a classH -module algebras which we call Leibnitz, and give a construction which yields a vaof different LeibnitzH -module algebras (Theorem 3.11).

We also show how LeibnitzH -module algebras are related to Nichols and Weisfer’s R/k-bialgebras [10], which arise in Hopf-algebra approaches to differential alg(Theorem 4.7). In Section 5 we also give a method for describing quotients of LeH -module algebras (Theorem 5.8).

* Corresponding author.

E-mail addresses:[email protected] (R.L. Grossman), [email protected] (R.G. Larson).

0196-8858/$ – see front matter 2005 Elsevier Inc. All rights reserved.doi:10.1016/j.aam.2005.01.001

98 R.L. Grossman, R.G. Larson / Advances in Applied Mathematics 35 (2005) 97–119

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Hopf algebras can be used to simplify computations of derivations [5]. In theway, LeibnitzH -module algebras can be used to simplify the symbolic computatioderivations acting on polynomials and other algebras of functions. This was used in [[2] to derive geometrically stable numerical integration algorithms, although the resupresented differently.

2. Bialgebras and trees

This section reviews some material from [4] on the Hopf algebra structure of treebackground material on Hopf algebras see [13]. Throughout this paperk is a field.

Let R be a commutativek-algebra. LetD be a vector space overk, and letS be a subseof D. Let T (S) denote the set of ordered trees in which each node other than this labeled with an element ofS . Let k{T (S)} be the vector space overk with basis theelements ofT (S).

When we speak of a “subtree of a tree” we also include the (unlabeled) node in thto which the subtree is attached as the root of the subtree. When we refer to the “cof a nodev” we will sometimes mean the nodes which are attached tov as immediatedescendants, and sometimes mean the full subtrees which are attached tov. Which senseis meant will be clear from context.

We have a grading onk{T (S)}: k{T (S)}r is spanned by the trees withr + 1 nodes.There is a bialgebra structure onk{T (S)} given in [4] which we summarize here.

Multiplication in k{T (S)} is defined as follows. LetT1 andT2 be rooted trees. Removthe root from the treeT1 to form a multisetF of rooted trees. Letd be a function fromFto the set of nodes ofT2. Let Td be the rooted tree formed by adding an edge to linkroot of T ′ ∈ F to the noded(T ′) of T2. The order on the children of a node ofTd is givenby declaring that the order of the children of a node of a tree inF is preserved, that thorder of the children of a node ofT2 is preserved, that if two roots of trees inF are linkedto the same node ofT2, they are given the order they had as children of the root ofT1, andthat the root ofT ′ precedes every child of the noded(T ′) to which it is linked. Now

T1 · T2 =∑d

Td,

where the sum ranges over all treesTd formed as described above. This product is assotive, and the tree with only one node is a multiplicative unit. See [4] for details and Ffor some examples. (Note that in [4] multiplication is defined so that the root ofT ′ ∈ Ffollows every child of the noded(T ′); this will not change the application of the resultsuse from there.)

The coproduct ink{T (S)} is defined as follows. IfT is a labeled ordered rooted tredefine

∆(T ) =∑

TX ⊗ TF\X ,

X

R.L. Grossman, R.G. Larson / Advances in Applied Mathematics 35 (2005) 97–119 99

tsular,

is not

Fig. 1. Some examples of multiplying labeled trees.

whereX ranges over all sub-multisets of the ordered multisetF described above. IfY is asub-multiset ofF , the labeled ordered treeTY is formed by adding edges to link the rooof the trees inY to a new root, preserving their original order and labels. In particTF = T andT∅ = 1 is the unit. The counit� : k{T (S)} → k is defined as follows

�(T ) ={

1 if T = 1,0 otherwise.

In this construction, each non-root node retains its original label; recall that the rootlabeled. See [4] for more details and Fig. 2 for some examples.

Sincek{T (S)} is a graded bialgebra with dimk{T (S)}0 = 1, it is a Hopf algebra.We summarize this discussion in the following

Proposition 2.1. Let k be a field and letS be a vector space overk. Thenk{T (S)} is acocommutative graded connected Hopf algebra.

Fig. 2. Some examples of co-multiplying labeled trees.


.

..

con-

3. H -module algebras

Let R be a commutativek-algebra, and letH be ak-bialgebra. Recall the followingdefinition [8, 4.1.1]:

Definition 3.1. The algebraR is aleft H -module algebraif R is a leftH -module for which

h · (rs) =∑(h)

(h(1) · r)(h(2) · s),

whereh ∈ H , ∆(h) = ∑(h) h(1) ⊗ h(2), andr, s ∈ R.Example 3.2. Let R = k[X1, . . . ,XN ]. Then the Lie algebraD of derivations ofR has{∂/∂X1, . . . , ∂/∂XN } as anR-basis. LetH be the bialgebrak{T (D)} defined in Section 2We define anH -module algebra structure onR as follows. LetT ∈ k{T (D)} be a labeledtree. Number the root ofT with 0, and number the other nodes 1, . . . ,m. Let nodei, i > 0,be labeled with

Ei =N∑

µi=1riµi

∂

∂Xµi, (1)

whereriµi ∈ R. Suppose that nodei, i � 0, has childrenj1, . . . ,jk . Define

R(i;µjk . . .µj1) =

∂∂Xµjk

· · · ∂∂Xµj1

s if i = 0,∂

∂Xµjk· · · ∂

∂Xµj1riµi otherwise.

