A PSEUDO-ANALYZER APPROACH TO FORMAL GROUP LAWS NOT OF OPERAD TYPE

RALF HOLTKAMP

Abstract. Formal group schemes, associated to affine group schemes or Lie groups by completion, can be described by classical formal group laws. More generally, cogroup objects in categories of complete algebras (e.g. associative) are described by group laws for operads or analyzers. M. Lazard has introduced analyzers to study formal group laws and group law chunks (truncated formal power series). A main example of a type of generalized formal group laws not given by an operad or analyzer are group laws corresponding to noncommutative complete Hopf algebras. To cover this case and other types of group laws, pseudo-analyzers are introduced. We point out differences to the (quadratic) operad case, e.g. there is no classification of group laws by Koszul duality. On the other hand we show how pseudo-analyzer cohomology can be used to describe extension of group law chunks.

Monomials, polynomials and power series with variables from a (finite) set X = {x_{1}, ..., x_{m}} are defined after fixing a ”type” or operad, e.g. associative commutative (type ℭ or ℭom, the classical case), associative noncommutative (type 𝔈), or nonassociative noncommutative (type 𝔎). We will be mostly interested in these, but we will also mention other examples.

Formal group laws over these operads were introduced (as group laws in analyzers) by Lazard (cf. [La1], [La2]). They correspond to cogroup objects in the corresponding categories of complete algebras. Over a field K of caracteristic 0, they are classified by Lazard-Lie theory, which can be described in terms of Koszul duality for operads (cf. [GK], [Fr1]).

Group laws corresponding to Hopf algebra structures on the free associative power series algebra K⟨⟨X⟩⟩ ( ˆ⊗-cogroup structures), see [Ho1], [Ho2], are not group laws in an operad or analyzer. Hopf algebras are not algebras over an operad, they are given by PROPs (cf. [ML1], [FM]). Here types 𝔈 and ℭ are mixed in a certain way. The same is true for group laws corresponding to ˆ⊗-cogroup structures (and ˆ∗-cogroup structures) on free nonassociative power series algebras.

To cover these mixed types, we will give a modification of Lazard’s theory. We introduce pseudo-analyzers and prove that the mixed types indicated above provide examples of pseudo-analyzers that are not analayzers. A main point is that Lazard’s concept of describing extension of group law chunks by analyzer cohomology also applies to group laws over pseudo-analyzers. We show that the torsion theorem, the key to the classification of group laws over analyzers (in the rational case) is not true for pseudo-analyzers, and we describe some cohomology modules in the case corresponding to noncommutative complete Hopf algebras. We will then review the classification of group laws over analyzers in the rational case and show that structure constants providing formal group laws of two (operad) types, e.g. 𝔈 and ℭ, also provide a (non-trivial) formal group law of the corresponding mixed type.

I thank the DFG for financial support. Research at MSRI is supported in part by NSF grant DMS-9701755.

In section 1 we recall the definition of an operad and define monomials and power series. Besides the ”usual” power series, we will include ”mixed types” like elements of (K⟨⟨X⟩⟩)^{ˆ⊗p}, p ≥ 2.

The definitions of pseudo-analyzers and analyzers together with first examples of analyzers are given in section 2. Lazard’s cohomology modules are introduced for pseudo-analyzers.

In section 3 we give examples for pseudo-analyzers that are not analyzers.

Section 4 contains the definition of group laws in pseudo-analyzers and the generalization of Lazard’s criterion for extension of group law chunks. We also look at the type corresponding to complete (in general noncommutative) Hopf algebras.

Over a ground field of characteristic 0, we show that every finitely generated complete associative algebra with Hopf algebra structure is isomorphic to a free complete algebra K⟨⟨X⟩⟩ modulo an ideal I that is contained in the ideal generated by all commutators. For I = 0 such structures are nothing else but formal group laws in the corresponding pseudo-analyzer.

In section 5 we prove that Lazard’s torsion theorem does not hold for pseudoanalyzers. We describe cohomology modules in the analyzer and pseudo-analyzers cases.

The last section relates results from section 5 to the concept of Koszul duality for operads and shows how the classification of group laws in rational analyzers provides examples for formal group laws not of operad type.

1. Power Series of Different Types.

We fix m ∈ ℕ^{∗}. For all p ∈ ℕ, let X^{⊔p} be the set {x_{1}^{(1)}, ..., x_{m}^{(1)}, ... , x_{1}^{(p)}, ..., x_{m}^{(p)}} of variables, ordered by x_{1}^{(1)} < ... < x_{m}^{(1)} < ... < x_{1}^{(p)} < ... < x_{m}^{(p)}. X^{⊔1} is identified with X = {x_{1}, ..., x_{m}}. K will denote a field of coefficients. (Later we will allow K to be a unitary commutative associative ring).

Over X, we will form monomials of the following types. The first case (classical case), type ℭ, is the case of associative commutative variables. Type 𝔈 will denote the associative noncommutative case, type ℌ the nonassociative commutative case and 𝔎 the nonassociative noncommutative case.

Definition 1.1. For n ∈ ℕ^{∗} let S_{n} be the symmetric group and S-Vect_{K} be the following category: Objects 𝔙 are sequences (𝔙(n))_{n∈ℕ} of vector spaces 𝔙(n) with S_{n}-action. Morphisms are given by homomorphisms compatible with the S_{n}-action.

To every S-vector space 𝔙 there is associated an endofunctor on Vect_{K} given by

F_{𝔙}(V) := ∑_{n=0}^{∞}𝔙(n) ⊗_{S_{n}} V^{⊗n} (where S_{n} acts on V^{⊗n} by place permutation). S-Vect_{K}

can be identified with the full subcategory of End(Vect_{K}) consisting of functors of the form F_{𝔙}. We note that Id_{Vect_{K}} is given by the S-vector space ℑ with ℑ(1) = K, ℑ(n) = 0 for n ≠ 1.

Then a (K-linear) operad 𝔓 is a monoid (F_{𝔓}, μ : F_{𝔓} ∘ F_{𝔓} → F_{𝔓}, 1 : Id → F_{𝔓}) with respect to the composition of functors in this subcategory (i.e. a monad, cf. [ML2], chap. VI), cf. [Fr1]. Clearly the unit is given by a homomorphism K → 𝔓(1). If we don’t require operad-associativity (i.e. the associativity of μ : F_{𝔓} ∘ F_{𝔓} → F_{𝔓}), we get the notion of a pseudo-operad, cf. [HL].

For any operad 𝔓, F_{𝔓}(V) is called the free 𝔓-algebra generated by the space V.

