A voyage round coalgebras

32 Pages·2010·0.324 MB·English

by  Marco Manetti

A VOYAGE ROUND COALGEBRAS
MARCO MANETTI
voyage.tex; version August 24, 2010
Abstract. I found the most ready way of explaining my employment was to ask
them how it was that they themselves were not curious concerning earthquakes and
volcanos?-whysomespringswerehotandotherscold?-whythereweremountainsin
Chile,andnotahillinLaPlata?Thesebarequestionsatoncesatisfiedandsilencedthe
greaternumber;some,however(likeafewinEnglandwhoareacenturybehindhand),
thoughtthatallsuchinquirieswereuselessandimpious;andthatitwasquitesufficient
that God had thus made the mountains.
Charles Darwin: The voyage of the Beagle.
Through all the chapter we work over a fixed field K of characteristic 0. Unless oth-
erwise specified all the tensor products are made over K.
Notation. G is the category of graded vector spaces over K.
The tensor algebra generated by V ∈ G is by definition the graded vector space
T(V) = (cid:76) (cid:78)nV
n≥0
endowed with the associative product (v ⊗···⊗v )(v ⊗···⊗v ) = v ⊗···⊗v .
1 p p+1 n 1 n
Let V,W ∈ G. The twist map tttwww: V ⊗W → W⊗V is defined by the ruletttwww(v⊗w) =
(−1)vww⊗v, for every pair of homogeneous elements v ∈ V, w ∈ W.
The following convention is adopted in force: let V,W be graded vector spaces and
F: T(V) → T(W) a linear map. We denote by
Fi: T(V) → (cid:78)iW, F : (cid:78)jV → T(W), Fi: (cid:78)jV → (cid:78)iW
j j
the compositions of F with the inclusion (cid:78)jV → T(V) and/or the projection T(W) →
(cid:78)iW.
1. Graded coalgebras
Definition 1.1. AcoassociativeZ-gradedcoalgebraisthedataofagradedvectorspace
C = ⊕n∈ZCn ∈ G and of a coproduct ∆: C → C ⊗C such that:
• ∆ is a morphism of graded vector spaces.
• (coassociativity) (∆⊗Id )∆ = (Id ⊗∆)∆: C → C ⊗C ⊗C.
C C
For simplicity of notation, from now on with the term graded coalgebra we intend a
Z-graded coassociative coalgebra.
Date: August 24, 2010.
1
2 MARCOMANETTI
Definition 1.2. Let (C,∆) and (B,Γ) be graded coalgebras. A morphism of graded
coalgebras f: C → B is a morphism of graded vector spaces that commutes with co-
products, i.e.
Γf = (f ⊗f)∆: C → B⊗B.
The category of graded coalgebras is denoted by GC.
Example 1.3. Let C = K[t] be the polynomial ring in one variable t (of degree 0). The
linear map
n
(cid:88)
∆: K[t] → K[t]⊗K[t], ∆(tn) = ti⊗tn−i,
i=0
gives a coalgebra structure (exercise: check coassociativity).
For every sequence f ∈ K, n > 0, it is associated a morphism of coalgebras f: C → C
n
defined as
n
(cid:88) (cid:88)
f(1) = 1, f(tn) = f f ···f ts.
i1 i2 is
s=1 (i1,...,is)∈Ns
i1+···+is=n
The verification that ∆f = (f ⊗ f)∆ can be done in the following way: Let {xn} ⊂
C∨ = K[[x]] be the dual basis of {tn}. Then for every a,b,n ∈ N we have:
(cid:88)
(cid:104)xa⊗xb,∆f(tn)(cid:105) = f ···f f ···f ,
i1 ia j1 jb
i1+···+ia+j1+···+jb=n
(cid:88) (cid:88) (cid:88)
(cid:104)xa⊗xb,f ⊗f∆(tn)(cid:105) = f ···f f ···f .
i1 ia j1 jb
s i1+···+ia=s j1+···+jb=n−s
Note that the sequence {f }, n ≥ 1, can be recovered from f by the formula f =
n n
(cid:104)x,f(tn)(cid:105).