We will usually abbreviateR(i;µjk . . .µj1) by R(i). Define

T · s =N∑

µ1,...,µm=1R(m) · · ·R(1)R(0)

for s ∈ R. It can be shown [6, Proposition 2] that this makesR into a leftH -module algebraWe will prove the existence of more complexH -module algebra structures in Section 5

We will generalize Example 3.2 in Proposition 3.9. Here is a specific case of thestruction in Example 3.2.

Example 3.3. Consider the following two vector fields onR8 introduced in [3]:

E1 = ∂∂x1

,

∂ ∂ 1 2 ∂ ∂ 1 3 ∂ 1 2 ∂ 1 2 ∂
E2 =∂x2
− x1∂x3

+2x1 ∂x4

+ x1x2∂x5

−6x1 ∂x6

−2x1x2∂x7

−2x1x2 ∂x8

.


rators

d a

eled

Fig. 3. This figure illustrates some of the trees which arise in Example 3.3. The Lie bracket[[E2,E1],E1] onlyshows some of the trees which arise in the expansion.

Then it is simple to check, for example, that

[E2,E1] = ∂∂x3

− x1 ∂∂x4

− x2 ∂∂x5

+ 12x21

∂

∂x6+ x1x2 ∂

∂x7+ 1

2x22

∂

∂x8,

[[E2,E1],E1] = ∂∂x4

− x1 ∂∂x6

− x2 ∂∂x7

,

[[E2,E1],E2] = ∂∂x5

− x1 ∂∂x7

− x2 ∂∂x8

.

See Fig. 3 for the corresponding trees. Note also that:

[E2,E1](x3) = 1,[[E2,E1],E1](x4) = 1, and [[E2,E1],E2](x5) = 1.

We could continue this example by checking that the actions of the differential opeis the same as the actions of the trees.

We will use the following definition in the sequel.

Definition 3.4. Let D be a vector space.

(a) LetE ∈ D. Denote by v(E) the labeled ordered tree with two nodes: the root, ansingle child which is labeled withE.

(b) LetT1, . . . , Tk be labeled ordered trees, and letE ∈ D. Denote by u(E;T1, . . . , Tk) thelabeled ordered tree whose root has one child, labeled withE, with which the roots ofthe subtreesT1, . . . , Tk are identified. The ordering on the children of the node labwith E in u(E;T1, . . . , Tk) is given by specifying that the children of the root ofT1

precede the children of the root ofT2, . . . , which precede the children of the root ofTk .


n

d-ofot

ion

s

n

Fig. 4. This figure illustrates the maps v(E), u(E;T1, . . . , Tk) and t(T1, . . . , Tk) which are used to define aaction of the algebra of labeled treesk{T (D)} onR.

(c) Let T1, . . . , Tk be labeled ordered trees. Denote by t(T1, . . . , Tk) the labeled orderetree formed by identifying the roots of the treesT1, . . . , Tk . The ordering on the children of the root t(T1, . . . , Tk) is given by specifying that the children of the rootT1 precede the children of the root ofT2, . . . , which precede the children of the roof Tk .

This definition is illustrated in Fig. 4. Note that u(E;T1, . . . , Tk) = u(E; t(T1, . . . , Tk)).We will need the following lemma.

Lemma 3.5. LetE,F ∈D. Thent(v(E),v(F )

) = v(E) · v(F ) − u(F ;v(E)).Proof. The proof of the lemma follows immediately from the definition of multiplicatfor trees. �

This lemma is illustrated in Fig. 5.

Definition 3.6. Let R be a commutativek-algebra, and letD be a Lie algebra of derivationof R. A mapD ×D → D sending(E,F ) ∈D ×D to ∇EF ∈ D satisfying

(a) ∇E1+E2F = ∇E1F + ∇E2F ,(b) ∇E(F1 + F2) = ∇EF1 + ∇EF2,(c) ∇f ·EF = f · ∇EF ,(d) ∇E(f · F) = f · ∇EF + E(f )F ,

whereE,F ∈D, f ∈ R, is called aconnection.

(For example,M could be aC∞-manifold,R be the algebra ofC∞ functions onM , Dcould be the Lie algebra of vector fields onM , and∇EF could be the Koszul connectio

(see [11, Chapter 5], and [12, Chapters 5 and 6]).)


nallowtion.

actsee

y a treehat of

Fig. 5. This figure illustrates the computation in Lemma 3.5.

Construction 3.7. We use the action ofD onR and a connectionD×D → D to constructan action of the algebra of labeled treesk{T (D)} on R. See Figs. 6 and 7. We give ainductive description of the action. The description of this construction is intended toan inductive proof of Theorem 3.9. We make the following assumptions about the ac

(a) The tree v(E) acts asE.(b) The tree u(E;v(F )) acts as∇F E.(c) Suppose thatT is a labeled ordered tree whose root has a single child and which

on R as the differential operatorET . Suppose further thatU is a labeled ordered trewhich containsT as a proper subtree. Denote byU(T |v(ET )) the labeled ordered treresulting from replacing the subtreeT with the tree v(ET ). In this construction, werequire thatU acts likeU(T |v(ET )).This assumption says that a subtree whose root has one child can be replaced bwhich has one non-root node which is labeled with a derivation whose action is tthe original subtree.

We make use of these assumptions as follows.

(d) If E,F ∈ D, by Lemma 3.5 the tree t(v(E),v(F )) acts as v(E) · v(F ) − u(F ;v(E)),whose action we know by (a) and (b).

Fig. 6. An illustration of the action defined by Construction 3.7.


umber

nowction

ly

the

of

ildren

ld.

Fig. 7. Another illustration of the action defined by Construction 3.7.

