On the logarithm of the identity on connected filtered bialgebras

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On the logarithm of the identity on connected filtered bialgebras

Darij Grinberg

Version 0.19, 18 June 2021

***

The purpose of this document is to prove several properties of coalgebras, bialgebras and Hopf algebras. The proofs given here are mostly not new, and often not optimal;

however, they are very detailed and don’t use Sweedler’s notation.

Remark (2017):

This “lab notebook” has been mostly written in 2011–2013 (when I was a graduate student), and collects various results I have encountered while exploring the theory of Hopf algebras (e.g., variants of the Cartier-Milnor- Moore and Leray theorems; properties of the Eulerian idempotent; facts like the invertibility of the antipode in a connected filtered Hopf algebra).

I expect few of these results to be new; the best I can claim is that they are stated more explicitly here than in the available literature. Unfortunately, this notebook is rather disorganized, and the results are written down more or less in the order in which I have found them. I have tried to give detailed proofs of all statements (if only to make sure that they are correct); these proofs should be “technically” readable but in practice you might have an easier time skimming them for their main ideas (which are, unfortunately, sometimes hidden well) and reconstructing the rest yourself. Needless to say, the notations used in this notebook are also not the best.

Acknowledgments: Thanks to Philipp Varˇso for finding an error in the proof of Proposition 5.13 (the statement of Corollary 5.14 was insufficiently general).

§ 1. Notations and definitions

In this document, we shall use some standard terminology from the theory of Hopf algebras (see, for example, [Schnei15] or [Mancho06]) along with the following notations:

Convention 1.1. In the following, the symbol N always denotes the set {0,1,2, . . .}.

Convention 1.2. A “ring” shall always mean an associative ring with unity.¹

Definition 1.3. Let k be a field, and U and V be two k-vector spaces.

Then,L(U, V) denotes the vector space of all k-linear maps U →V. (This vector space is commonly denoted by Hom_k(U, V).)

Convention 1.4. In the following, whenever a commutative ring k exists in the context², the ⊗ sign will always mean⊗_k.

1We will sometimes say “ring with unity” to additionally stress this, but even if we do not say

“with unity” we still mean unital rings only.

2Most of the time we will be working over a ground field, which will be denoted byk. Occasionally (e.g., in§14), we will be working over a ground ring, but it will also be denoted by k.

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Convention 1.5. IfQis any set andn is any nonnegative integer, then we shall denote the Cartesian product Q×Q× · · · ×Q

| {z }

ntimes

by Q^×n. This Carte- sian product is usually denoted byQⁿin the literature, but we shall instead reserve the notation Qⁿ for another meaning (which will be introduced in Convention 15.2).

Convention 1.6. Whenever k is a field, A, B, C and D are four k-vector spaces, and f : A → B and g : C → D are two k-linear maps, then the notationf⊗g can mean two different things: On the one hand, it can mean the k-linear map f⊗g :A⊗C →B⊗D(which mapsa⊗ctof(a)⊗g(c) for every a ∈ A and c ∈ C). On the other hand, it can mean the tensor f⊗g ∈ L(A, B)⊗ L(C, D) (sincef ∈ L(A, B) and g ∈ L(C, D)). Let us agree that in the following, whenever a term likef⊗g (with f andg being two k-linear maps) occurs, it will mean the first thing (i. e., the k-linear map f ⊗g : A⊗C → B ⊗D) and not the second one (i. e., the tensor f⊗g ∈ L(A, B)⊗ L(C, D)).

Definition 1.7. Let k be a field and A be a k-algebra. Then, µ_A will always denote the multiplication map A⊗A →A of the k-algebra A, and η_Awill always denote the unity map k →Aof the k-algebra A. When it is clear which algebra we are talking about, we will abbreviateµ_A and η_A as µand η, respectively.

Definition 1.8. Let k be a field and C be a k-coalgebra. Then, ∆_C will always denote the comultiplication map C →C⊗C of the k-coalgebraC, and ε_C will always denote the counit map C → k of the k-coalgebra C.

When it is clear which coalgebra we are talking about, we will abbreviate

∆_C and ε_C as ∆ and ε, respectively.

Definition 1.9. Let k be a field, let A be a k-algebra, and let C be a k-coalgebra. Then, the k-vector space L(C, A) becomes a k-algebra (L(C, A),∗) by setting

f∗g =µA◦(f ⊗g)◦∆C for any f ∈ L(C, A) and g ∈ L(C, A). (1) This k-algebra (L(C, A),∗) has unity η_A◦ε_C and is called theconvolution algebra of C and A. In the following, we will simply refer to this algebra asL(C, A).

The binary operation∗defined in (1) is calledconvolution. In particular, for any f ∈ L(C, A) and g ∈ L(C, A), we will refer tof∗g as theconvolution of the maps f and g.

Convention 1.10. Let k be a field, let A be a k-algebra, and let C be a k-coalgebra. For any n ∈ N and any f ∈ L(C, A), we will denote by f^∗n the n-th power of the elementf in the convolution algebra L(C, A).

If an element f ∈ L(C, A) has a multiplicative inverse in the k-algebra

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Definition 1.11. Let k be a field. Let H be a k-Hopf algebra. Then, SH will always denote the antipode of the k-Hopf algebra S (that is, the

∗-inverse of the identity map id_H ∈ L(H, H)).

Definition 1.12. Let k be a field. Let C be a k-coalgebra. Let A be a k-algebra. Then, eC,A shall denote the map ηA◦εC : C → A. This map e_C,A is the unity of the convolution algebraL(C, A).