Example 1.4. Let A be a graded associative algebra with product µ: A⊗A → A and
C a graded coassociative coalgebra with coproduct ∆: C → C ⊗C.
Then Hom∗(C,A) is a graded associative algebra by the convolution product
fg = µ(f ⊗g)∆.
We left as an exercise the verification that the product in Hom∗(C,A) is associative.
InparticularHom (C,A) = Hom0(C,A)isanassociativealgebraandC∨ = Hom∗(C,K)
G
is a graded associative algebra.
Remark 1.5. The above example shows in particular that the dual of a coalgebra is an
algebra. In general the dual of an algebra is not a coalgebra (with some exceptions, see
e.g.Example2.3).Heuristically,thisasymmetrycomesfromthefactthat,foraninfinite
dimensional vector space V, there exist a natural map V∨⊗V∨ → (V ⊗V)∨, while does
not exist any natural map (V ⊗V)∨ → V∨⊗V∨.
Example 1.6. The dual of the coalgebra C = K[t] (Example 1.3) is exactly the algebra
of formal power series A = K[[x]] = C∨. Every coalgebra morphism f: C → C induces a
localhomomorphismofK-algebrasft: A → A.Themorphismft isuniquelydetermined
by the power series ft(x) = (cid:80) f xn and then every morphism of coalgebras f: C →
n>0 n
C is uniquely determined by the sequence f = (cid:104)ft(x),tn(cid:105) = (cid:104)x,f(tn)(cid:105).
n
The map f (cid:55)→ ft is functorial and then preserves the composition laws.
Definition 1.7. Let (C,∆) be a graded coalgebra; the iterated coproducts ∆n: C →
C⊗n+1 are defined recursively for n ≥ 0 by the formulas
∆0 = Id , ∆n: C −∆→ C ⊗C −I−dC−−⊗−∆−n−−→1 C ⊗C⊗n = C⊗n+1.
C
A VOYAGE ROUND COALGEBRAS 3
Lemma 1.8. Let (C,∆) be a graded coalgebra. Then:
(1) For every 0 ≤ a ≤ n−1 we have
∆n = (∆a⊗∆n−1−a)∆: C → (cid:78)n+1C.
(2) For every s ≥ 1 and every a ,...,a ≥ 0 we have
0 s
(∆a0 ⊗∆a1 ⊗···⊗∆as)∆s = ∆s+Pai.
(3) If f: (C,∆) → (B,Γ) is a morphism of graded coalgebras then, for every n ≥ 1
we have
Γnf = (⊗n+1f)∆n: C → (cid:78)n+1B.
Proof. [1] If a = 0 or n = 1 there is nothing to prove, thus we can assume a > 0 and use
induction on n. we have:
(∆a⊗∆n−1−a)∆ = ((Id ⊗∆a−1)∆⊗∆n−1−a)∆ =
C
= (Id ⊗∆a−1⊗∆n−1−a)(∆⊗Id )∆ =
C C
= (Id ⊗∆a−1⊗∆n−1−a)(Id ⊗∆)∆ = (Id ⊗(∆a−1⊗∆n−1−a)∆)∆ = ∆n.
C C C
[2] Induction on s, being the case s = 1 proved in item 1. If s ≥ 2 we can write
(∆a0 ⊗∆a1 ⊗···⊗∆as)∆s = (∆a0 ⊗∆a1 ⊗···⊗∆as)(Id ⊗∆s−1)∆ =
C
(∆a0 ⊗(∆a1 ⊗···⊗∆as)∆s−1)∆ = (∆a0 ⊗∆s−1+Pi>0ai)∆ = ∆s+Pai.
[3] By induction on n,
Γnf = (Id ⊗Γn−1)Γf = (f ⊗Γn−1f)∆ = (f ⊗(⊗nf)∆n−1)∆ = (⊗n+1f)∆n.
B
(cid:3)
Definition 1.9. Let (C,∆) be a graded coalgebra and p: C → V a morphism of graded
vector spaces. We shall say that p is a system of cogenerators of C if for every c ∈ C
there exists n ≥ 0 such that (⊗n+1p)∆n(c) (cid:54)= 0 in (cid:78)n+1V.