(e) We define the action of a tree whose root has only one child by induction on the nof children of the child of the root:• If the child of the root has one child, by induction on the number of nodes we k

how that child acts, since application of (c) and (b) allows us to determine the aof the tree onR.

• Suppose thatT = u(E;T1, . . . , Tn+1), where eachTi is a tree whose root has onone child. Then

T1 · u(E;T2, . . . , Tn+1) = t(T1,u(E;T2, . . . , Tn+1)

) + T + ∑j

Uj ,

where theUj are trees in whichT1 has been linked to various nodes (other thanroot) in the treesTj in t(E;T2, . . . , Tn+1). By induction we know the action ofT1,of u(E;T2, . . . , Tn+1), and of theUj on R. By (c) and (d) we know the actiont(T1,u(E;T2, . . . , Tn+1)). Therefore we know the action ofT .

This gives the action of a tree whose root has only one child.(f) We now determine the action of a general tree by induction on the number of ch

of the root:• The case where the root has one child follows from (e).• Suppose thatT = t(T1, . . . , Tn+1), where eachTi is a tree whose root has one chi

Now

T1 · t(T2, . . . , Tn+1) = T +∑j

Vj ,

where theVj are trees whose roots haven children. We know the action ofT1 by (e),and of t(T2, . . . , Tn+1) andVj by induction. Therefore we know the action ofT .

Note that this construction includes Example 3.2, upon letting

∇Ei Ej =∑

riµi∂rjµj ∂

µi,µj∂Xµi ∂Xµj


3.6

ra of

e

ts.es with

ildrencase

e

f any

ose

atto

where theEi are the differential operators given in (1). This follows from Definitionand the fact that

∇ ∂∂Xi

∂

∂Xj= 0.

Note that Construction 3.7 gives the basic description of the action of a bialgebordered trees whose non-root nodes are labeled with derivations ofR on the commutativealgebraR.

Definition 3.8. If T1, T2 ∈ k{T (D)} act identically onR for T1, T2 ∈ k{T (D)}, writeT1 ∼ T2.

Theorem 3.9. Let R be a commutativek-algebra, and let∇EF be a connection on thLie algebraD of derivations ofR. Then Construction3.7gives ak{T (D)}-module struc-ture onR. This module structure induces a mapψ : k{T (D)} → End(R). The followingconditions are satisfied:

(1) h · r = ψ(h) · r for all h ∈ k{T (D)}, r ∈ R;(2) thek{T (D)}-module structure onR is a k{T (D)}-module algebra structure;(3) Imψ ⊆ Diff (R).

Proof. (1) Assumption (a) in Construction 3.7 guarantees that the actions ofh andψ(h)are the same ifh is a tree with one non-root node.

Assumption (b) describes how a tree with two nodes whose root has one child acCondition (c) says that subtrees which act as derivations can be replaced by tre

one non-root node which act as the same derivation.Condition (d) says that the actions ofh andψ(h) are the same ifh is the product of two

trees which have only one non-root node.Conditions (e) and (f) are used to prove the result by induction on the number of ch

of the root of any tree, and on the number of children of the child of the root in thethat the root has only one child.

(2) The fact that the action gives ak{T (D)}-module algebra structure follows from thdefinition of the coalgebra structure ofk{T (D)}.

(3) Diff(R) is generated by derivations. Since Construction 3.7 gives the action otree as a sum of products of derivations, it follows thatψ(h) ∈ Diff (R). �Definition 3.10. Let T be a labeled ordered tree. Suppose that a non-root nodei of Tis labeled withrE, wherer ∈ R andE ∈ D. Denote the labeled ordered subtree wh(unlabeled) root isi by Ti . Denote byT (i,G,T ′) the tree identical toT , except that thenode i of T (i,G,T ′) is labeled withG ∈ D, and the labeled ordered subtree rootednodei is T ′ ∈ k{T (D)}. (Note thatT (i, rE,Ti) = T .) Extending this notation allows us

replaceTi with a linear combination of trees.


f

ledes. See

(c)

ees the

Let R be a commutativek-algebra, letD = Der(R), and suppose thatR is a Hopfmodule algebra over thek-bialgebrak{T (D)}. Thek{T (D)}-module algebra structure oR is calledLeibnitzif

T · s =∑(Ti )

(Ti (1) · r)(T (i,E,Ti(2)) · s

)

for all treesT , for all non-root nodesi of T , and for all factorizationsrE, wherer ∈ R,E ∈ D, of the label of nodei. (Note the coproduct in the formula above is over the treeTi ,not over the treeT .)

Basically thek{T (D)}-module algebra structure is Leibnitz if the action of a labetree is consistent in that subtrees act consistently with how they act as separate treFig. 8 for a simple example.

Theorem 3.11. Let R be a commutativek-algebra, andD = Der(R), and let∇ :D ×D → D be a connection. Then thek{T (D)}-module algebra structure onR given in Con-struction3.7 is Leibnitz.

Proof. Let T ∈ k{T (D)}, and suppose that the subtree rooted at nodei is Ti =t(Ti1, . . . , Tik), where eachTij is a tree whose root has only one child. By assumptionof Construction 3.7 the subtreeTij can be replaced byUj = v(Fj ) whereTij acts asFj ,so thatTi can be replaced byU = t(U1, . . . ,Uk).

We show by induction onk that

T · s =∑(U)

(U(1) · r)(T (i,E,U(2)) · s

).

Fig. 8. This figure illustrates the defining formula for a LeibnitzH -module algebra structure for a simple tracting on a functions. Note that in this simple case, the formula defining a Leibnitz structure generalize

following standard formula for connections:(∇F (rE))(s) = r((∇F E)s) + F(r)E(s).