Next, let us define some concepts related to filtrations on vector spaces. Different authors define the words “filtered” and “filtration” in different (often non-equivalent) ways, so some care should be taken when consulting the literature.

Definition 1.13. Let k be a field. A filtered k-vector space means a k- vector spaceV equipped with a family (V_≤`)_`≥0 of k-vector subspaces ofV satisfyingV≤0 ⊆V≤1 ⊆V≤2 ⊆ · · · andV = S

`≥0

V≤`. Such a filteredk-vector space will often be denoted simply by V (that is, the family (V≤`)_`≥0 will not be explicitly mentioned). The family (V≤`)_`≥0 is called thefiltration of this filtered k-vector space. For each m ∈N, the k-vector subspace V≤m of V is called the m-th part of the filtration (V≤`)_`≥0.

Convention 1.14. In the following, wheneverV is a filtered vector space, we will denote the filtration onV by (V≤`)_`≥0. (This is a general convention, so it does not only pertain to filtered vector spaces called V, but pertains to any filtered vector space. For instance, if we have a filtered vector space calledC, then this convention yields that the filtration on C is denoted by (C≤`)_`≥0.)

Furthermore, whenever V is a filtered vector space and ` is a negative integer, we define V_≤` to mean the k-vector subspace 0 of V. (Thus, V_≤` is defined for each `∈Z, not only for ` ∈N.)

Definition 1.15. Letkbe a field. Afiltered k-coalgebrameans ak-coalgebra Cthat is simultaneously a filteredk-vector space (i.e., that is equipped with a family (C≤`)_`≥0 ofk-vector subspaces ofC satisfyingC≤0 ⊆C≤1 ⊆C≤2 ⊆

· · · and C = S

`≥0

C≤`) and has the property that

eachn ∈N satisfies ∆ (C≤n)⊆

n

X

u=0

C≤u⊗C≤n−u.

Definition 1.16. Letk be a field, and let C be a filteredk-coalgebra. We say that the filtered k-coalgebra C is connected if and only if the map ε_C |_C_≤0:C≤0 →k is a k-vector space isomorphism³.

3Recall that (C_≤`)_`≥0 denotes the filtration of the filtered k-coalgebra C (according to Conven- tion 1.14).

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§ 2. Unital coalgebras

Next we shall define the notion of a unital coalgebra. This notion is an intermediate step between the (rather well-known) notions of a coalgebra and of a bialgebra; it also is an intermediate step between the notions of a coalgebra and of a filtered connected coalgebra. Its definition is very simple:

Definition 2.1. Let k be a field. Let C be a k-coalgebra, and let i be an element ofC. Then, (C, i) is said to be aunital coalgebra⁴ if ∆_C(i) = i⊗i and ε_C(i) = 1.

When (C, i) is a unital coalgebra, we will denote the elementi by 1_(C,i) and call it the unity of the unital coalgebra (C, i).

When C is a k-coalgebra, there may be several elements i ∈ C for which (C, i) is a unital coalgebra⁵ (but there also may be no such elements). Hence, a unital coalgebra (C, i) is not uniquely determined by the coalgebra C. However, there are many cases where we have some additional structure onC (like ak-bialgebra structure or a connected filteredk-coalgebra structure) which gives rise to one preferred canonical i. First we consider the case when C is a k-bialgebra. In this case, we have:

Proposition 2.2. Letk be a field. Let C be a k-bialgebra. Then, (C,1_C) is a unital coalgebra. (Here, 1_C denotes the unity of the k-algebra C, as usual.)

Proof of Proposition 2.2. By the axioms of a k-bialgebra, we have ∆_C(1_C) = 1_C ⊗1_C and εC(1C) = 1. By the definition of a unital coalgebra, this means that (C,1C) is a unital coalgebra. Proposition 2.2 is thus proven.

Note that Proposition 2.2 really needs the condition that C is a k-bialgebra, and not just some k-vector space with a k-algebra structure and a k-coalgebra structure.

Now we consider the case of connected filtered coalgebras:

Proposition 2.3. Let k be a field. Let C be a connected filtered k- coalgebra. Then,

C, ε_C |_C_≤0−1

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is a unital coalgebra.

Proof of Proposition 2.3. SinceC is connected, the mapε_C |_C_≤0:C≤0 →k is ak-vector space isomorphism. Hence, the map ε_C |_C_≤0⁻¹

:k →C_≤0 is well-defined.

Let us define an element i ∈ C byi = ε_C |_C_≤0−1

(1). We are going to show that (C, i) is a unital coalgebra.

Since ε_C |_C_≤0:C≤0 →k is an isomorphism, we have C≤0 = ε_C |_C_≤0−1

k

|{z}

=k·1

!

= ε_C |_C_≤0−1

(k·1) =k· ε_C |_C_≤0−1

(1)

| {z }

=i

=k·i.

4To be completely honest, we would have to call this “unitalk-coalgebra” to make clear that this notion depends on the fieldk. However, since we are not going to change the field kanytime soon,

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Since i = ε_C |_C_≤0−1

(1), we have 1 = ε_C |_C_≤0

(i) = ε_C

|{z}

=ε

(i) = ε(i). Now, i ∈ C≤0, so that ∆ (i)∈ ∆ (C≤0) ⊆

0

P

u=0

C≤u ⊗C≤0−u (since C is a filtered coalgebra).

Since

0

X

u=0

C≤u⊗C≤0−u =C≤0⊗C≤0−0

| {z }

=C≤0

=C≤0⊗C≤0 = (k·i)⊗(k·i) (since C≤0 =k·i)

=k·(i⊗i),

this rewrites as ∆ (i) ∈ k·(i⊗i). Thus, there exists some λ ∈ k such that ∆ (i) = λ·(i⊗i). Consider this λ.