Example 1.2. The functor F_{ℭom}(V) := ∑_{n=1}^{∞}(V^{⊗n})_{S_{n}} (non-unitary symmetric alge-

bra) defines the operad ℭom given by ℭom(n) = K · x^{(1)} · ... · x^{(n)} ≅ K (all n ≥ 1),

this is the classical case. F_{𝔄s}(V) := ∑_{n=1}^{∞}V^{⊗n} defines an operad where 𝔄s(n) is the n!-

dimensional K-space generated by the (type 𝔈-)monomials x^{(σ(1))} · ... · x^{(σ(n))}, σ ∈ S_{n}.

𝔏ie is an operad given by the (n−1)!-dimensional K-spaces generated by multilinear bracket monomials (with respect to e.g. the Jacobi identity), cf. [GK]. Types ℌ and 𝔎 correspond to free quadratic operads generated by one (resp. two) quadratic monomial(s) and the obvious S_{n}-actions, compare [GK].

Remark 1.3. We can regard free 𝔓-algebras with 1 generated by the vector space V, where V has X as basis. Clearly they are polynomial rings K[X], K⟨X⟩, if the operads are ℭom (type ℭ), 𝔄s (type 𝔈). A K-basis is given in each case by the monomials of the corresponding type (together with 1).

We remark, that K⟨X^{⊔p}⟩ together with the p injections i^{(n)} : K⟨X⟩ → K⟨X^{⊔p}⟩, x_{i} ↦ x_{i}^{(n)} is the coproduct (or copower, cf. [ML2] p.64) (K⟨X⟩)^{∗p} := K⟨X⟩ ⊔𝔈 ... ⊔𝔈 K⟨X⟩ in the category of unitary associative K-algebras. The coproduct ⊔ℭ of commutative unitary K-algebras is the tensor product ⊗ = ⊗_{K}. The commutative associative polynomial algebra K[X^{⊔p}] corresponds to (K[X])^{⊗p} := K[X] ⊔ℭ ... ⊔ℭ K[X] via x_{i}^{(n)} ↦ 1 ⊗ ... ⊗ 1 ⊗ x_{i} ⊗ 1 ⊗ ... ⊗ 1 (x_{i} in the n-th place).

Similar assertions hold for types 𝔎 and ℌ. We denote the associated coproducts by ⊔𝔎 and ⊔ℌ. For all these copowers we can use monomials with variables from X^{⊔p}, p ≥ 2, as a K-basis.

Now the coproduct-functors may have extensions to bigger categories. For example one can consider the (non-free) associative algebra K⟨X⟩ ⊗_{K} ... ⊗_{K} K⟨X⟩ or form K{X} ∗_{K} K{X} for the free nonassociative K-algebra K{X}.

This is the reason why we will make further distinctions when we speak of monomials with variables from X^{⊔p}, p ≥ 2. We may allow relations (of commutativity or associativity) between variables x_{i}^{(n)}, x_{j}^{(n′)}, n ≠ n′, i, j arbitrary, by specifying the type of (categorical) coproduct, that induces the type of monomials over X^{⊔p}.

Definition 1.4. Let 𝔄 be one of the types 𝔎, ℌ, 𝔈, ℭ defined above and let either 𝔅 ∈ {𝔈, ℭ} in case 𝔄 = 𝔈, or 𝔅 ∈ {𝔎, ℌ, 𝔈, ℭ} in case 𝔄 = 𝔎, or 𝔅 ∈ {ℌ, ℭ} in case 𝔄 = ℌ. Let A be the free algebra of type 𝔄 with variables from X. There is a (canonical) K-module basis consisting of homogeneous elements of \underbrace{A ⊔𝔅 ... ⊔𝔅 A}_{p}.

We call the basis elements (≠ 1) ⊔𝔅-^{m}𝔄^{(p)}-monomials.

We will sometimes abbreviate ⊔𝔅-^{m}𝔅 by ^{m}𝔅.

We count for every n = 1, ..., p, how many variables x_{i}^{(n)} occur in the monomial. This gives us a degree-tuple α = (α_{1}, ..., α_{p}) ∈ ℕ^{p}. Then |α| := ∑_{n=1}^{p} α_{n} is the total degree of the given monomial, while α_{n} is the degree with respect to the variables x_{1}^{(n)}, ..., x_{m}^{(n)}. By ⊔𝔅-^{m}𝔄_{α}^{(p)}-monomials we mean ⊔𝔅-^{m}𝔄^{(p)}-monomials with multidegree α = (α_{1}, ..., α_{p}).

Let |𝔄|_{α}^{p} be the K-module freely generated by all 𝔄_{α}^{(p)}-monomials.

Let |𝔄|_{r}^{p} = ⨁_{{α∈ℕ^{p}: |α|=r}} |𝔄|_{α}^{p}, |𝔄|^{p} := ⨁_{r∈ℕ^{∗}} |𝔄|_{r}^{p} and ˆ|𝔄|^{p} := ∏_{r∈ℕ^{∗}} |𝔄|_{r}^{p}. Here

ˆ|𝔄|^{p} is considered as the completion of |𝔄|^{p} with respect to the topology given by the basis ( ⨁_{r≥R} |𝔄|_{r}^{p})_{R∈ℕ^{∗}} of 0-neighborhoods.

If α < β reverse lexicographically, i.e. if (α_{p}, ..., α_{1}) is lexicographically smaller than (β_{p}, ..., β_{1}), all monomials with multidegree (α_{1}, ..., α_{p}) are smaller than any monomial of multidegree (β_{1}, ..., β_{p}). Representatives of monomials of the same multidegree are compared reverse lexikographically (starting on the rigth) with respect to submonomials, where especially w_{1} < w_{2} if deg(w_{1}) < deg(w_{2}). Monomials will be written as concatenation of increasing submonomials (recursively).

Example 1.5. In the classical case (commutative associative, ⊗ as coproduct), we write the monomials as concatenation of increasing variables from X^{⊔p}.

Each ℭ_{α}^{(p)}-monomial is of the form x_{i_{1}}^{(1)}... x_{i_{α_{1}}}^{(1)} ... x_{i_{r−αp+1}}^{(p)}... x_{i_{r}}^{(p)}, where r = |α|, with 1 ≤ i_{1} ≤ ... ≤ i_{α_{1}} ≤ m, ... , 1 ≤ i_{r−α_{p}+1} ≤ ... ≤ i_{r} ≤ m.

Any ⊗_{K}-^{m}𝔄^{(p)}-monomial can be viewed as a product w^{(1)} · ... · w^{(p)}, of p separated (possibly empty) monomials w^{(n)}, each in m variables (for n = 1, ..., p) x_{1}^{(n)}, ..., x_{m}^{(n)}. In the ⊗_{K}-^{m}𝔈^{(p)}-case, every monomial (with given degree-tuple α) is of the form x_{i_{1}}^{(n_{1})}...x_{i_{α_{1}}}^{(n_{1})} · ... · x_{i_{r−αp+1}}^{(n_{r})}...x_{i_{r}}^{(n_{r})}, 1 ≤ n_{1} ≤ ... ≤ n_{r} ≤ p.