Example 1.10. In the notation of Example 1.3, the natural projection K[t] → K⊕Kt
is a system of cogenerators.
Proposition 1.11. Let p: B → V be a system of cogenerators of a graded coalgebra
(B,Γ).
Then every morphism of graded coalgebras φ: (C,∆) → (B,Γ) is uniquely determined
by its composition pφ: C → V.
Proof. Let φ,ψ: (C,∆) → (B,Γ) be two morphisms of graded coalgebras such that
pφ = pψ. In order to prove that φ = ψ it is sufficient to show that for every c ∈ C and
every n ≥ 0 we have
(⊗n+1p)Γn(φ(c)) = (⊗n+1p)Γn(ψ(c)).
By Lemma 1.8 we have Γnφ = (⊗n+1φ)∆n and Γnψ = (⊗n+1ψ)∆n. Therefore
(⊗n+1p)Γnφ = (⊗n+1p)(⊗n+1φ)∆n = (⊗n+1pφ)∆n =
= (⊗n+1pψ)∆n = (⊗n+1p)(⊗n+1ψ)∆n = (⊗n+1p)Γnψ.
(cid:3)
4 MARCOMANETTI
Definition 1.12. Let (C,∆) be a graded coalgebra. A linear map d ∈ Homn(C,C) is
called a coderivation of degree n if it satisfies the coLeibniz rule
∆d = (d⊗Id +Id ⊗d)∆.
C C
A coderivation d is called a codifferential if d2 = d◦d = 0.
More generally, if θ: C → D is a morphism of graded coalgebras, a morphism of graded
vector spaces d ∈ Homn(C,D) is called a coderivation of degree n (with respect to θ) if
∆ d = (d⊗θ+θ⊗d)∆ .
D C
In the above definition we have adopted the Koszul sign convention: i.e. if x,y ∈
C, f,g ∈ Hom∗(C,D), h,k ∈ Hom∗(B,C) are homogeneous then (f ⊗ g)(x ⊗ y) =
(−1)gxf(x)⊗g(y) and (f ⊗g)(h⊗k) = (−1)ghfh⊗gk.
The coderivations of degree n with respect a coalgebra morphism θ: C → D form a
vector space denoted Codern(C,D;θ).
For simplicity of notation we denote Codern(C,C) = Codern(C,C;Id).
θ ρ
Lemma 1.13. Let C−→D−→E be morphisms of graded coalgebras. The compositions
with θ and ρ induce linear maps
ρ : Codern(C,D;θ) → Codern(C,E;ρθ), f (cid:55)→ ρf;
∗
θ∗: Codern(D,E;ρ) → Codern(C,E;ρθ), f (cid:55)→ fθ.
Proof. Immediate consequence of the equalities
∆ ρ = (ρ⊗ρ)∆ , ∆ θ = (θ⊗θ)∆ .
E D D C
(cid:3)
θ
Lemma 1.14. Let C−→D be morphisms of graded coalgebras and let d: C → D be a
θ-coderivation. Then:
(1) For every n
n
(cid:88)
∆n ◦d = ( θ⊗i⊗d⊗θ⊗n−i)◦∆n.
D C
i=0
(2) If p: D → V is a system of cogenerators, then d is uniquely determined by its
composition pd: C → V.
Proof. The first item is a straightforward induction on n, using the equalities ∆n =
Id⊗∆n−1 and θ⊗i∆i−1 = ∆i−1θ.
C D
For item 2, we need to prove that pd = 0 implies d = 0. Assume that there exists c ∈ C
such that dc (cid:54)= 0, then there exists n such that p⊗n+1∆ndc (cid:54)= 0. On the other hand
D
n
(cid:88)
p⊗n+1∆ndc = ( (pθ)⊗i⊗pd⊗(pθ)⊗n−i)◦∆nc = 0.