If k = 1 then letV be the subtree rooted at parent node ofi which has nodei as its onechild, which is labeled withrE and which has one child node labeled withF1, that is, thetree u(rE;v(F1)). By assumption (b) of Construction 3.7, the subtreeV acts as

∇F1(rE) = r∇F1E + F1(r)E

so that

T · s =∑(U)

(U(1) · r)(T (i,E,U(2)) · s

)

in this case.Now consider the casek > 1. DefineG� = ∇F1F�. Here

u(rE;v(F1), . . . ,v(Fk)

)∼ v(F1) · u

(rE;v(F2), . . . ,v(Fk)

)− t(v(F1),u(rE;v(F2), . . . ,v(Fk)))

−k∑

�=2u(rE;v(F2), . . . ,v(G�),v(Fk)

)

∼ v(F1) ·(∑

(V )

(V(1) · r)u(E;V(2)))

−∑(V )

(V(1) · r)t(v(F1),u(E;V(2))

)

−∑�,(V )

(V�(1) · r)u(E;V�(2))

∼∑(U)

(U(1) · r)u(E;U(2))

+∑(V )

(V(1) · r)t(v(F1),u(E;V(2))

)

+∑�,(V )

(V�(1) · r)u(E;V�(2))

−∑(V )

(V(1) · r)t(v(F1),u(E;V(2))

)

−∑�,(V )

(V�(1) · r)u(E;V�(2))

∼∑

(U(1) · r)u(E;U(2)). (2)

(U)


ies,llows

-of the

.

t

].

s

are

(Note that we are usingV as a “local variable” in each sum, and that the value vardepending on context.) The identity of the actions of the terms in expression (2) fofrom the induction hypothesis. Since u(rE;v(F1), . . . ,v(Fk)) ∼ ∑(U)(U(1) · r)u(E;U(2))the theorem now follows from assumption (c) of Construction 3.7.�

4. R/k-bialgebras

The notions ofR/k-bialgebraandR/k-Hopf algebraas described in [9] and [10] capture many of the essential aspects of differential algebra. We first review somematerial found there.

Let R be ak-algebra. AR/k-algebra is ak-algebraB into which R is embeddedNote that this makesB into a left and rightR-module, and that(rb)s = r(bs) for all r ,s ∈ R, b ∈ B, and that also(rb)c = r(bc), (br)c = b(rc), and (bc)r = b(cr) for all b,c ∈ B, r ∈ R. When we refer to theR-module structure ofB, we will understand the lefR-module structure.

We denote byB ⊗R B the tensor product ofB with itself using the leftR-modulestructure ofB. That is, with(rb) ⊗ c = b ⊗ (rc). Note that in generalB ⊗R B is not anR-algebra.

The material we present here is related to the×R-bialgebra construction given in [14There a×R-bialgebra is defined in terms of maps betweenB andB ×R B, but the spaceB ×R B in [14] is only ak-subspace ofB ⊗R B.

Definition 4.1. A R/k-bialgebra is a R/k-algebraB together withR-module maps∆ :B → B ⊗R B and� :B → R satisfying

(a) B together with the maps∆ and� is a coalgebra overR.(b) ∆(1) = 1⊗ 1.(c) For allb, c ∈ B, if ∆(b) = ∑i bi ⊗ b′i and∆(c) = ∑j cj ⊗ c′j are any representation

of ∆(b),∆(c) ∈ B ⊗R B, then∆(bc) = ∑i,j bicj ⊗ b′ic′j .(d) �(1) = 1.(e) �(bc) = �(b�(c)).

Note that condition (c) of Definition 4.1 implies that

∆(br) =∑(b)

b(1)r ⊗ b(2) =∑(b)

b(1) ⊗ b(2)r

for b ∈ B andr ∈ R. It can be shown (see [9] for details) that conditions (b) and (c)equivalent to the assertion that the action ofB onB ⊗R B defined by

def

b · (c ⊗ d) = ∆(b)(c ⊗ d),


i-
for b, c, d ∈ H , givesB ⊗R B a well-defined leftB-module structure, and that condtions (d) and (e) are equivalent to the assertion that the action ofB defined onR by
b · r def= �(br),

for b ∈ B andr ∈ R, givesR a well-defined leftB-module structure.If B is aR/k-algebra, thenB is a (R,R)-bimodule via the left and right actions ofR

onB induced by the embedding ofR in B. Denote byB ⊗r B the tensor product ofB withitself using this(R,R)-bimodule structure. That is,B ⊗r B is an (R,R)-bimodule withr(b ⊗r c) = (rb)⊗r c, (b ⊗r c)r = b ⊗r (cr), and thatbr ⊗r c = b ⊗r rc. The multiplicationonB induces a mapµ :B ⊗r B → B.

Definition 4.2. Let B be aR/k-bialgebra. Anantiproduct forB is ak-linear mapE :B →B ⊗r B satisfying

(a) E(rb) = rE(b) = E(b)r for all r ∈ R, b ∈ B;(b)

∑(b) E(b(1))b(2) = b ⊗r 1 for all b ∈ B;

(c) (I ⊗R E) ◦ ∆(b) = (∆ ⊗r I ) ◦ E(b) for all b ∈ B;(d) µ ◦ E(b) = �(b)1 for all b ∈ B.

A R/k-bialgebra which has an antiproduct is called aR/k-Hopf algebra.