To combine 𝔎 with ⊔𝔈 we proceed similarly and get expressions w_{1}^{(n_{1})} · ... · w_{r}^{(n_{r})}, n_{i} ≠ n_{i+1}. The representatives for the other types are chosen similarly.

Remark 1.6. In general the semigroup structure of the set of 𝔄^{(p)}-monomials induces the structure of a K-algebra with 1 on K ⊕ |𝔄|^{p}, without 1 on |𝔄|^{p}.

(K⟨X⟩)^{∗p} is identified with K⊕|∗-^{m}𝔈|^{p}, (K⟨X⟩)^{⊗p} with K⊕|⊗-^{m}𝔈|^{p} and (K[X])^{⊗p} with K ⊕ |^{m}ℭ|^{p}.

If ˆ∗ and ˆ⊗ denote the appropriate completions, we can identify K ⊕ { \atop | ∗ -^{m}𝔈|}^{p}, K ⊕ { \atop | ⊗ -^{m}𝔈|}^{p} and K ⊕ ˆ|^{m}ℭ|^{p} with the (complete) power series algebras K⟨⟨X^{⊔p}⟩⟩ = (K⟨⟨X⟩⟩)^{ˆ∗p}, (K⟨⟨X⟩⟩)^{ˆ⊗p} and K[[X^{⊔p}]] = (K[[X]])^{ˆ⊗p}, respectively. More generally, we have a structure of a topological K-algebra with 1 on K ⊕ |𝔄̂|^{p}. There exist surjective continuous K-linear maps { \atop | ∗ -^{m}𝔈|}^{p} → { \atop | ⊗ -^{m}𝔈|}^{p} and { \atop | ∗ -^{m}𝔈|}^{p} → ˆ|^{m}ℭ|^{p} induced by the projections of the corresponding semigroups. In the nonassociative cases one has to regard magmas instead of semigroups.

For ℭ, 𝔈, ℌ, 𝔎, we recover the usual notion of power series over the given operads, defined as follows.

Definition 1.7. If 𝔓(0) = 0, 𝔓(1) = K, the free complete 𝔓-algebra (without 1)

generated by V is defined by F̂_{𝔓}(V) := ∏_{n=0}^{∞}𝔓(n) ⊗_{S_{n}} V^{⊗n}. If V = ⟨x^{(1)}, ..., x^{(n)}⟩ is the

vector space with basis x^{(1)}, ..., x^{(n)}, then the elements of F̂_{𝔓}(V) are called 𝔓-power series, cf. [Fr1] or [GK].

For A an algebra (of any type) with an augmentation ε : A → K, Ā its augmentation ideal, we say that A is complete iff Ā = \underleftarrow{lim} Ā/J_{n}, (J_{n})_{n∈ℕ} a sequence of ideals s.t. all Ā/J_{n} are nilpotent.

2. Analyzers and Pseudo-Analyzers

The modern notion of an operad has Lazard’s notion of an analyzer, see [La2], as a precursor. In our setting, this concept has some advantages. We want to modify the definition of (complete or incomplete) analyzers over K replacing two axioms by weaker axioms.

From now on, K denotes a commutative associative ring with 1.

Let (𝔄^{n})_{n∈ℕ} be a sequence of K-modules, where in each 𝔄^{n} there are n distinct elements e_{n,1}, ..., e_{n,n} selected. For every p, q ∈ ℕ and all g_{1}, ..., g_{p} ∈ 𝔄^{q}, let a K-linear map η_{g_{1},...,g_{p}}^{p,q} : 𝔄^{p} → 𝔄^{q}, called composition map (or insertion map), be given.

If f ∈ 𝔄^{p}, we say that f has p arguments. For 1 ≤ i ≤ p, we say f is neutral with respect to the i-th argument, iff η_{g_{1},...,g_{i},...,g_{p}}^{p,q}(f) = η_{g_{1},...,g′_{i},...,g_{p}}^{p,q}(f) for all g_{1}, ..., g_{p}, g′_{i}. We denote by supp_{{1,...,p}}(f) the set {1, ..., p}−{i : f is neutral w.r.t. i-th argument}. Two elements f and g of 𝔄^{p} are said to be pairwise compatible if supp_{{1,...,p}}(f) ∩ supp_{{1,...,p}}(g) = ∅.

Definition 2.1. Let (𝔄^{n})_{n∈ℕ}, e_{n,i}, η_{g_{1},...,g_{p}}^{p,q} be given as above and assume C1 For all p ∈ ℕ, f ∈ 𝔄^{p} : η_{e_{p,1},...,e_{p,p}}^{p,p}(f) = f.

For all p, q, i ∈ ℕ with 1 ≤ i ≤ p and all f_{1}, ..., f_{p} ∈ 𝔄^{q} : η_{f_{1},...,f_{p}}^{p,q}(e_{p,i}) = f_{i}. C2a For all n, p, q ∈ ℕ, all f ∈ 𝔄^{n}, all pairwise compatible g_{1}, ..., g_{n} ∈ 𝔄^{p} and all

pairwise compatible h_{1}, ..., h_{p} ∈ 𝔄^{q} it holds: η_{h_{1},...,h_{p}}^{p,q}(η_{g_{1},...,g_{n}}^{n,p}(f)) = η_{l_{1},...,l_{n}}^{n,q}(f),

where l_{i} := η_{h_{1},...,h_{p}}^{p,q}(g_{i}) ∈ 𝔄^{q}(all i).

D1 𝔄^{n} is a multigraded K-module for every n:

𝔄^{n} = ⨁_{r∈ℕ} 𝔄_{r}^{n} with 𝔄_{r}^{n} = ⨁_{{α∈ℕ^{n}: |α|=r}} 𝔄_{α}^{n}, where |α| := α_{1} + ... + α_{n}. As in [La1] we furthermore assume 𝔄_{0}^{n} = 0 (all n).

D2 The e_{n,i} (all n, i) are multihomogeneous of multidegree (0, ..., 0, 1, 0, ..., 0) (1 at

the i-th position). D3 For all f ∈ 𝔄_{α}^{p}, α ∈ ℕ^{p}, g_{i} ∈ 𝔄^{q}, λ_{i} ∈ K:

η_{λ_{1}g_{1},...,λ_{n}g_{n}}^{p,q}(f) = λ_{1}^{α_{1}} · ... · λ_{n}^{α_{n}} · η_{g_{1},...,g_{n}}^{n,p}(f), where 0^{0} := 1.