D C
i=0
(cid:3)
Exercise 1.15. A counity of a graded coalgebra is a morphism of graded vector spaces
(cid:15): C → K such that ((cid:15)⊗Id )∆ = (Id ⊗(cid:15))∆ = Id . Prove that if a counity exists, then
C C C
it is unique (Hint: ((cid:15)⊗(cid:15)(cid:48))∆ =?).
Exercise 1.16. Let (C,∆) be a graded coalgebra. A graded subspace I ⊂ C is called a
coideal if ∆(I) ⊂ C ⊗I +I ⊗C. Prove that a subspace is a coideal if and only if is the
kernel of a morphism of coalgebras.
A VOYAGE ROUND COALGEBRAS 5
Exercise 1.17. Let C be a graded coalgebra and d ∈ Coder1(C,C) a codifferential of
degree 1. Prove that the triple (L,δ,[,]), where:
L = ⊕n∈ZCodern(C,C), [f,g] = fg−(−1)gfgf, δ(f) = [d,f]
is a differential graded Lie algebra.
2. Connected coalgebras
Definition 2.1. A graded coalgebra (C,∆) is called nilpotent if ∆n = 0 for n >> 0.
It is called locally nilpotent if it is the direct limit of nilpotent graded coalgebras or
equivalently if C = ∪ ker∆n.
n
Example 2.2. The vector space
K[t] = {p(t) ∈ K[t] | p(0) = 0} = (cid:76) Ktn
n>0
with the coproduct
n−1
(cid:88)
∆: K[t] → K[t]⊗K[t], ∆(tn) = ti⊗tn−i,
i=1
is a locally nilpotent coalgebra. The projection K[t] → K[t], p(t) → p(t) − p(0), is a
morphism of coalgebras.
Example 2.3. Let A = ⊕A be a finite dimensional graded associative commutative
i
K-algebra and let C = A∨ = Hom∗(A,K) be its graded dual.
SinceAandC arefinitedimensional,thepairing(cid:104)c1⊗c2,a1⊗a2(cid:105) = (−1)a1c2(cid:104)c1,a1(cid:105)(cid:104)c2,a2(cid:105)
gives a natural isomorphism C ⊗C = (A⊗A)∨ commuting with the twisting maps T;
we may define ∆ as the transpose of the multiplication map µ: A⊗A → A.
Then (C,∆) is a coassociative cocommutative coalgebra. Note that C is nilpotent if and
only if A is nilpotent.
Exercise 2.4. Let (C,∆) be a graded coalgebra. Prove that for every a,b ≥ 0
a+1
∆a(ker∆a+b) ⊂ (cid:78)(ker∆b).
(Hint: prove first that ∆a(ker∆a+b) ⊂ ker∆b⊗C⊗a.)
Exercise 2.5. Let (C,∆) be a locally nilpotent graded coalgebra. Prove that every
projection p: C → ker∆ is a system of cogenerators.
Definition 2.6 ([8, p. 282]). A graded coalgebra (C,∆) is called connected if there is
an element 1 ∈ C such that ∆(1) = 1⊗1 (in particular deg(1) = 0) and C = ∪+∞F C,
r=0 r
where F C is defined recursively by the formulas
r
F C = K 1, F C = {x ∈ C | ∆(x)−1⊗x−x⊗1 ∈ F C ⊗F C}.
0 r+1 r r
Example 2.7. Every locally nilpotent coalgebra is connected (with 1 = 0, see Exer-
cise 2.4). If f: C → D is a surjective morphism of coalgebras and C is connected, then
also D is connected.
Lemma 2.8. Let C be a connected coalgebra and e ∈ C such that ∆(e) = e⊗e. Then
either e = 0 or e = 1.
In particular the idempotent 1 as in Definition 2.6 is determined by C.
6 MARCOMANETTI
Proof. Let r be the minimum integer such that e ∈ F C. If r = 0 then e = t1 for some
r
t ∈ K; if 1 (cid:54)= 0 then t2 = t and t = 0,1.
If r > 0 we have
(e−1)⊗(e−1) = ∆(e)−1⊗e−e⊗1+1⊗1 ∈ F C ⊗F C
r−1 r−1
and then e−1 ∈ F C which is a contradiction. (cid:3)
r−1
The reduction of a connected coalgebra C is defined as its quotient C = C/K1; it is
a locally nilpotent coalgebra.