We recall from [9, Proposition 8]

Proposition 4.3. LetR be ak-algebra. IfH is a cocommutativek-Hopf algebra over whichR if a H -bimodule algebra, then

R #k Hdef= R ⊗k H

is aR/k-Hopf algebra, with theR-coalgebra structure given by

R ⊗k H IH ⊗∆H−−−−−→ R ⊗k H ⊗k H ∼= (R ⊗k H) ⊗R (R ⊗k H)

and with multiplication given by

(r # h)(s # k)def=

∑(h)

r(h(1) · s) # h(2)k

wherer, s ∈ R, h, k ∈ H , and with antiproduct given by

E(r # h) =∑(h)

(r # h(1)) ⊗r(1 #S(h(2))

)

wherer ∈ R, h ∈ H , andS is the antipode ofH .


rara

o-

icive.5].

Proposition 4.4. Let R be ak-algebra, letH and H̄ be cocommutativek-Hopf algebrasover whichR is a module algebra, and letϕ :H → H̄ be ak-Hopf algebra homomorphismwhich is consistent with theH andH̄ -module algebra structures onR. Then the map

R #k H → R #k H̄

given byr # h �→ r # ϕ(h) is aR/k-Hopf algebra homomorphism.

Proof. Omitted. �Theorem 4.5. LetR be a commutativek-algebra, letD = Der(R), and letS ⊆ D.

(a) Let Diff k(S) denote thek-algebra of higher order derivations generated byS , thatis, the subalgebra ofU(D) generated byS . Then the smash productR #k Diff k(S) isa R/k-Hopf algebra. The subalgebra ofDiff (R) generated byS is a homomorphicimage ofR #k Diff k(S).

(b) Let k{T (S)} denote the Hopf algebra of trees labeled with elements ofS defined inSection2. Then the smash productR #k k{T (S)} is aR/k-Hopf algebra.

(c) The map

R #k k{T (S)

} → R #k Diff k(S)is aR/k-bialgebra homomorphism.

Proof. The action ofk{T (D)} on R given in Construction 3.7 defines a Hopf algebhomomorphismk{T (D)} → Diff (R) which is consistent with the Hopf module-algebstructures onR, and so induces aR/k-Hopf algebra homomorphismR #k k{T (D)} →R #k Diff (R), which allows us to use the differential algebra structure onR #k k{T (D)} tostudy the differential algebra structure onR #k Diff (R).

Diff k(S) is a cocommutativek-Hopf algebra, soR #k Diff k(S) is aR/k-Hopf algebra.By Proposition 2.1,k{T (S)} is a cocommutativek-Hopf algebra. Therefore, by Prop

sition 4.3,R #k k{T (S)} is aR/k-Hopf algebra.Part (c) follows immediately from Proposition 4.4.�If R is a k-bialgebra (for example ifR is the coordinate ring of an affine algebra

group), then the set DiffR(R) of right invariant differential operators is a cocommutatk-Hopf algebra. The following proposition follows immediately from [7, Theorem 2.4

Proposition 4.6. Let R be ak-bialgebra, and letDiff R(R) be thek-Hopf algebra of rightinvariant differential operators. Then

Diff (R) ∼= R #k Diff R(R)

is aR/k-Hopf algebra.


tash

is

-

at an

If R is ak-bialgebra, and ifB is ak-basis for DerR(R), the Lie algebra of right invarianderivations ofR to itself, then we will see that Proposition 4.6 allows us to use the smproductR #k k{T (B)} to do formal computations in Diff(R). (For example, such a basalways exists ifR is the coordinate ring of an affine algebraic group.)

We can use theR/k-bialgebraR #k k{T (S)} to do formal computations involving elements of a subsetS ⊆ Der(R).

Let S ⊆ Der(R), and letF(S) be the free associative algebra generated byS . Recallingthat the elements ofS are primitive, we get a Hopf algebra structure onF(S). SinceF(S)is freely generated byS , we have mapsF(S) → Diff k(S) mappingE ∈ S to E ∈ Diff k(S),andF(S) → k{T (S)} mappingE ∈ S to v(E) ∈ k{T (S)}. These induce mapsp :R #kF (S) → R #k Diff k(S) and i :R #k F (S) → R #k k{T (S)}. If R is an algebra for whichthere is a connection on Der(R) (for example, ifR is the algebra ofC∞ functions on aRiemannian manifold), there is a mapϕ :R #k k{T (S)} → R #k Diff k(S) induced by themap described in Construction 3.7. We have

Theorem 4.7. LetR be a commutative algebra for which there is a connection onDer(R),and letp, i, andϕ be the maps described above. Then the diagram

R #k k{T (S)}ϕ

R #k F (S)

i

pR #k Diff k(S)

commutes.

This theorem allows us to do formal computations involving elements ofS in the alge-braR #k k{T (S)} rather than inR #k F (S).

5. Quotients of R/k-bialgebras

In this section we discuss certain quotients of theR/k-Hopf algebraR #k k{T (D)},whereD = Der(R). The main result in this section is Theorem 5.8, which says thLeibnitz action of theR/k-Hopf algebraR #k k{T (D)} can be computed from the actioof R #k k{T (B)} if B is anR-basis ofD.

Let B be aR/k-bialgebra. AR/k-biideal is an idealI in the R/k-algebraB, suchthat �(I ) = 0, and such that ifπ :B → B/I is the projection ofB onto B/I , we have(π ⊗R π) ◦ ∆(I) = 0. If I is a R/k-biideal in B, thenB/I is a R/k-bialgebra. IfB isa R/k-Hopf algebra with antiproductE, and if (π ⊗r π) ◦ E(I) = 0, thenI is called aR/k-Hopf ideal, andB/I is aR/k-Hopf algebra.