D4 If f ∈ 𝔄_{(α_{1},...,α_{n})}^{n} and (for all i = 1, ..., n) g_{i} ∈ 𝔄_{(β_{i,1},...,β_{i,p})}^{p} then

η_{g_{1},...,g_{n}}^{n,p}(f) ∈ 𝔄_{(γ_{1},...γ_{n})}^{p} with γ_{j} = ∑_{i=1}^{n} α_{i}β_{i,j} (all j = 1, ..., p). D5a If, for β ∈ ℕ^{n}, P_{β} denotes the projection 𝔄^{n} → 𝔄_{β}^{n} onto the homogeneous

component of multidegree β, we require for every f ∈ 𝔄_{α}^{p}:

P_{β}(η_{(e_{p+1,1}+e_{p+1,2}), e_{p+1,3}, ..., e_{p+1,p+1}}^{p,p+1}(f)) = 0, if β_{1} + β_{2} ≠ α_{1} or β_{i} ≠ α_{i−1} for any i = 3, ..., p + 1.

Then 𝔄 will be called incomplete pseudo-analyzer over K. The incomplete pseudo-

analyzer 𝔄 is called incomplete analyzer over K, if furthermore C2b and D5b hold: C2b Axiom C2 holds without the assumptions on compatibility. D5b In case β_{1} + β_{2} = α_{1} and β_{i} = α_{i−1} one has

η_{e_{p,1},e_{p,1},e_{p,2},...,e_{p,p}}^{p+1,p}(P_{β}(η_{(e_{p+1,1}+e_{p+1,2}), e_{p+1,3}, ..., e_{p+1,p+1}}^{p,p+1}(f))) = \frac{α_{1}!}{β_{1}!β_{2}!}f.

Let (𝔄̂^{n})_{n∈ℕ} be a sequence of (multi)graded K-modules 𝔄̂^{n} = ∏_{r∈ℕ^{∗}} 𝔄_{r}^{n} with 𝔄_{r}^{n} = ⨁_{{α∈ℕ^{n}:|α|=r}} 𝔄_{α}^{n}. Assume 𝔄 := ⋃_{n∈ℕ} 𝔄^{n} with 𝔄^{n} = ⨁_{r∈ℕ} 𝔄_{r}^{n} is an incomplete (pseudo-)analyzer and that the composition maps of 𝔄 are continuously extended to 𝔄̂ := ⋃_{n∈ℕ} 𝔄̂^{n} with respect to the topology given by the basis ( ⨁_{r≥R} 𝔄_{r}^{p})_{R∈ℕ^{∗}} of 0-neighborhoods.

Then 𝔄̂ (together with these maps) is called complete (pseudo-)analyzer.

A morphism 𝔄 → 𝔄′ of pseudo-analyzers is a sequence φ of K-linear maps φ_{n} : 𝔄^{n} → (𝔄′)^{n} that respects the structure: We require φ_{n}(e_{n,i}) = e′_{n,i}, φ_{n}(𝔄_{α}^{n}) ⊆ (𝔄′)_{α}^{n}

and η_{g′_{1},...,g′_{n}}^{𝔄′,n,p}(f′) := φ_{p}(η_{g_{1},...,g_{n}}^{𝔄,n,p}(f)) for every f, g_{i} with φ_{n}(f) = f′, φ_{p}(g_{i}) = g′_{i}.

Example 2.2. The sequence (^{1}ℭ^{n})_{n∈ℕ^{∗}} of K-modules (K[x^{(1)}, ..., x^{(n)}])_{n∈ℕ^{∗}}, i.e. the sequence (|ℭ|^{n})_{n∈ℕ^{∗}} for m = 1 of section 1, together with the elements e_{n,i} := x^{(i)}, defines an incomplete analyzer, where the composition maps are (canonically) defined by insertion of polynomials. This analyzer ^{1}ℭ := ⋃_{n∈ℕ} ^{1}ℭ^{n} is called the classical analyzer, see [La1], p.332(example c). ^{1}ℭ̂ given by the sequence (K[[x^{(1)}, ..., x^{(n)}]])_{n∈ℕ^{∗}} is the associated complete analyzer. The composition maps are given by insertion of power series without constant term.

Similarly defined are analyzers ^{1}𝔈 (occuring in [La1], p.332, as example b), ^{1}ℌ, ^{1}𝔎 ([La1], example a) and the corresponding complete analyzers. ^{1}𝔎 is called Kurosch’s analyzer, cf. also [La1], p.391. There is an inclusion morphism from the Lie-analyzer ([La1], p.333, example d) into the analyzer ^{1}𝔈.

We are going to look at pseudo-analyzers that are not analyzers in the next section.

Remark 2.3. For every operad 𝔓, the sequence (F̂_{𝔓}(⟨x^{(1)}, ..., x^{(n)}⟩))_{n∈ℕ} forms an analyzer over K, cf. [Fr1], and we can get the operad back as a subspace (generated in each component by the multilinear monomials, i.e. the monomials having degree 1 in each variable). The corresponding analyzer of ℭom-power series is ^{1}ℭ̂, the analyzer of 𝔄s-power series is ^{1}𝔈̂, and so on.

There are analyzers that do not correspond to operads, cf. [La1], p.333.

Note that, while associativity of composition is not required for pseudo-operads, we do require a partial associativity for pseudo-analyzers (C2a).

We will finish this section with some results from [La1] which hold for pseudoanalyzers.

Definition 2.4. Let 𝔄 be a pseudo-analyzer.

For R ∈ ℕ^{∗} we will say that elements f, g from 𝔄^{p}, are equivalent up to degree R, written f ≡ g mod J^{R+1}, if ∑_{r=1}^{R} P_{r}f = ∑_{r=1}^{R} P_{r}g, where P_{r} denotes the projection onto the homogeneous component of total degree r.

We can identify the elements of 𝔄[R]^{p} := ⨁_{r=1}^{R} 𝔄_{r}^{p} with elements from 𝔄^{p} mod J^{R+1}.

Remark 2.5. It is easy to see, that 𝔄[R] = ⋃_{n∈ℕ} 𝔄[R]^{n} is also a pseudo-analyzer if 𝔄 is. 𝔄[R] is both incomplete and complete. There is a canonically defined projection morphism from 𝔄 onto 𝔄[R].

Lemma 2.6. (”Composition Lemma” of [La74])

Let 𝔄 be a pseudo-analyzer, n ∈ ℕ^{∗}, r ∈ ℕ and F ∈ 𝔄^{n} with F ≡ 0 mod J^{r+1} be given.

We regard, for G^{(1)}, ..., G^{(n)} ∈ 𝔄̂^{p} (p ∈ ℕ^{∗}), the element η_{G^{(1)},...,G^{(n)}}^{n,p}(F) ∈ 𝔄̂^{p}.