3. The reduced tensor coalgebra
Given a graded vector space V, we denote T(V) = (cid:76) (cid:78)nV. When considered as
n>0
a subset of T(V) it becomes an ideal of the tensor algebra generated by V.
The reduced tensor coalgebra generated by V is the graded vector space T(V) endowed
with the coproduct a: T(V) → T(V)⊗T(V):
n−1
(cid:88)
a(v ⊗···⊗v ) = (v ⊗···⊗v )⊗(v ⊗···⊗v ).
1 n 1 r r+1 n
r=1
We can also write
+∞ n−1
(cid:88) (cid:88)
a = a ,
a,n−a
n=2 a=1
where a : (cid:78)nV → (cid:78)aV ⊗(cid:78)n−aV is the inverse of the multiplication map.
a,n−a
The coalgebra (T(V),a) is coassociative, it is locally nilpotent and the projection
p1: T(V) → V is a system of cogenerators: in fact, for every s > 0,
(cid:88)
as−1(v ⊗···⊗v ) = (v ⊗···⊗v )⊗···⊗(v ⊗···⊗v )
1 n 1 i1 is−1+1 n
1≤i1<i2<···<is=n
and then
s−1
keras−1 = (cid:76) V⊗i, (⊗sp1)as−1 = ps: T(V) → V⊗s.
i=1
Exercise 3.1. Let µ: (cid:78)sT(V) → T(V) be the multiplication map. Prove that for
every v ,...,v ∈ V
1 n
(cid:18) (cid:19)
n−1
µas−1(v ⊗···⊗v ) = v ⊗···⊗v .
1 n 1 n
s−1
For every morphism of graded vector spaces f: V → W the induced morphism of
graded algebras
T(f): T(V) → T(W), T(f)(v ⊗···⊗v ) = f(v )⊗···⊗f(v )
1 n 1 n
is also a morphism of graded coalgebras.
If (C,∆) is a locally nilpotent graded coalgebra then, for every c ∈ C, there exists
n > 0 such that ∆n(c) = 0 and then it is defined a morphism of graded vector spaces
∞
1 (cid:88)
= ∆n: C → T(C).
1−∆
n=0
Proposition 3.2. Let (C,∆) be a locally nilpotent graded coalgebra, then:
1
(1) The map = (cid:80) ∆n: C → T(C) is a morphism of graded coalgebras.
1−∆ n≥0
A VOYAGE ROUND COALGEBRAS 7
(2) ForeverygradedvectorspaceV andeverymorphismofgradedcoalgebrasφ: C →
T(V), there exists a unique morphism of graded vector spaces f: C → V such
that φ factors as
∞
1 (cid:88)
φ = T(f) = (⊗nf)∆n−1: C → T(C) → T(V).
1−∆
n=1
Proof. [1] We have
   
n
(cid:88) (cid:88) (cid:88)(cid:88)
 ∆n⊗ ∆n∆ = (∆a⊗∆n−a)∆
n≥0 n≥0 n≥0a=0
 
n
(cid:88)(cid:88) (cid:88)
= aa+1,n+1−a∆n+1 = a ∆n
n≥0a=0 n≥0
where in the last equality we have used the relation a∆0 = 0.
[2] The unicity of f is clear, since by the formula φ = T(f)((cid:80) ∆n) it follows that
n≥0
f = p1φ.
To prove the existence of the factorization, take any morphism of graded coalgebras
φ: C → T(V), denote by f = p1φ and by ψ: C → T(V) the coalgebra morphism
ψ = T(f)(1 − ∆)−1. Since p1ψ = p1φ and p1 is a system of cogenerators we have
φ = ψ. (cid:3)
It is useful to restate part of the Proposition 3.2 in the following form
Corollary 3.3. Let V be a fixed graded vector space; for every locally nilpotent graded
coalgebra C the composition with the projection p1: T(V) → V induces a bijection
∼
Hom (C,T(V))−→Hom (C,V).