We will use the following definition in the sequel.


n).

ne case

of of

any

d to

Definition 5.1. Let R be a Leibnitzk{T (D)}-module algebra. LetI(k{T (D)}) be theR-linear span of the elements of the form

1 #T −∑(Ti )

Ti (1) · r # T (i,E,Ti(2)), (3)

whereT ∈ k{T (D)}, with non-root nodei labeled withrE, with r ∈ R, E ∈ D (we includeall possible factorizations of the label of nodei).

Lemma 5.2. The subspaceI(k{T (D)}) defined in Definition5.1 is a two-sided ideal inR #k k{T (D)}.

Proof. Let J be the ideal ofR #k k{T (D)} generated byI(k{T (D)}).Let Z be an element of the form (3), and letT ′ be any tree. It follows from the definitio

of the product of trees thatZ(1#T ′) is anR-linear combination of elements of the form (3We now show that

(1 #T ′

)Z = 1 #T ′ ·

(1 #T −

∑(Ti )

Ti (1) · r # T (i,E,Ti(2)))

= 1 #T ′ · T −∑

(T ′),(Ti )T ′(1) · Ti(1) · r # T ′(2) · T (i,E,Ti(2))

is anR-linear combination of elements of the form (3). Sincek{T (D)} is generated as aalgebra by trees whose root has one child (see [4]), it is sufficient to show this in ththat the root of the treeT ′ has one child. Terms in the tree productT ′ ·T pair in an obviousfashion with terms in

∑(T ′),(Ti ) T

′(1) ·Ti(1) · r #T ′(2) ·T (i,E,Ti(2)). Therefore(1 #T ′) ·Z

is again anR-linear combination of elements of the form (3). This completes the prothe lemma. �Proposition 5.3. Let R be a commutativek-algebra, letD = Der(R), and suppose thatRis a Leibnitzk{T (D)}-module algebra. Then the idealI(k{T (D)}) is aR/k-Hopf ideal.

Proof. It is immediate that� is zero on any element of the form (3), since it is zero ontree with more than one node.

Let π be the projection ofk{T (D)} ontok{T (D)}/I(k{T (D)}). To see that(π ⊗R π)◦∆ is zero on any element of the form (3), write

∆(T ) =∑j

T ′j ⊗ T ′′j ∈ k{T (D)

} ⊗k k{T (D)}.

If node i occurs inT ′j , then the corresponding term arising in the coproduct applie

element (3) is


ng

e

s

t

e

thercur

s

1 #T ′j ⊗R 1 #T ′′j −∑(Ti )

Ti (1) · r # T ′j (i,E,Ti (2)) ⊗R 1 #T ′′j

=(

1 #T ′j −∑(Ti )

Ti (1) · r # T ′j (i,E,Ti (2)))

⊗R 1 #T ′′j ,

which is clearly in Ker(π ⊗R I) ⊆ Ker(π ⊗R π). Similarly, if nodei occurs in treeT ′′j ,then the corresponding term of the coproduct applied to this element is in Ker(I ⊗R π) ⊆Ker(π ⊗R π). It follows that(π ⊗R π)◦∆ vanishes onI , since it vanishes on a generatiset for it as an ideal in the algebraR #k k{T (D)}.

To show thatI(k{T (D)}) is compatible with the antiproduct ofk{T (D)}, we will needto work with a restricted generating set ofI(k{T (D)}).

Lemma 5.4. Let R be a commutativek-algebra, and letD = Der(R). Suppose that ware given a Leibnitzk{T (D)}-module action onR. Let J be the ideal inR #k k{T (D)}generated by all elements of the form(3) whereT ranges over all labeled ordered treewhose root has only one child. ThenJ = I(k{T (D)}).

Proof. It is immediate thatJ ⊆ I(k{T (D)}). Let T be any labeled ordered tree, and leibe a node ofT which is labeled withrE. We will prove that

1 #T −∑(Ti )

Ti (1) · r # T (i,E,Ti(2)) (4)

is in J by induction on the number of children of the root ofT .If the root ofT has one child then the element (4) is inJ by definition.Suppose that the root ofT hasn + 1 children. LetT0 be the tree consisting of the tre

whose root has one child, which is the first child of the root of the treeT , in the orderin which they occur inT . Let T1 be the tree whose root has as children all of the ochildren of the root of the treeT and their descendents, in the order in which they ocin T . The root of the treeT1 hasn children. To show that element (4) is inJ , we considertwo cases.

Node i ∈ T0. In this case letT0 · T1 = T + ∑j Uj , where eachUj is a tree whose root haonly n children. Then, sinceT = T0 · T1 − ∑j Uj ,

1 #T −∑(Ti )

Ti (1) · r # T (i,E,Ti(2))

=(

1 #T0 −∑(Ti )

Ti (1) · r # T0(i,E,Ti(2)))

· (1 #T1)

−∑(

1 #Uj −∑

Ti(1) · r # Uj (i,E,Ti(2)))

.

j (Ti )


t

Note that, since the treeT0 precedesT1 in the product, the labeled ordered subtreeTi rootedat nodei is the same inT0 as inUj . The term

(1 #T0 −

∑(Ti )

Ti (1) · r # T0(i,E,Ti (2)))

· (1 #T1)

is in J since the root ofT0 has one child andJ is an ideal. The terms

1 #Uj −∑(Ti )

Ti (1) · r # Uj (i,E,Ti(2))

are inJ by induction on the number of children of the root of the tree.