Then, given L^{(i)} ≡ G^{(i)} mod J^{s+1} (s ∈ ℕ), all i = 1, ..., n, we have:

η_{G^{(1)},...,G^{(n)}}^{n,p}(F) ≡ η_{L^{(1)},...,L^{(n)}}^{n,p}(F) mod J^{r+s+1}.

Proof. The proof is given in [La1] for analyzers. We note that we can replace in this proof the assumptions (C2) and (D5) by (C2a) and (D5a). Thus the same proof fits for pseudo-analyzers.

Definition 2.7. Let 𝔄 be a pseudo-analyzer, n, i ∈ ℕ^{∗}, i < n.

We define the map ∂_{i} : 𝔄^{n} → 𝔄^{n+1} by η_{e^{(1)},...,e^{(i−1)},(e^{(i)}+e^{(i+1)}),e^{(i+2)},...,e^{(n+1)}}^{𝔄,n,n+1}

−η_{e^{(1)},...,e^{(i)},e^{(i+2)},...,e^{(n+1)}}^{𝔄,n,n+1} − η_{e^{(1)},...,e^{(i−1)},e^{(i+1)},...,e^{(n+1)}}^{𝔄,n,n+1} where e_{n+1,i} is denoted by e^{(i)} for short.

A differential δ = δ_{n} : 𝔄^{n} → 𝔄^{n+1} is defined by

\begin{aligned}δ_{n} & = ∑_{i=1}^{n}(−1)^{i}∂_{i} \\ & = η_{e^{(2)},...,e^{(n+1)}}^{𝔄,n,n+1} + ∑_{i=1}^{n}(−1)^{i}η_{e^{(1)},...,(e^{(i)}+e^{(i+1)}),...,e^{(n+1)}}^{𝔄,n,n+1} + (−1)^{n+1}η_{e^{(1)},...,e^{(n)}}^{𝔄,n,n+1}. \end{aligned}

Remark 2.8. The same formula is given for analyzers in [La1], §8. One checks δ_{n+1} ∘ δ_{n} = 0 (all n) for pseudo-analyzers by the same direct computation as for analyzers, because all inserted elements are compatible.

Let us call f ∈ 𝔄^{n} pseudolinear with respect to the i-th argument, if ∂_{i}f = 0. f is called pseudolinear, if f is pseudolinear with respect to all arguments. f is called multilinear if f is homogeneous of degree (1, ..., 1), especially any such f is pseudolinear and no argument is neutral. Note that f ∈ 𝔄^{n} is neutral with respect to the i-th argument iff η_{e_{n,1},...,e_{n,i−1},0,e_{n,i+1},...,e_{n,n}}^{n,n}(f) = f.

f ∈ 𝔄^{n} is called symmetric if σf = f for all σ ∈ S_{n}, where σf := η_{e_{n,σ(1)},...,e_{n,σ(n)}}^{n,n}(f). f is called anti-symmetric if σf = (signσ)f, all σ.

Lemma 2.9. It holds δ_{n}(𝔄_{r}^{n}) ⊆ 𝔄_{r}^{n+1}. Thus 𝔄 = ⨁_{n∈ℕ^{∗}} 𝔄^{n} = ⨁_{n,r∈ℕ^{∗}} 𝔄_{r}^{n} together with δ is a (co)complex of K-modules and there are cohomology modules H^{n}(𝔄) = ⨁_{r∈ℕ^{∗}} H_{r}^{n}(𝔄), where H_{r}^{n}(𝔄) = ker(δ_{n}|𝔄_{r}^{n})/ im(δ_{n−1}|𝔄_{r}^{n−1}).

H is functorial on pseudo-analyzers.

Proof. To verify the last assertion, let φ be a morphism 𝔄 → 𝔄′. Then we have δ′_{n} ∘ φ_{n} = φ_{n+1} ∘ δ_{n}, because for all f ∈ 𝔄^{n} (with φ_{n}(e_{n,i}) = e′_{n,i} and denoting φ_{n}(f) by f′):

\begin{aligned}δ′_{n}(φ_{n}(f)) & = η_{e′_{n+1,2},...,e′_{n+1,n+1}}^{𝔄′,n,n+1}(f′) + ... + (−1)^{n+1}η_{e′_{n+1,1},...,e′_{n+1,n}}^{𝔄′,n,n+1}(f′) \\ & = φ_{n+1}(δ_{n}(f)). \end{aligned}

3. Examples of Pseudo-Analyzers

Let p, r ∈ ℕ^{∗} and α ∈ ℕ^{p}, let 𝔄^{(p)} be one of the types defined in (1.4), i.e. 𝔄 ∈ { ^{m}ℭ, ⊗-^{m}𝔈, ^{m}𝔈, ⊗-^{m}ℌ, ^{m}ℌ, ⊗-^{m}𝔎, ∗-^{m}𝔎, ⊔ℌ-^{m}𝔎, ^{m}𝔎}. For any K-module V, we denote by (V)^{m} the K-module of m-tuples with entries from M (often written as column vectors). If φ is a homomorphisms V → V, then (φ)^{m} denotes the induced homomorphism (φ, ..., φ) : (V)^{m} → (V)^{m}.

Let 𝔄_{α}^{p} = (|𝔄|_{α}^{p})^{m}, 𝔄_{r}^{p} = (|𝔄|_{r}^{p})^{m} and 𝔄^{p} := ⨁_{r∈ℕ^{∗}} 𝔄_{r}^{p}, 𝔄̂^{p} := ∏_{r∈ℕ^{∗}} 𝔄_{r}^{p}.

Let 𝔄 = ⋃_{n∈ℕ} 𝔄^{n}, where 𝔄̂^{0} = 𝔄^{0} = 0. By e_{p,i} = e𝔄_{p,i} (1 ≤ i ≤ p ) we denote the elements (x_{1}^{(i)}, ..., x_{m}^{(i)}) ∈ 𝔄_{(0,...,0,1,0,...,0)}^{p} (1 in the i-th place).

Forming direct products (see [La1], p.323) of the analyzers ^{1}ℭ̂, ^{1}𝔈̂, ^{1}ℌ̂ and ^{1}𝔎̂ given in (2.2) we can show, that the analyzers we get as m-th power ∏^{m} are given by ^{m}ℭ̂, ^{m}𝔈̂, ^{m}ℌ̂ and ^{m}𝔎̂. Such a construction does not work for the ”mixed” cases like ⊗-^{m}𝔈, as for example ⊗-^{1}𝔈 = ^{1}ℭ. Moreover, there is an algebra homomorphism (”multiplication”), e.g. μ^{2,1} : K⟨X⟩ ⊔𝔈 K⟨X⟩ → K⟨X⟩, defined by x_{i}^{(n)} ↦ x_{i}, while it is well-known that in the ”mixed cases” we get only K-module homomorphisms (not algebra homomorphisms) like the map K⟨X⟩ ⊔ℭ K⟨X⟩ → K⟨X⟩.