Let can : C⊗k →C be the canonicalk-module isomorphism (sendingc⊗xtocxfor allc∈Candx∈k). Then, by the axioms of a coalgebra, we have can◦(id⊗ε)◦∆ = id.

But

(can◦(id⊗ε)◦∆) (i) = can





(id⊗ε)





∆ (i)

| {z }

=λ·(i⊗i)











= can





(id⊗ε) (λ·(i⊗i))

| {z }

=λ·id(i)⊗ε(i)







= can (λ·id (i)⊗ε(i)) =λ·id (i)

| {z }

=i

·ε(i)

|{z}=1

(by the definition of can)

=λi, so that

λi= (can◦(id⊗ε)◦∆)

| {z }

=id

(i) = id (i) = i.

Now, ∆ (i) = λ·(i⊗i) = λi

|{z}

=i

⊗i=i⊗i.

So we have ∆C(i) = ∆ (i) = i⊗i and εC(i) = 1. By the definition of a unital coalgebra, this shows that (C, i) is a unital coalgebra. Since i = ε_C |_C_≤0−1

(1), this means that

C, εC |C≤0

−1

(1)

is a unital coalgebra. Proposition 2.3 is proven.

Proposition 2.2 gives us a unital coalgebra when we start with a k-bialgebra.

Proposition 2.3 gives us a unital coalgebra when we start with a connected filtered k-coalgebra. One might wonder what happens if we start with a connected filteredk- bialgebra: In this case, each of Propositions 2.2 and 2.3 gives us a unital coalgebra. Are these two unital coalgebras the same? The answer is yes, as the following proposition shows:

Proposition 2.4. Let k be a field, and let C be a connected filtered⁶ k- bialgebra. Then, 1_C = ε_C |_C_≤0−1

(1) (where 1_C denotes the unity of the k-algebra C). As a consequence, the unital coalgebras (C,1_C) and

C, ε_C |_C_≤0−1

(1)

are identic.

6Of course, when we say that the filtered k-bialgebra C is connected, we mean that the filtered k-coalgebra Cis connected.

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Proof of Proposition 2.4. By the definition of a connected filteredk-coalgebra, the map εC |C≤0: C≤0 → k is a k-vector space isomorphism (since C is connected). Moreover,

ε_C |_C_≤0

(1_C) = ε_C(1_C) = 1, so that 1_C = ε_C |_C_≤0−1

(1). This proves Proposi- tion 2.4.

Since it is cumbersome to explicitly mention the unity every time we are referring to a unital coalgebra, we make the following convention:

Convention 2.5. Let k be a field. Let (C, i) be a unital coalgebra. Then, we will often abbreviate the “unital coalgebra (C, i)” as “unital coalgebra C”. This abbreviation is an abuse of notation, since a unital coalgebra (C, i) is not uniquely determined by the coalgebra C; but we will only use this abbreviation when it is clear what i we mean. In particular, we will use this abbreviation when there is a canonical unital coalgebra structure onC obtained from either Definition 2.6 or Definition 2.7 (below), or when the unital coalgebra is just being defined⁷.

Definition 2.6. Letk be a field. LetC be ak-bialgebra. Then, according to Proposition 2.2, the unital coalgebra (C,1_C) is well-defined (where 1_C denotes the unity of thek-algebraC). This unital coalgebra (C,1C) is called theunital coalgebra canonically induced by the k-bialgebra C. Whenever we just speak of “the unital coalgebra C” (where C is a k-bialgebra), we will always mean this unital coalgebra (C,1C).

Definition 2.7. Letkbe a field. LetCbe a connected filteredk-coalgebra.

Then, the unital coalgebra

C, ε_C |_C_≤0−1

(1)

(this is well-defined according to Proposition 2.3) is called theunital coalgebra canonically induced by the connected filtered k-coalgebra C. Whenever we just speak of “the unital coalgebra C” (where C is a connected filtered k-coalgebra), we will always mean this unital coalgebra

C, ε_C |_C_≤0⁻¹ (1)

.

Remark 2.8. Let k be a field. Let C be a connected filtered k-bialgebra.

Then, the “unital coalgebra C” as understood according to Definition 2.6 is identic with the “unital coalgebra C” as understood according to Defini- tion 2.7. (This is just a rewording of Proposition 2.4.) Thus, the notations introduced in Definitions 2.6 and 2.7 don’t conflict with each other.⁸ Remark 2.9. Letkbe a field. LetCbe a unital coalgebra. Then, the unity

of this unital coalgebraC is denoted by 1_C. This is not a new notation we introduce; it is just a consequence of our Definition 2.1 (where we stipulated

7For instance, when we write “LetC be a unital coalgebra”, we are using this abbreviation; this is okay, because there are no elementsiofC defined yet which we can confuse.

8This only pertains to the case when C is a connected filtered k-bialgebra. If C would be a vector space with a connected filtered k-coalgebra structure on one hand and a (totally unrelated!) k-bialgebra structure on the other, then, of course, then the “unital coalgebra C” in the sense of Definition 2.6 would not necessarily be identic to the “unital coalgebraC” in the sense of Definition 2.7,

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that the unity of a unital coalgebra (C, i) will be denoted by 1_(C,i)) because we abbreviate (C, i) by C.