Node i ∈ T1. In this case first note that

1 #T1 −∑(V )

V(1) · r # T1(i,E,V(2)), (5)

whereV is the labeled subtree ofT1 rooted at nodei, is in J by induction, since the rooof T1 has onlyn children. Let

T0 · T1 = T +∑

k

Uk +∑

�

U ′�,

where theUk are the trees in the product in whichT0 is not attached to the nodei or to anyof its descendents, and theU ′� are the trees in the product in whichT0 is attached to thenodei or to one of its descendents. Now the element

(1 #T0) ·(

1 #T1 −∑(V )

V(1) · r # T1(i,V(2),E))

= 1 #T0 · T1 −∑(V )

T0 · V(1) · r # T1(i,V(2),E)

−∑(V )

V(1) · r # T0 · T1(i,V(2),E)

= 1 #T +∑

k

1 #Uk +∑

�

1 #U ′�

−∑(V )

T0 · V(1) · r # T1(i,V(2),E)

−∑(V )

V(1) · r # T (i,V(2),E)

−∑

V(1) · r # Uk(i,V(2),E)

(V )


)

t.

al

−∑(V )

V(1) · r # T1(i, T0 · V(2),E)

= 1 #T −∑(V )

V(1) · r # T (i,V(2),E) (6)

+∑

k

(1 #Uk −

∑(V )

V(1) · r # Uk(i,V(2),E))

(7)

+∑

�

(1 #U ′� −

∑(V ′)

V ′(1) · r # U ′�(i,V ′(2),E

))(8)

whereV ′ is the labeled subtree rooted at nodei in U ′�, is in the idealJ , since the element (5is in the ideal. Since the roots of the treesUk andU ′� have onlyn children, by inductionthe terms (7) and (8) are inJ . Therefore the term (6) is in the idealJ .

This completes the proof of the lemma.�We now use Lemma 5.4 to show thatI(k{T (D)}) is compatible with the antiproduc

Denote byπ the projection ofR #k k{T (D)} onto R #k k{T (D)}/I(k{T (D)}). We needto show that(π ⊗r π) ◦ E(I(k{T (D)})) = 0. By [9, Proposition 6], sinceR #k k{T (D)}is pointed, it is sufficient to show that(π ⊗r π) ◦ E = 0 on a generating set of the ideI(k{T (D)}). By Lemma 5.4 it is sufficient to consider the value of(π ⊗r π) ◦ E on ele-ments of the form (3), when the root of the treeT has only one child. In this case

E

(1 #T −

∑(Ti )

Ti (1) · r # T (i,E,Ti(2)))

= 1 #T ⊗r 1 # 1− 1 # 1⊗r 1 #T−

∑(Ti )

Ti (1) · r # T (i,E,Ti(2)) ⊗r 1 # 1

+∑(Ti )

Ti (1) · r # 1⊗r 1 #T (i,E,Ti(2))

=(

1 #T −∑(Ti )

Ti (1) · r # T (i,E,Ti(2)))

⊗r 1 # 1

− 1 # 1⊗r(

1 #T −∑(Ti )

Ti (1) · r # T (i,E,Ti(2)))

,

and this is clearly annihilated byπ ⊗r π . This proves Proposition 5.3.�


dese is anorderdenceebra

Definition 5.5. Suppose that thek{T (D)}-module algebra structure ofR is Leibnitz. LetB be anR-basis forD. Repeated applications of the substitution

1 #T =∑u

∑(Ti )

Ti (1) · ru # T (i,Xu,Ti(2)), (9)

where the nodei in the treeT is labeled with∑

u ruXu, with ru ∈ R, Xu ∈ B, gives a mapαB :R #k k{T (D)} → R #k k{T (B)}.

Lemma 5.6. Suppose that thek{T (D)}-module algebra structure ofR is Leibnitz, andlet B be anR-basis forD. Then the mapαB :R #k k{T (D)} → R #k k{T (B)} given inDefinition5.5 is well defined.

Proof. It is sufficient to show that application of the substitution rule (9) to two nodoes not depend on the order in which the rule is applied to the nodes. If neither nodancestor of the other, then it follows immediately that the result is independent of thein which the rule is applied. If one node is an ancestor of the other, then the indepenof the result of the order of application follows immediately from the fact that the algmodule structure is Leibnitz. This completes the proof of the lemma.�

Let βB :R #k k{T (B)} → R #k k{T (D)} be the inclusion map. ThenαB ◦ βB is theidentity onR #k k{T (B)}.

Lemma 5.7. Suppose that thek{T (D)}-module algebra structure ofR is Leibnitz, and letB be anR-basis forD. Then

KerαB = I({

kT (D)})

.

Proof. Note that KerαB is the linear span of elements of the form

1 #T −∑u,(Ti )

Ti(1) · su # T (i,Xu,Ti(2))

where the nodei in the treeT is labeled with∑

u suXu, with su ∈ R, Xu ∈ B. Theseelements are all of the form (3), which spanI(k{T (D)}), so that KerαB ⊆ I(k{T (D)}).

The idealI(k{T (D)}) is the linear span of elements of the form

1 #T −∑(Ti )

Ti (1) · r # T (i,E,Ti(2))

= 1 #T −∑u,(Ti )

(Ti (1) · r)(Ti (2) · su) # T (i,Xu,Ti (3))

= 1 #T −∑

Ti(1) · (rsu) # T (i,Xu,Ti(2)),

u,(Ti )


itz

byof

of la-

res for

er dif-is, theodes,se

and

whererE, r, su ∈ R, E ∈ D, is a factorization of the label of a nodei, Xu ∈ B, andE =∑u suXu, so thatI(k{T (D)}) ⊆ KerαB.This completes the proof of the lemma.�We have the following theorem, which says that we can useR #k k{T (B)} to do com-

putations inR #k k{T (D)}.