Definition 3.1. For 𝔄 ∈ { ^{m}ℭ, ^{m}𝔈, ^{m}ℌ, ^{m}𝔎}, we define a (unique, continuous)

K-linear map μ^{ˆ|𝔄|,p·s,s} : ˆ|𝔄|^{p·s} → ˆ|𝔄|^{s} by:

μ^{ˆ|𝔄|,p·s,s}(x_{i}^{(n)}) = x_{i}^{(\overline{n})}, where \overline{n} ∈ \{1, ..., s\} with n̄ mod s = n mod s
μ^{ˆ|𝔄|,p·s,s}(w) = μ^{ˆ|𝔄|,p·s,s}(v_{1}) · μ^{ˆ|𝔄|,p·s,s}(v_{2}) if w = v_{1} · v_{2}.

A PSEUDO-ANALYZER APPROACH TO FORMAL GROUP LAWS NOT OF OPERAD TYPE 9

Using the unique representatives as indicated in (1.5) and by inserting brackets as in v_{1}(v_{2}(v_{3}...)..) on the right (if necessary), we define μ^{ˆ|𝔅|,p·s,s} also if 𝔅 is ⊗-^{m}ℌ̂, ⊗-^{m}𝔎̂, ∗-^{m}𝔎̂, ⊔ℌ-^{m}𝔎̂, all m, or ⊗-^{m}𝔈̂, m ≥ 2.

Here a ⊗-^{m}𝔈^{(p·s)}-monomial w^{(1)}...w^{(p·s)} will be identified with the corresponding ∗-^{m}𝔈^{(p·s)}-monomial and we make a similar identification in the nonassociative cases before applying one of the μ^{ˆ|𝔄|,p·s,s} defined above. The last step is to re-order the letters to get the correct representative. Lemma 3.2. Let p, q ∈ ℕ^{∗} and 𝔄 be as in (1.4).

(i) For s, q ∈ ℕ, q ≥ p + s, there is an algebra homomorphism ζ_{s}^{ˆ|𝔄|,p,q} mapping x_{i}^{(k)} to x_{i}^{(k+s)} (all i, k).

(ii) Let s ∈ ℕ^{∗} and G^{(k)} = (G_{1}^{(k)}, ..., G_{m}^{(k)}) ∈ 𝔄̂^{s} for k = 1, ..., p be given. There exists a unique K-linear map Δ_{G^{(1)},...,G^{(p)}}^{𝔄̂,p,p·s} = (Δ_{G^{(1)},...,G^{(p)}}^{|𝔄̂|,p,p·s})^{m} : 𝔄̂^{p} → 𝔄̂^{p·s}, where Δ_{G^{(1)},...,G^{(p)}}^{|𝔄̂|,p,p·s} : |𝔄̂|^{p} → |𝔄̂|^{p·s} is the algebra homomorphism given by

x_{i}^{(k)} ↦ ζ_{(k−1)s}^{|𝔄̂|,s,p·s}(G_{i}^{(k)}) for i = 1, ..., m and k = 1, ..., p.

Especially, Δ_{e_{1,1},...,e_{1,1}}^{𝔄̂,p,p} = id𝔄̂^{p}, and

for 1 ≤ k ≤ p, e_{p,k} is mapped on (ζ_{(k−1)s}^{|𝔄̂|,s,p·s})^{m}(G^{(k)}).

Definition 3.3. For 𝔄, p, q as above and G^{(1)}, ..., G^{(p)} ∈ 𝔄̂^{q}, let η_{G^{(1)},...,G^{(p)}}^{|𝔄̂|,p,q} : |𝔄̂|^{p} → |𝔄̂|^{q} be the (continuous) K-linear map given by μ^{|𝔄̂|,p·q,q} ∘ Δ_{G^{(1)},...,G^{(p)}}^{|𝔄̂|,p,p·q}.

Let η_{G^{(1)},...,G^{(p)}}^{𝔄̂,p,q} := (η_{G^{(1)},...,G^{(p)}}^{|𝔄̂|,p,q})^{m} : 𝔄̂^{p} → 𝔄̂^{q}. Similarly defined are maps η_{G^{(1)},...,G^{(p)}}^{𝔄,p,q} for the polynomial cases. Example 3.4. Let m ≥ 3, 𝔄 as above, and let L^{(1)} = e_{2,2}, L^{(2)} = L^{(3)} = e_{2,1} ∈ 𝔄^{2}. Then η_{L^{(1)},L^{(2)},L^{(3)}}^{|𝔄|,3,2}(x_{1}^{(1)}(x_{2}^{(2)}x_{3}^{(3)})) = μ^{|𝔄|,6,2}(x_{1}^{(2)}(x_{2}^{(3)}x_{3}^{(5)})) is given by x_{1}^{(2)}(x_{2}^{(1)}x_{3}^{(1)}) if 𝔄 ∈ {𝔈, 𝔎, ∗-𝔎}, and x_{2}^{(1)}x_{3}^{(1)}x_{1}^{(2)} if 𝔄 ∈ {ℭ, ℌ, ⊗-𝔈, ⊗-ℌ, ⊗-𝔎, }.

Theorem 3.5. Together with the selected elements e_{n,i} = (x_{1}^{(i)}, ..., x_{m}^{(i)}) and the composition maps from (3.3), ⊗-^{m}ℌ̂, ⊗-^{m}𝔎̂, ∗-^{m}𝔎̂, ⊔ℌ-^{m}𝔎̂, all m, and ⊗-^{m}𝔈̂ (m ≥ 2) are complete pseudo-analyzers over K that are not analyzers. Proof. Let 𝔄 be as above or, more generally, as in (1.4). We are going to show, that all 𝔄 are incomplete pseudo-analyzers. Then (using lemma (2.6)) it is clear that all 𝔄̂ are complete pseudo-analyzers.

1) From lemma (3.2) one gets (all 1 ≤ i ≤ p): η_{G^{(1)},...,G^{(p)}}^{𝔄,p,p}(e_{p,i}) = G^{(i)} and η_{e_{p,1},...,e_{p,p}}^{𝔄,p,p} = id𝔄^{p}, as Δ_{e_{p,1},...,e_{p,p}}^{𝔄̂,p,p^{2}}(e_{p,k}) = e_{p^{2},(k−1)p+k}.