This notation can conflict with the notation 1_A for the unity of ak-algebra A: In fact, if we have a vector space V which happens to be a k-algebra and a unital coalgebra at the same time, then in general it might occur that the unity of the unital coalgebraV is not the same as the unity of the k-algebraV, so the notation 1_V would be ambiguous (it could mean each of these two unities). However, when we have ak-bialgebraC, then the unity of the unital coalgebraC (which is understood according to Definition 2.6) is the same as the unity of the k-algebra C ⁹, so there is no conflict in this case. Fortunately, we are going to have this case all of the time, so we won’t have to care about possible conflicts between these notations.

Remark 2.10. Let k be a field, and let C be a connected filtered k- coalgebra. Then,C is a unital coalgebra (according to Definition 2.7) with unity 1_C = ε_C |_C_≤0⁻¹

(1).

Proof of Remark 2.10. The unital coalgebraC(as defined in Definition 2.7) is

C, ε_C |_C_≤0⁻¹ (1)

. Hence, the unity of this coalgebra is ε_C |_C_≤0−1

(1). Since we denote the unity of the unital coalgebra C by 1_C, this means that 1_C = ε_C |_C_≤0−1

(1). Remark 2.10 is proven.

Remark 2.11. Let k be a field, and let C be a connected filtered k- coalgebra. Then,C≤0 =k·1C. (Here, as usual, 1C denotes the unity of the unital coalgebra C, which unital coalgebra is defined as in Definition 2.7.) Proof of Remark 2.11. In the proof of Proposition 2.3, we showed that C≤0 = k ·i, where i = ε_C |_C_≤0−1

(1). But this i is equal to 1_C (since i = ε_C |_C_≤0−1

(1) = 1_C by Remark 2.10), so that C_≤0 = k ·i rewrites as C_≤0 = k·1_C. Remark 2.11 is now proven.

Remark 2.12. Letk be a field, and let H be a filtered k-bialgebra. Then, H is connected if and only if H≤0 =k·1_H (where 1_H denotes the unity of the k-algebra H).

Proof of Remark 2.12. a) Let us prove that ifH is connected, thenH_≤0 =k·1_H. Proof. Assume thatH is connected. Then,H becomes a unital coalgebra according to Definition 2.7. Thus, the notation 1_H can mean two different things: On the one hand, it can mean the unity of the k-algebra H, but on the other hand, it can mean the unity of the unital coalgebra H (defined by Definition 2.7). Fortunately, this does not yield a conflict because these two things are the same (by Remark 2.9, since H is a k-bialgebra). Remark 2.11 (applied to C =H) now yields H_≤0 =k·1_H.

We thus have proven that if H is connected, thenH≤0 =k·1_H. b) Let us prove that if H≤0 =k·1_H, then H is connected.

9Proof. Let C be a k-bialgebra. According to Definition 2.6, the unital coalgebra C is (C,1C), where 1C denotes the unity of the k-algebra C. Hence, the unity of the unital coalgebra C is 1C, where 1C denotes the unity of thek-algebraC. In other words, the unity of the unital coalgebra Cis

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Proof. Assume thatH≤0 =k·1_H. Then, everyα∈H≤0satisfying ε_H |_H_≤0

(α) = 0 must satisfy α = 0 ¹⁰. Hence, the k-linear map εH |H≤0 must be injective. Since this map is also surjective¹¹, it thus follows that the map ε_H |_H_≤0 is bijective. Hence, ε_H |_H_≤0 is an isomorphism. By the definition of “connected”, this yields that H is connected.

We have thus proven that if H≤0 =k·1_H, thenH is connected.

Combining the above points a) and b), we obtain Remark 2.12.

Remark: The above Remark 2.12 is often used as an alternative definition of the notion of a connected filtered k-bialgebra. However, we prefer Definition 1.16, since it works for filtered k-coalgebras as well (and not only for filtered k-bialgebras).

Definition 2.13. Let k be a field, and let C be a unital coalgebra. Then, we denote by ηC the map k → C which sends every λ ∈ k to λ·1C ∈ C.

This map is called the unity map of the unital coalgebra C.

This notation could sometimes conflict with the notation η_A for the unity map of a k-algebra A. In fact, such a conflict might emerge when we have a k-vector space H which is both a unital coalgebra and a k-algebra at the same time; in this case, η_H might mean two different things (namely, the unity map of the k-algebra H on the one hand, and the unity map of the unital coalgebra H on the other), just as 1_H might mean two different things. However, when H is a k-bialgebra, both meanings of 1_H are the same, and therefore both meanings ofη_H are the same¹². Thus, no conflict can occur as long as H is a k-bialgebra.

10Proof. Letα∈H_≤0 satisfying ε_H |_H_≤0

(α) = 0 be arbitrary. Then,ε(α) = ε_H|_H_≤0

(α) = 0.

On the other hand, α ∈ H_≤0 = k·1H, so that there exists some λ ∈ k such that α = λ·1H. Consider this λ. Then, ε(α) = ε(λ·1H) = λ ε(1H)

| {z }

=1

=λ. Thus, ε(α) = 0 becomes λ= 0, so that α= λ

|{z}

=0

·1_H= 0, qed.

11In fact, every β ∈ k satisfies β ∈ ε_H |H≤0

(H_≤0) (because β · 1_H

|{z}

∈H≤0

∈ H_≤0 and εH|H_≤0

(β·1H) =ε(β·1H) =β ε(1H)

| {z }

=1

=β, so thatβ = εH |H_≤0

(β·1H)∈ εH |H_≤0

(H_≤0)).