GC G
In other words, every morphism of graded vector spaces C → V has a unique lifting to
a morphism of graded coalgebras C → T(V).
When C is a reduced tensor coalgebra, Proposition 3.2 takes the following more
explicit form
Corollary 3.4. Let U,V be graded vector spaces. the projection. Given f: T(U) → V,
the linear map F: T(U) → T(V):
n
(cid:88) (cid:88)
F(v ⊗···⊗v ) = f(v ⊗···⊗v )⊗···⊗f(v ⊗···⊗v ),
1 n 1 i1 is−1+1 is
s=1 1≤i1<i2<···<is=n
is the morphism of graded coalgebras lifting f.
Example3.5. LetAbeanassociativegradedalgebra.Considertheprojectionp: T(A) →
A, the multiplication map µ: T(A) → A and its conjugate
µ∗ = −µT(−1), µ∗(a ⊗···⊗a ) = (−1)n−1µ(a ⊗···⊗a ) = (−1)n−1a a ···a .
1 n 1 n 1 2 n
The two coalgebra morphisms T(A) → T(A) induced by µ and µ∗ are isomorphisms,
the one inverse of the other.
In fact, the coalgebra morphism F: T(A) → T(A)
n
(cid:88) (cid:88)
F(a ⊗···⊗a ) = (a a ···a )⊗···⊗(a ···a )
1 n 1 2 i1 is−1+1 is
s=1 1≤i1<i2<···<is=n
8 MARCOMANETTI
is induced by µ (i.e. pF = µ), µ∗F(a) = a for every a ∈ A and for every n ≥ 2
n
(cid:88) (cid:88)
µ∗F(a ⊗···⊗a ) = (−1)s−1 a a ···a =
1 n 1 2 n
s=1 1≤i1<i2<···<is=n
n (cid:18) (cid:19) (cid:32)n−1 (cid:18) (cid:19)(cid:33)
(cid:88) n−1 (cid:88) n−1
= (−1)s−1 a a ···a = (−1)s a a ···a = 0.
1 2 n 1 2 n
s−1 s
s=1 s=0
This implies that µ∗F = p and therefore, if F∗: T(A) → T(A) is induced by µ∗ then
pF∗F = µ∗F = p and by Corollary 3.3 F∗F is the identity.
Proposition 3.6. Let (C,∆) be a locally nilpotent graded coalgebra, V a graded vector
space and
∞
(cid:88)
θ = (⊗nf)∆n−1: C → T(V)
n=1
the morphism of coalgebras induced by pθ = f ∈ Hom0(C,V). For every n and every
q ∈ Homk(C,V), the linear map
∞ n
(cid:88) (cid:88)
Q = ( (f⊗i⊗q⊗f⊗n−i)∆n: C → T(V)
n=0 i=0
is the θ-coderivation induced by pQ = q. In particular the map
Coderk(C,T(V);θ) → Homk(C,V), Q (cid:55)→ pQ,
is bijective.
Proof. The map Q is the composition of the coalgebra morphism (cid:80)∆n: C → T(C) and
the map
(cid:88)
R: T(C) → T(V), R = f⊗i⊗q⊗f⊗j.
i,j≥0
It is therefore sufficient to prove that R is a T(f)-coderivation, i.e. that satisfies the
coLeibniz rule
(R⊗T(f)+T(f)⊗R)a = aR.
Denoting R = (cid:80) f⊗i⊗q⊗f⊗j we have, for every a,n
n i+j=n−1
a R = (R ⊗f⊗n−a+f⊗a⊗R )a .
a,n−a n a n−a a,n−a
Taking the sum over a,n−a we get the proof. (cid:3)
Corollary 3.7. Let V be a graded vector space. Every q ∈ Homk(T(V),V) induce a
coderivation Q ∈ Coderk(T(V),T(V)) given by the explicit formula
Q(a ⊗···⊗a ) =
1 n
(cid:88)
= (−1)k(a1+···+ai)a ⊗···⊗a ⊗q(a ⊗···⊗a )⊗···⊗a .