For λ_{1}, ..., λ_{p} ∈ K and F ∈ 𝔄̂_{α}^{p}, α = (α_{1}, ..., α_{p}):

Δ_{λ_{1}e_{1,1},...,λ_{p}e_{1,1}}^{|𝔄̂|,p,p} = λ_{1}^{α_{1}} · ... · λ_{p}^{α_{p}}F, where 0^{0} = 1. Similarly,

η_{λ_{1}G^{(1)},...,λ_{p}G^{(p)}}^{𝔄,p,p}(F) = λ_{1}^{α_{1}} · ... · λ_{p}^{α_{p}}η_{G^{(1)},...,G^{(p)}}^{𝔄,p,p}(F) for all G^{(i)} ∈ 𝔄^{q}. For homogeneous G^{(k)}, k = 1, ..., p, G^{(k)} ∈ 𝔄_{(β_{k,1},...,β_{k,q})}^{q}, it holds

Δ_{G^{(1)},...,G^{(p)}}^{𝔄̂,p,p·q}(𝔄̂_{(α_{1},...,α_{p})}^{p}) ⊆ 𝔄̂_{(α_{1}β_{1,1},...,α_{1}β_{1,q}, ... ,α_{p}β_{p,1}...,α_{p}β_{p,q})}^{p·q}.

Hence η_{G^{(1)},...,G^{(p)}}^{𝔄,p,q}(𝔄_{(α_{1},...,α_{p})}^{p}) ⊆ 𝔄_{(γ_{1},...,γ_{q})}^{q}, where γ_{j} = ∑_{k=1}^{p} α_{k}β_{k,j}. We conclude that the conditions (C1), (D1)-(D4) are fulfilled.

2) Assume L^{(1)}, ..., L^{(n)} ∈ 𝔄^{p}, M^{(1)}, ..., M^{(p)} ∈ 𝔄^{q}, G^{(i)} = η_{M^{(1)},...,M^{(p)}}^{𝔄,p,q}(L^{(i)}). For 𝔄 ∉ { ^{m}ℭ, ^{m}𝔈, ^{m}ℌ, ^{m}𝔎} assume furthermore that L^{(1)}, ..., L^{(n)} are pairwise compatible and also that M^{(1)}, ..., M^{(p)} are pairwise compatible. Then: η_{G^{(1)},...,G^{(n)}}^{𝔄,n,q} = (η_{M^{(1)},...,M^{(p)}}^{𝔄,p,q}) ∘ (η_{L^{(1)},...,L^{(n)}}^{𝔄,n,p}). In fact, as all maps are algebra homomorphisms if the assumptions on compatibility are fulfilled, it suffices to apply both sides to the e_{n,i} and the equation is trivially fulfilled. Thus (C2a) holds for all 𝔄 and furthermore (C2b) holds for 𝔄 ∈ { ^{m}ℭ, ^{m}𝔈, ^{m}ℌ, ^{m}𝔎}. Let F ∈ 𝔄_{α}^{p} and let G = η_{L^{(1)},e_{p+1,3},...,e_{p+1,p+1}}^{p,p+1}(F), where L^{(1)} := e_{p+1,1} + e_{p+1,2} =

(x_{1}^{(1)} + x_{1}^{(2)}, ..., x_{m}^{(1)} + x_{m}^{(2)}). Then in fact G ∈ ⨁_{{β_{1},β_{2}∈ℕ: β_{1}+β_{2}=α_{1}}} 𝔄_{β_{1},β_{2},α_{2},...,α_{p}}^{p+1} (direct sum of α_{1} + 1 submodules), which shows (D5a) for all 𝔄.

3) For a counter example to (C2b) for 𝔄 = ⊗-𝔈, m ≥ 2, let p = 2 and L^{(1)} = e_{2,2}, L^{(2)} = e_{2,1} ∈ 𝔄^{2}, M^{(1)} = M^{(2)} = e_{1,1} ∈ 𝔄^{1}. Then η_{L^{(1)},L^{(2)}}^{|𝔄|,2,2}(x_{1}^{(1)}x_{2}^{(2)}) =

x_{2}^{(1)}x_{1}^{(2)}, which is mapped by η_{M^{(1)},M^{(2)}}^{|𝔄|,2,1} onto x_{2}^{(1)}x_{1}^{(1)}. Now G^{(i)} := η_{M^{(1)},M^{(2)}}^{𝔄,2,1}(L^{(i)})

= e_{1,1} for i = 1, 2, and η_{e_{1,1},e_{1,1}}^{|𝔄|,2,1}(x_{1}^{(1)}x_{2}^{(2)}) = x_{1}^{(1)}x_{2}^{(1)} ≠ x_{2}^{(1)}x_{1}^{(1)}. For 𝔄 ∈ {⊗-

mℌ, ⊗-^{m}𝔎, ∗-^{m}𝔎}, m ≥ 1, similarly η_{e_{1,1},e_{1,1}}^{|𝔄|,2,1}(η_{e_{2,1},e_{2,1},e_{2,2}}^{|𝔄|,3,2}(x_{1}^{(1)}x_{1}^{(2)}x_{1}^{(3)}x_{1}^{(3)})) =

(x_{1}^{(1)}x_{1}^{(1)})(x_{1}^{(1)}x_{1}^{(1)}) ≠ η_{e_{1,1},e_{1,1},e_{1,1}}^{|𝔄|,3,1}(x_{1}^{(1)}x_{1}^{(2)}x_{1}^{(3)}x_{1}^{(3)}), compare (3.1). If the representatives are chosen as indicated in (1.5) and v ≠ w are monomials of type 𝔎 over X, we use counter examples of the form η_{e_{1,1},e_{1,1}}^{|𝔄|,2,1}(η_{e_{2,2},e_{2,1}}^{|𝔄|,2,2}(v^{(1)}w^{(2)})) = wv

≠ η_{e_{1,1},e_{1,1}}^{|𝔄|,2,1}(v^{(1)}w^{(2)}) for ⊔ℌ-^{m}𝔎.

Corollary 3.6. For R ∈ ℕ^{∗}, the assertions of (3.5) remain true for the corresponding truncations 𝔄[R].

Remark 3.7. From the proof above we can get a direct proof that ^{m}ℭ̂, ^{m}𝔈̂, ^{m}ℌ̂, ^{m}𝔎̂ are analyzers. We still have to verify that (D5b) holds. The main reason for this is that for the (operad) types ℭ, 𝔈, ℌ, 𝔎 a simple support argument shows that we only have to check (D5b) in the case where f = F - see part 2) of the proof - is a single (normed) monomial. Then \binom{α_{1}}{β_{1}}F, the righthandside of (D5b) in (2.1), is obtained similarly to the fact, that P_{β_{1},α_{1}−β_{1}}((x_{j}^{(1)} + x_{j}^{(2)})^{α_{1}}) is given by

∑_{I={i_{1},...,i_{β_{1}}}⊂{1,2,...,α_{1}}} x_{j}^{(δ_{1})} · ... · x_{j}^{(δ_{α_{1}})}, δ_{s} replaced by 1 if s ∈ I and by 2 if s ∉ I.