12Proof. LetH be ak-bialgebra. Then, the notationηH might mean two different things: namely, the unity map of thek-algebra H on the one hand, and the unity map of the unital coalgebraH on the other. However, these two things are the same, since

(the unity map of the unital coalgebra H)

= (the mapk→H which sends everyλ∈kto λ·1H, where 1H denotes the unity of the unital coalgebraH) (because this is how the unity map of the unital coalgebraH was defined)

= (the mapk→H which sends everyλ∈kto λ·1H, where 1H denotes the unity of thek-algebraH) since Remark 2.9 (applied toC=H) says that the unity of the unital coalgebraH is the same

as the unity of thek-algebraH

= (the unity map of thek-algebraH)

(because this is how the unity map of thek-algebraH was defined).

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Now that we have defined the notion of unital coalgebras and cleared up some possible and impossible confusions, let us continue introducing notation. First, we observe that if we replace the words “k-algebra” by “unital coalgebra” in the definition of e_H,A (Definition 1.12), then this definition still makes sense (because the expression 1Amakes sense not only whenAis ak-algebra, but also whenAis a unital coalgebra), although of course the map e_H,A it defines is no longer the unity of the convolution algebraL(H, A) (since this convolution algebra does not exist in this situation). Thus, we obtain the following definition:

Definition 2.14. Let k be a field. Let C be a k-coalgebra. Let A be a unital coalgebra. Then, e_C,A shall denote the map η_A◦ε_C :C→A.

Definition 2.14 does not conflict with Definition 1.12 in the case when A is a k- bialgebra (because both interpretations of the expression 1_A mean the same thing in this case); these definitions also do not conflict in the case when A is a connected filtered coalgebra (for the same reason). Of course, if A is simultaneously a k-algebra and a unital coalgebra with two completely unrelated unities, then the two definitions can conflict.

§ 3. Logarithms and exponentials in convolution al- gebras

Definition 3.1. Let k be a field, let A be a k-algebra, and let H be a connected filteredk-coalgebra.¹³

(a) We denote by g(H, A) the subspace {f ∈ L(H, A) | f(1_H) = 0} of L(H, A). (Here, 1H denotes the unity of the unital coalgebra H, which is defined according to Definition 2.7.)

(b)For everyn∈N, we denote byLⁿ(H, A) the subspace

f ∈ L(H, A) | f |_H_≤n−1= 0

14. Then,

L⁰(H, A) =

f ∈ L(H, A) | f |_H_≤0−1= 0 =L(H, A)

(since every f ∈ L(H, A) satisfies f |_H_≤0−1= 0 (because H≤0−1 =H≤−1 =

13We denote thisk-coalgebra byH rather than by the (more appropriate) letterC because this is how it is often called in this context in standard literature.

14Recall that (H≤`)_`≥0 denotes the filtration of the filteredk-coalgebra H (according to Conven- tion 1.14).

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0)) and L¹(H, A) =

f ∈ L(H, A) | f |_H_≤1−1= 0 =

f ∈ L(H, A) | f |_{λ·1_H_|λ∈k}= 0

since H≤1−1 =H≤0 =k·1_H (by Remark 2.11, applied to C =H) and thus H≤1−1 =k·1_H ={λ·1_H |λ∈k}

=







f ∈ L(H, A) | f(λ·1H)

| {z }

=λf(1H)

= 0 for everyλ ∈k







=







f ∈ L(H, A) | λf(1_H) = 0 for every λ ∈k

| {z }

this is equivalent tof(1H)=0







={f ∈ L(H, A) | f(1_H) = 0}=g(H, A).

(c)Let us denote byG(H, A) the subset {f ∈ L(H, A) | f(1_H) = 1_A}of L(H, A). (Here, 1_A denotes the unity ofA.)

Definition 3.2. (a) In Definition 3.1 (a), we have defined g(H, A) when H is a connected filteredk-coalgebra andAis a k-algebra. In the same way (that is, by the formula

g(H, A) = {f ∈ L(H, A) | f(1_H) = 0}

), we can defineg(H, A) whenHis a unital coalgebra andAis anyk-vector space. In particular, g(H, A) is thus defined when H is a k-bialgebra and A is any k-vector space (because according to Definition 2.6, when H is a k-bialgebra,H canonically becomes a unital coalgebra).

(b)In Definition 3.1 (c), we have definedG(H, A) when H is a connected filteredk-coalgebra and A is a k-algebra. In the same way (that is, by the formula

G(H, A) ={f ∈ L(H, A) | f(1_H) = 1_A}

), we can define the set G(H, A) when H is a unital coalgebra and A is a k-algebra. Moreover, in the same way (i.e., by the same formula), we can define the setG(H, A) whenH is a unital coalgebra and A is a unital coalgebra (because the notation 1A makes sense whenever A is a unital coalgebra).

Remark 3.3. Let k be a field. Let A be a k-algebra. Let H be a unital coalgebra. Recall (from Definition 1.12) thateH,A denotes the mapηA◦εH : H →A. Recall the sets g(H, A) and G(H, A) defined in Definition 3.2.

(a) We have e_H,A ∈G(H, A).

(b) We haveG(H, A) =e_H,A+g(H, A).

Proof of Remark 3.3. We know that (H,1_H) is a unital coalgebra. In other words, His

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We have e_H,A =η_A◦ε_H. Thus,

e_H,A(1_H) = (η_A◦ε_H) (1_H) =η_A



ε_H(1_H)

| {z }

=1



=η_A(1)

= 1·1_A (by the definition of the map η_A)

= 1A. Recall that

g(H, A) = {f ∈ L(H, A) | f(1_H) = 0} (2) and

G(H, A) ={f ∈ L(H, A) | f(1_H) = 1_A}. (3) (a) Now, e_H,A is an element of L(H, A) satisfying e_H,A(1_H) = 1_A. In other words, e_H,A is an f ∈ L(H, A) satisfying f(1_H) = 1_A. In other words, e_H,A ∈ {f ∈ L(H, A) | f(1H) = 1A}. In view of (3), this rewrites as eH,A ∈ G(H, A). This proves Remark 3.3 (a).