1 i i+1 i+l n
i,l
Proof. Apply Proposition 3.6 with the map f: T(V) → V equal to the projection (and
then θ = Id). (cid:3)
Exercise 3.8. Let p: T(V) → T(V) be the projection with kernel K = (cid:78)0V and
φ: T(V) → T(V)⊗T(V)theuniquehomomorphismofgradedalgebrassuchthatφ(v) =
v⊗1+1⊗v for every v ∈ V. Prove that pφ = ap.
A VOYAGE ROUND COALGEBRAS 9
Exercise 3.9. Let A be an associative graded algebra over the field K, for every local
homomorphism of K-algebras γ: K[[x]] → K[[x]], γ(x) = (cid:80)γ xn, we can associate a
n
coalgebra morphism F : T(A) → T(A) induced by the linear map
γ
f : T(A) → A, f(a ⊗···⊗a ) = γ a ···a .
γ 1 n n 1 n
Prove the composition formula F = F F . (Hint: Example 1.6.)
γδ δ γ
Exercise 3.10. A graded coalgebra morphism F: T(U) → T(V) is surjective (resp.:
injective, bijective) if and only if F1: U → V is surjective (resp.: injective, bijective).
1
(Hint: F preserves the filtrations of kernels of iterated coproducts.)
4. Rooted trees
Definition 4.1. An unreduced1 rooted forest is the data of a finite set of vertices V and
a flow map f: V → V such that:
(cid:92)
Fix(f) = fn(V −Fix(f)),
n>0
where Fix(f) = {v ∈ V | f(v) = v} is the subset of fixed points of f.
Theverticesofanunreducedrootedforest(V,f)aredividedintothreedisjointclasses:
• V = {root vertices} = Fix(f).
r
• V = {tail vertices} = V −f(V).
t
• V = {internal vertices} = f(V)−{root vertices}.
i
Every unreduced rooted forest (V,f) can be described by a directed graph with set
of vertices V and oriented edges v→f(v) for every v (cid:54)∈ {root vertices}. In our pictures
internal vertices will be denoted by a black dot, while tail and root vertices will be
denoted by a circle.
As an example, the pair (V,f), where V = {1,2,3,4} and f(i) = min(4,i + 1) is an
almost rooted forest described by the oriented graph
(cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47)
◦ •• •• ◦
tail root
Note that the map f: V ∪ V → V ∪ V is surjective and then the number of tail
t i i r
vertices is always greater than or equal to the number of root vertices.
The set of edges {(v,f(v)) | v (cid:54)∈ {roots}} is divided into types. An edge (v,f(v)) is
called a root edge if f(v) is a root vertex; it is called a tail edge if v is a tail vertex and it
is called an internal edge if both v,f(v) are both internal vertices. Notice that an edge
may be tail and root at the same time.
The arity (also called valence in literature) |v| of a vertex v is the number of incoming
edges; equivalently
|v| = |{w (cid:54)= v | f(w) = v}|.
A rooted forest is an almost rooted forest such that every root has arity 1 and every
internal vertex has arity ≥ 2.
A rooted tree is a rooted forest with exactly one root.
1Here unreduced means not necessarily reduced.
10 MARCOMANETTI
Every rooted forest is then a disjoint union of rooted trees; the following picture
represents a rooted tree with 5 tail vertices and 4 internal vertices.
◦◦(cid:119)(cid:71)(cid:119)(cid:71)(cid:119)(cid:71)(cid:119)(cid:71)(cid:119)(cid:71)(cid:119)◦(cid:35)(cid:35)•(cid:59)(cid:59)(cid:119)(cid:71)(cid:71)(cid:119)(cid:71)(cid:119)(cid:71)(cid:119)(cid:71)(cid:119)(cid:71)(cid:59)(cid:59)•(cid:71)(cid:35)(cid:35) (cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:35)(cid:35)
◦ ◦(cid:119)(cid:119)(cid:119)(cid:119)(cid:119)•(cid:59)(cid:59)•◦(cid:119)(cid:71)(cid:119)(cid:71)(cid:119)(cid:71)(cid:119)(cid:71)(cid:119)(cid:71)(cid:59)(cid:59)•(cid:35)(cid:35)•• (cid:47)(cid:47)◦root
◦
An automorphism of a rooted forest (V,f) is a bijective map φ: V → V such that
fφ = φf. The group of automorphisms will be denoted by Aut(V,f).