Theorem 5.8. Let R be a commutativek-algebra. Assume thatD = Der(R) is free as anR-module. Suppose that we have a Leibnitzk{T (D)}-module algebra structure onR. LetB be anR-basis ofD. Then

αB :R #k k{T (D)

}/I(k{T (D)

}) ∼= R #k k{T (B)}.Proof. From Lemma 5.7 KerαB = I(k{T (D)}) so that the mapαB in injective.

From the fact thatαB ◦ βB is the identity onR #k k{T (B)} it follows theαB is surjec-tive. �

If B is an R-basis ofD = Der(R), then there is a bijection between the Leibnk{T (D)}-module algebra structures ofR and thek{T (B)}-module algebra structures ofR.According to [4, Theorem 5.1],k{T (B)} is freely generated as an associative algebrathe setX of trees whose root has a single child, which are labeled with elementsB.Therefore, there is a bijection between Leibnitzk{T (D)}-module algebra structures onRand functions fromX to Der(R).

In particular, there are Leibnitz module algebra structures for the Hopf algebrabeled ordered trees, labeled with elements of Der(R), on R = k[X1, . . . ,XN ] other thanthe example given in Example 3.2. In particular, there exist module algebra structuthe Hopf algebra of trees labeled with elements of{∂/∂X1, . . . , ∂/∂XN } under which treeswith more than two nodes whose root has only one child act as non-zero first-ordferential operators. We will see in the next section that under an additional hypothesk{T (D)}-module algebra structure is determined by the actions of trees with two nwhich correspond to the actions of the elements ofD, and trees with three nodes whoroots have only a single child, which correspond to the connection onR.

6. Coherent actions and connections

In this section we discuss how certain actions ofk{T (D)} on R are determined bythe action ofE ∈ D = Der(R), and of the action of the connection∇EF for E,F ∈ D.Throughout this sectionR is a commutativek-algebra.

We consider actions ofk{T (D)} on R under which v(E) acts asE, and under whichu(F ;v(E)) acts as∇EF .

Definition 6.1. Suppose thatU is a labeled ordered tree whose root has a single child

which acts onR as the differential operatorEU , and suppose thatT is a labeled ordered


e-

all

ratedhat is

t-

on

ee

co-

tree which containsU as a subtree. Denote byT (U |v(EU)) the labeled ordered tree rsulting from replacing the subtreeU with the tree v(EU). The action ofk{T (D)} on Ris calledcoherentif for all labeled ordered treesU whose root has a single child, andlabeled ordered treesT which containU as a subtree, the actions onR of the treesT andT (U |v(EU)) are identical, that is,T ∼ T (U |v(EU)).

The actions defined in Example 3.2 and in Construction 3.7 are coherent.Note thatk{T (D)} is isomorphic as an algebra to the free associative algebra gene

by the labeled ordered trees whose roots have only one child ([4, Theorem 5.1]—wcalledLOT in [4] is calledT here) so non-coherent actions ofk{T (D)} onR can be easilyconstructed.

Theorem 6.2. LetR be a commutativek-algebra, and letD = Der(R). Suppose a coherenaction ofk{T (D)} on R is given. Then the action ofk{T (D)} on R is completely determined by the actionE of the treesv(E), and the action∇EF of the treesu(F ;v(E)) forall E,F ∈D.

Proof. The proof uses two lemmas.

Lemma 6.3. LetE1, . . . ,En ∈ D. Then

u(F ;v(E1), . . . ,v(En)

)= v(E1) · u

(F ;v(E2), . . . ,v(En)

)

−n∑

i=2u(F ;v(E2), . . . ,u

(Ei;v(E1)

), . . . ,v(En)

)

− t(v(E1),u(F ;v(E2), . . . ,v(En))).Proof. The proof of this lemma follows immediately from the definition of multiplicatifor trees. �Lemma 6.4. Suppose thatk{T (D)} acts coherently onR. Let the action of the treu(F ;v(E2), . . . ,v(En)) on R be denoted byG ∈ D, and let the action of the treu(Ei;v(E1)) onR be denoted byHi ∈ D. Then

u(F ;v(E1), . . . ,v(En)

)

= v(E1) · v(G) −n∑

i=2u(F ;v(E2), . . . ,v(Hi), . . . ,v(En)

) − t(v(E1),v(G)).

Proof. This lemma follows immediately from Lemma 6.3 and from the definition of

herence. �


minedt the

hed

y-thesis.proof

aper.

. Non-

orithms,

ctly, in:p. 29–

4–210.mbolic

stages,

.

.uston,

uston,

We now prove Theorem 6.2. The action of labeled trees with two nodes is deterby the action ofD onR. Repeated application of the definition of coherence shows thaaction of trees with more than two nodes inT (D) is determined by the action of trees of tform u(F ;v(E1), . . . ,v(En)). We prove by induction onn that this action is determineby the actions ofE and∇F E for all E, F ∈D.

For n = 1 this is simply the assertion that the action of u(E;v(F )) ∼ ∇F E is deter-mined. Suppose that the action is determined forn. We prove that it is determined forn+1.Lemma 6.4 implies that the action of a tree of the form u(F ;v(E1), . . . ,v(En+1)) is deter-mined by the action of trees of the form v(E), of trees of the form u(F ;v(E1), . . . ,v(En)),and of trees of the form t(v(E),v(F )). The action of trees of the first form is given by hpothesis. The action of trees of the second form is determined by the induction hypoThe action of trees of the third form is determined by Lemma 3.5. This completes theof the theorem. �

Acknowledgment

The authors wish to thank the referee for various suggestions that improved the p

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