(b) Let g ∈ G(H, A). Thus, g ∈ G(H, A) = {f ∈ L(H, A) | f(1_H) = 1_A}.

In other words, g is an element f ∈ L(H, A) satisfying f(1H) = 1A. In other words, g is an element of L(H, A) and satisfies g(1_H) = 1_A. Now, (g−e_H,A) (1_H) = g(1_H)

| {z }

=1_A

−e_H,A(1_H)

| {z }

=1A

= 1_A−1_A= 0. Thus, g−e_H,Ais an element ofL(H, A) and satisfies (g−eH,A) (1H) = 0. In other words,g−eH,A is anf ∈ L(H, A) satisfying f(1H) = 0.

In other words, g−e_H,A ∈ {f ∈ L(H, A) | f(1_H) = 0}. In view of (2), this rewrites as g−e_H,A∈g(H, A). Thus, g ∈e_H,A+g(H, A).

Now, forget that we fixedg. We thus have shown thatg ∈eH,A+g(H, A) for each g ∈G(H, A). In other words,

G(H, A)⊆e_H,A+g(H, A). (4)

On the other hand, let h ∈ e_H,A +g(H, A). Thus, h ∈ L(H, A) and h−e_H,A ∈ g(H, A). We have h − e_H,A ∈ g(H, A) = {f ∈ L(H, A) | f(1_H) = 0}. In other words, h−e_H,A is an element f of L(H, A) satisfying f(1_H) = 0. In other words, h − e_H,A is an element of L(H, A) and satisfies (h−e_H,A) (1_H) = 0. Comparing (h−e_H,A) (1_H) = 0 with (h−e_H,A) (1_H) = h(1_H)− e_H,A(1_H)

| {z }

=1A

= h(1_H)− 1_A, we obtainh(1_H)−1_A= 0. In other words, h(1_H) = 1_A. Now,his an element of L(H, A) satisfying h(1_H) = 1_A. In other words, h is an f ∈ L(H, A) satisfying f(1_H) = 1_A. In other words, h ∈ {f ∈ L(H, A) | f(1_H) = 1_A}. In view of (3), this rewrites as h∈G(H, A).

Now, forget that we fixed h. We thus have shown that h ∈ G(H, A) for each h∈e_H,A+g(H, A). In other words, e_H,A+g(H, A)⊆G(H, A). Combining this with (4), we obtain G(H, A) = e_H,A+g(H, A). This proves Remark 3.3 (b).

Remark 3.4. If we replace the word “k-algebra” by “unital coalgebra” in Remark 3.3, then Remark 3.3 still holds. In fact, the same proof given above still applies in this situation.

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Remark 3.5. Let k be a field, let A be a k-algebra, and let H be a connected filteredk-coalgebra. Everyi∈N, n∈Nand f ∈g(H, A) such that i > n satisfy f^∗i(H≤n) = 0.

Proof of Remark 3.5. We prove Remark 3.5 by induction over i:

Induction base: Fori= 0, Remark 3.5 is vacuously true (becausei > ncannot hold (since i= 0 andn ∈N)).

Induction step: Let j ∈ N be arbitrary. Assume that Remark 3.5 holds for i = j.

Now, we must prove that Remark 3.5 also holds for i=j+ 1.

Let n∈N and f ∈g(H, A) be such that j+ 1 > n. Then, ∆_H(H≤n)⊆

n

P

u=0

H≤u⊗ H≤n−u (since H is a filtered coalgebra) and

f^∗(j+1)

| {z }

=f^∗j∗f

(H≤n)

= f^∗j∗f

(H≤n) = µ_A◦ f^∗j ⊗f

◦∆_H (H≤n)

=µ_A







f^∗j ⊗f







∆_H(H≤n)

| {z }

⊆

n

P

u=0

H≤u⊗H≤n−u













⊆µ_A f^∗j ⊗f

n

X

u=0

H≤u⊗H≤n−u

!!

=

n

X

u=0

µ_A







f^∗j ⊗f

(H≤u⊗H≤n−u)

| {z }

⊆f^∗j(^H^≤u)^⊗f(^H^≤n−u)







⊆

n

X

u=0

µ_A f^∗j(H≤u)⊗f(H≤n−u)

=

n−1

X

u=0

µA f^∗j(H≤u)⊗f(H≤n−u)

+µA f^∗j(H≤n)⊗f(H≤n−n)

. (5)

Now, everyu∈ {0,1, . . . , n−1}satisfiesj > u(sincej+ 1 > n= (n−1)

| {z }

≥u

+1 ≥u+ 1).

Thus, for every u∈ {0,1, . . . , n−1}, we can apply Remark 3.5 toj and u instead ofi and n (since we assumed that Remark 3.5 holds fori =j), and obtain f^∗j(H_≤u) = 0.

Besides,f ∈g(H, A) = L¹(H, A) yieldsf |_H_≤1−1= 0 (by the definition ofL¹(H, A)), so that f(H≤1−1) = 0. Thus, f(H≤n−n) = f(H≤0) =f(H≤1−1) = 0. Now, (5) becomes

f^∗(j+1)(H≤n)⊆

n−1

X

u=0

µ_A



f^∗j(H≤u)

| {z }

=0

⊗f(H≤n−u)



+µ_A



f^∗j(H≤n)⊗f(H≤n−n)

| {z }

=0





=

n−1

Xµ_A(0⊗f(H_≤n−u))+µ_A f^∗j(H_≤n)⊗0

= 0 + 0 = 0.