Definition 4.2. Let(V,f)bearootedforest.Anorientation of(V,f)isatotalordering
≤ on the set V of tail vertices such that if v ≤ u ≤ w and fk(v) = fh(w), for some
t
h,k ≥ 0, then there exists l ≥ 0 such that fk(v) = fl(u) = fh(w).
It is often convenient to describe an orientation ≤ by the order-preserving bijection
ν: {1,...,n} → V , where |V | = n. Therefore, an oriented rooted forest is a triple
t t
(V,f,ν) where ν is an orientation of (V,f).
For instance, there are (up to isomorphism) exactly three oriented rooted trees with
3 tails:
12◦◦(cid:119)(cid:119)(cid:71)(cid:119)(cid:71)(cid:119)(cid:35)(cid:35)•(cid:59)(cid:59)◦•(cid:119)(cid:119)(cid:71)(cid:71)(cid:119)••(cid:35)(cid:35)•(cid:59)(cid:59) ◦(cid:47)(cid:47) 32◦◦(cid:119)(cid:71)(cid:119)(cid:71)(cid:119)(cid:119)•(cid:35)(cid:35)•(cid:59)(cid:59)◦(cid:119)(cid:119)(cid:71)(cid:71)(cid:119)(cid:59)(cid:59)••(cid:35)(cid:35)• ◦(cid:47)(cid:47) 12◦◦(cid:71)(cid:119)(cid:71)(cid:119)(cid:71)(cid:119)(cid:71)(cid:119)(cid:71)(cid:119)(cid:35)(cid:35)(cid:59)(cid:59)•(cid:47)(cid:47)••• (cid:47)(cid:47)◦
3◦ 1◦ 3◦
Lemma 4.3. Let V be a rooted tree. Then the number of isomorphism classes of orien-
tations on V is equal to
1 (cid:89)
|v|!
|Aut(V)|
v∈Vi
Proof. The group of automorphisms of V acts freely on the set of orientations. We note
that every orientation is uniquely determined by:
(1) A total ordering of root edges.
(2) For every internal vertex, a total ordering of incoming edges.
Thereforetheproduct(cid:81) |v|!isequaltothenumberoforientationsonthetreeV. (cid:3)
v∈Vi
Denote by F(n,m) the set of isomorphism classes of oriented rooted forests with n
tails and m roots. Notice that F(n,m) = ∅ for m > n and F(n,n) contains only one
element, denoted by I .
n
There are defined naturally two binary operations:
◦: F(l,m)×F(n,l) → F(n,m) composition
⊗: F(n,m)×F(a,b) → F(n+a,m+b) tensor product
The tensor product V ⊗ W is the disjoint union of V and W with the orientation
{1,2,...} → W shifted by the number of tail vertices of V. For instance I ⊗I = I
t a b a+b
and
◦(cid:79)(cid:79) ◦(cid:79)(cid:79) ◦(cid:79)(cid:79) ◦(cid:79)(cid:79)
(cid:10)(cid:10)•••(cid:68)(cid:68) (cid:52)(cid:90)(cid:90)(cid:52)(cid:52) ⊗ (cid:10)(cid:10)•••(cid:68)(cid:68) (cid:52)(cid:90)(cid:90)(cid:52)(cid:52) = (cid:10)(cid:10)•••(cid:68)(cid:68) (cid:52)(cid:90)(cid:90)(cid:52)(cid:52) (cid:10)(cid:10)•••(cid:68)(cid:68) (cid:52)(cid:90)(cid:90)(cid:52)(cid:52)
(cid:10) (cid:10) (cid:10) (cid:10)
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦
1 2 1 2 1 2 3 4
Given (V,f) ∈ F(n,l) and (W,g) ∈ F(l,m) we define W ◦V in the following way:
first we take the unique bijection η: V → W such that for i >> 0 the map
r t
ηfi
V −−→ W
t t