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In other words, f^∗(j+1)(H≤n) = 0. We have thus proven that Remark 3.5 also holds for i=j + 1. This completes the induction step.

We thus have completed the induction proof of Remark 3.5.

Definition 3.6. Let k be a field of characteristic 0, let A be a k-algebra, and let H be a connected filtered k-coalgebra. For every f ∈ g(H, A), let us define a map e^∗f :H →A by the formula

e^∗f(x) =X

i≥0

f^∗i(x)

i! for every x∈H

!

. (6)

This map e^∗f is well-defined, because for every x ∈ H the infinite sum P

i≥0

f^∗i(x)

i! converges with respect to the discrete topology¹⁵. Besides, e^∗f is a k-linear map¹⁶, so that e^∗f ∈ L(H, A). More precisely, e^∗f ∈G(H, A).

17

Remark. Theein the notatione^∗f has nothing to do with theein the notatione_H,A. The ein the notatione^∗f is a pure symbol (in particular, e^∗f is not an “f-th power” of any e (whatever thise would be) with respect to convolution) which has been chosen to suggest similarity with the exponential function known from analysis; despite this

15Proof. Let x ∈ H. Then, there exists some n ∈ N such that x ∈ H_≤n (since H is filtered).

Consider such ann. Then, every integeri > n satisfiesf^∗i(x)∈f^∗i(H_≤n) = 0 (by Remark 3.5) and thusf^∗i(x) = 0. Hence, for every integeri > n, thei-th addend of the infinite sum P

i≥0

f^∗i(x)

i! is zero.

Hence, this infinite sum P

i≥0

f^∗i(x)

i! has only finitely many nonzero addends. Thus, this sum converges with respect to the discrete topology.

16Proof. Letα∈k,β ∈k, x∈H andy ∈ H be arbitrary. Then, (6) (applied toy instead of x) yieldse^∗f(y) = P

i≥0

f^∗i(y)

i! . But (6) (applied toαx+βyinstead ofx) yields e^∗f(αx+βy) =X

i≥0

f^∗i(αx+βy)

i! =X

i≥0

αf^∗i(x) +βf^∗i(y) i!

| {z }

=αf^∗i(x) i! ^+β

f^∗i(y) i!

since for everyi∈N, we havef^∗i(αx+βy) =αf^∗i(x) +βf^∗i(y) (becausef^∗i is ak-linear map)

=αX

i≥0

f^∗i(x) i!

| {z }

=e^∗f(x)

+βX

i≥0

f^∗i(y) i!

| {z }

=e^∗f(y)

=αe^∗f(x) +βe^∗f(y).

Since this holds for allα∈k,β ∈k,x∈H and y∈H, we thus see thate^∗f isk-linear, qed.

17Proof. Every integer i > 0 satisfies f^∗i(H_≤0) = 0 (by Remark 3.5, applied to n= 0) and thus f^∗i





 1H

|{z}

∈H≤0





∈f^∗i(H_≤0) = 0, so that f^∗i(1H) = 0. Hence, every integeri >0 satisfies f^∗i(1H) 0

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similarity, there is (in general) no “Euler number” e≈ 2.718. . .in L(H, A). The e in the notation eH,A simply stands for “neutral element” (just as the neutral element of a group is often denoted by e).

Definition 3.7. Let k be a field of characteristic 0, let A be a k-algebra, and let H be a connected filtered k-coalgebra. For every f ∈ g(H, A), let us define a map Log₁f :H →A by the formula

(Log₁f) (x) =X

i≥1

(−1)ⁱ⁻¹

i f^∗i(x) for every x∈H

!

. (8) This map Log₁f is well-defined, because for every x ∈H the infinite sum P

i≥1

(−1)ⁱ⁻¹

i f^∗i(x) converges with respect to the discrete topology¹⁸. Be- sides, Log₁f is a k-linear map¹⁹, so that Log₁f ∈ L(H, A). More precisely,

But applying (6) tox= 1_H, we get e^∗f(1H) =X

i≥0

f^∗i(1H)

i! = f^∗0(1H) 0!

| {z }

=f^∗0(1H)

1 ^=f

∗0(1_H)

+X

i>0

f^∗i(1H) i!

| {z }

=0 (by (7))

(here, we have split off the addend fori= 0 from the sum)

=f^∗0(1_H) +X

i>0

0

| {z }

=0

= f^∗0

|{z}

=e_H,A

(1_H) =e_H,A(1_H) = 1_A.

Thus,e^∗f ∈G(H, A) (by the definition ofG(H, A)).

18Proof. Let x ∈ H. Then, there exists some n ∈ N such that x ∈ H≤n (since H is filtered).

Consider thisn. Then, every integeri > nsatisfiesf^∗i(x) = 0 (this is proven just as in Definition 3.6).

Therefore, every integeri > nsatisfies (−1)ⁱ⁻¹ i f^∗i(x)

| {z }

=0

= 0. In other words, for every integeri > n, the

i-th addend of the infinite sum P

i≥1

(−1)ⁱ⁻¹

i f^∗i(x) is zero. Hence, this infinite sum P

i≥1

(−1)ⁱ⁻¹ i f^∗i(x) has only finitely many nonzero addends. Thus, this sum converges with respect to the discrete topology, qed.

19Proof. Letα∈k,β ∈k, x∈H andy ∈ H be arbitrary. Then, (8) (applied toy instead of x) yields (Log₁f) (y) = P

i≥1

(−1)ⁱ⁻¹

i f^∗i(y). But (8) (applied toαx+βyinstead ofx) yields (Log₁f) (αx+βy) =X

i≥1

(−1)ⁱ⁻¹

i f^∗i(αx+βy)

| {z }

=αf^∗i(x)+βf^∗i(y) (sincef^∗iis ak-linear map)

=X

i≥1

(−1)ⁱ⁻¹

i αf^∗i(x) +βf^∗i(y)

| {z }

=α(−1)ⁱ⁻¹

i ^f

∗i(x)+β(−1)ⁱ⁻¹

i ^f

∗i(y)

=αX

i≥1

(−1)ⁱ⁻¹ i f^∗i(x)

| {z }

=(Log₁f)(x)

+βX

i≥1

(−1)ⁱ⁻¹ i f^∗i(y)

| {z }

=(Log₁f)(y)

=α(Log₁f) (x) +β(Log₁f) (y).

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Log₁f ∈g(H, A). ²⁰

Definition 3.8. Let k be a field of characteristic 0, let A be a k-algebra, and letH be a connected filtered k-coalgebra. For every F ∈G(H, A), let us define an element LogF ∈g(H, A) by LogF = Log₁(F −e_H,A). ²¹

§ 4. Log id in a connected filtered cocommutative bial- gebra

We are now ready to state a first interesting result:

Theorem 4.1. Let k be a field of characteristic 0, and let H be a connected filtered cocommutative bialgebra over k. Consider the convolution algebra L(H, H). The map Log id ∈ L(H, H) is a projection from H to the subspace PrimH of all primitive elements of H.

The map Log id ∈ L(H, H) defined in Theorem 4.1 is known as the Eulerian idempotent of H.

Theorem 4.1 is a classical fact about the Eulerian idempotent in a bialgebra. In particular, it has been used in [PatReu98] (more precisely, in the Example in §2 of [PatReu98]).²²

§ 5. Basic properties of Log and e

^∗

Before we start proving Theorem 4.1, we shall study the concepts of exponentiation and logarithm closer – first, as formal power series, but then as operators on the space L(H, A) of linear maps from a coalgebra H to an algebra A. (To be precise, they do not act on the full space L(H, A); but we will make everything precise when we state the results.)

20Proof. Every integer i > 0 satisfies f^∗i(H≤0) = 0 (by Remark 3.5, applied to n= 0) and thus f^∗i





 1_H

|{z}

∈H≤0





∈f^∗i(H_≤0) = 0, so that f^∗i(1_H) = 0. But applying (8) tox= 1_H, we get

(Log₁f) (1H) =X

i≥1

(−1)ⁱ⁻¹

i f^∗i(1H)

| {z }

=0

=X

i≥1

(−1)ⁱ⁻¹ i 0 = 0.

Thus, Log₁f ∈g(H, A) (by the definition ofg(H, A)).

21This is well-defined since F

|{z}

∈G(H,A)=eH,A+g(H,A)

−eH,A∈e_H,A+g(H, A)−e_H,A=g(H, A).

22Actually, our Theorem 4.1 is stronger than the fact used in the Example in §2 of [PatReu98], because [PatReu98] considers only connected graded cocommutative bialgebras, whereas our Theo- rem 4.1 is stated (and proven) for any connectedfiltered cocommutative bialgebra.

Note that theL(H, H) in our Theorem 4.1 is not theL(H) of [PatReu98] in the case when H is graded. In fact, theL(H) of [PatReu98] contains only the gradedk-linear maps, while ourL(H, H)

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§ 5.1. log and exp as power series

Let us first study the logarithm and the exponentials as they act on formal power series:

Definition 5.1. Letk be a field of characteristic 0. For every power series P ∈ k[[X]] whose coefficient before X⁰ is 0, let expP denote the power series in k[[X]] defined by expP = P

i≥0

Pⁱ

i!. For every power series Q ∈ k[[X]] whose coefficient before X⁰ is 1, let logQdenote the power series in k[[X]] defined by logQ= P

i≥1

(−1)ⁱ⁻¹

i (Q−1)ⁱ. Theorem 5.2. Let k be a field of characteristic 0.

(a) Every power seriesP ∈k[[X]] whose coefficient beforeX⁰ is 0 satisfies log (expP) =P.

(b)Every power seriesQ∈k[[X]] whose coefficient before X⁰ is 1 satisfies exp (logQ) = Q.

Theorem 5.2 is an important result in mathematics; in particular, it is frequently applied in combinatorics (e.g., in dealing with generating functions) and in algebra.

Often it is derived from the analogous fact from complex analysis (where P and Q, rather than being formal power series, are required to be holomorphic functions). Let us instead give an elementary proof. The proof will be based on some basic properties of derivatives of power series:

Lemma 5.3. Let k be a field of characteristic 0. Let P ∈ k[[X]] be a power series such that d

dXP = 0. Assume furthermore that the coefficient of P before X⁰ is 0. Then, P = 0.

Proof of Lemma 5.3. Write the power series P ∈k[[X]] in the formP = P

n∈N

p_nXⁿ for some sequence (p₀, p₁, p₂, . . .)∈k^N of elements of k. Then,

d

dX P

|{z}

=P

n∈N

pnXⁿ

= d dX

X

n∈N

pnXⁿ

!

= X

n∈N; n≥1

np_nXⁿ⁻¹

by the definition of the operator d dX

=X

n∈N

(n+ 1)p_n+1 X^(n+1)−1

| {z }

=Xⁿ (since (n+1)−1=n)

(here, we have substituted n+ 1 for n in the sum)

=X

(n+ 1)pn+1Xⁿ.

On the logarithm of the identity on connected filtered bialgebras