Remark 3.8. The projection K⟨⟨X^{(n)}⟩⟩ → K[[S^{(n)}]] induces a surjective morphism of analyzers 𝔈̂ → ℭ̂. The projection (K⟨⟨X⟩⟩)^{⊗n} → K[[S^{(n)}]] induces a surjective morphism of pseudo-analyzers ⊗-𝔈̂ → ℭ̂. The projection π = (π^{(n)} : K⟨⟨X^{(n)}⟩⟩ → (K⟨⟨X⟩⟩)^{⊗n})_{n∈ℕ} does not induce a morphism of pseudo-analyzers: For g_{1} = g_{2} = e_{1,1}

and f = x_{2}^{(2)}x_{1}^{(1)} we have g′_{i} := π^{(1)}(g_{i}) = g_{i}, f′ := π^{(2)}(f) = x_{1}^{(1)}x_{2}^{(2)}, η_{g_{1},g_{2}}^{𝔈,2,1}(f) =

x_{2}^{(1)}x_{1}^{(1)}, and η_{g′_{1},g′_{2}}^{⊗-𝔈,2,1}(f′) = x_{1}^{(1)}x_{2}^{(1)} is not given by π^{(1)}(η_{g_{1},g_{2}}^{𝔈,2,1}(f)).

4. Group Laws

Cogroups in categories of complete algebras can be described by formal group laws (for analyzers or operads). If we replace in the definition of a cogroup object in a category of algebras (of a given operad type 𝔄), see [ML2] or [BH], the categorical coproduct ⊔𝔄 by some ⊔𝔅 as in (1.4), we get the notion of a ⊔𝔅-cogroup in the category of algebras of type 𝔄. This can be done similarly for complete augmented algebras, where we require the counit to be the augmentation map ε. Instead of elaborating the full definition, we define formal group laws in pseudo-analyzers and describe then how they are connected to ˆ⊔𝔅-cogroups.

Definition 4.1. Let 𝔄 be a complete pseudo-analyzer over the ring K and F ∈ 𝔄^{2}. We define F^{<} := η_{e_{3,1},e_{3,2}}^{𝔄,2,3}(F), F^{>} := η_{e_{3,2},e_{3,3}}^{𝔄,2,3}(F), and ΓF := η_{F^{<},e_{3,3}}^{𝔄,2,3}(F)−η_{e_{3,1},F^{>}}^{𝔄,2,3}(F)

= Δ_{F,e_{1,1}}^{𝔄,2,3}(F) − Δ_{e_{1,1},F}^{𝔄,2,3} (F) ∈ 𝔄^{3}. F is called group law in 𝔄, iff

(a) η_{e_{1,1},0}^{𝔄,2,1}(F) = e_{1,1} = η_{0,e_{1,1}}^{𝔄,2,1}(F)

(b) ΓF = 0. Group laws in 𝔄[R] are considered as R-chunks F (determined mod J^{R+1}) in 𝔄 by identifying an R-chunk with its canonical representative having zero (degree ≥ R+1)- components.

The group laws in ℭ̂ are also called classical formal group laws.

Homomorphisms of group laws (or chunks) are defined as for classical group laws, cf. [Ha], [La1]: φ ∈ 𝔄^{1} is a homomorphism F → G if Θφ := η_{F}^{𝔄,1,2}(φ)−η_{φ^{<},φ^{>}}^{𝔄,2,2}(G) = 0

in 𝔄^{2}, where φ^{<} := η_{e_{2,1}}^{𝔄,1,2}(φ), φ^{>} := η_{e_{2,2}}^{𝔄,1,2}(φ).

Such a homomorphism φ is called strict isomorphism, if φ ≡ e_{1,1} mod J^{2}.

Example 4.2. For every pseudo-analyzer, e_{2,1} + e_{2,2} is a group law and it is clearly the only possible 1-chunk.

The Campbell-Hausdorff series x^{(1)} + x^{(2)} + \frac{1}{2}[x^{(1)}, x^{(2)}] + ... is a group law in ^{1}𝔈̂. It is also a formal group law over 𝔏ie. For operads 𝔓, see (1.1), group laws over 𝔓 are defined as group laws in the associated analyzer, cf. [Fr1], [GK].

Example 4.3. Let m = n^{2} and write X = {x_{ij} : 1 ≤ i, j ≤ n}. Then the m-tuple (F_{ij}) with entries F_{ij} = x_{ij}^{(1)} + x_{ij}^{(2)} + ∑_{l=1}^{n} x_{il}^{(1)}x_{lj}^{(2)} defines a group law in the pseudoanalyzer ⊗-𝔈̂, cf. [Ho2]. The algebra homomorphism given by x_{ij} ↦ F_{ij} defines a ˆ⊗-cogroup structure (i.e. structure of a complete Hopf algebra) on K⟨⟨X⟩⟩. It is the completion of the well-known bialgebra-structure on the coordinate ring on n × n-matrices, or its Hopf envelope, which is also called the general linear quantum group.

We remark that the existence of the antipode is always given for complete bialgebras. It is known for complete 𝔓-algebras, 𝔓 an operad over a field K of characteristic 0, that they are isomorphic to a free complete 𝔓-algebra if they are endowed with a cogroup structure, see [Fr2].

The analogous result for ⊗-𝔈̂ is the following:

Theorem 4.4. Let A be a complete associative algebra, finitely generated, over a field K of characteristic 0. If A is a complete Hopf algebra, then A is as complete algebra isomorphic to the quotient of a free complete algebra K⟨⟨x_{1}, ..., x_{m}⟩⟩ modulo a closed ideal I that is contained in the ideal generated by all commutators [x_{i}, x_{j}].

Proof. Since the ideal [A, A] of all commutators in A is a Hopf ideal, A/[A, A] is a complete commutative Hopf algebra or cogroup object, which is isomorphic to some K[[X]] by the classical result for 𝔓 = ℭom. For A we can obtain the form K⟨⟨x_{1}, ..., x_{m}⟩⟩/I, I a closed ideal, for some m and we assume that m is minimal. I cannot contain elements of lower than quadratic order, because relations involving linear terms would allow the cancellation of a variable in contradiction to the minimality of m, see [GH] section 6. Thus A/[A, A] = K[[x_{1}, ..., x_{m}]] for the same m, and I must be contained in the ideal generated by the commutators.

Example 4.5. For every Hopf algebra, its completion with respect to the augmentation ideal ker ε is a complete Hopf algebra, see [Ho2]. In the case where I = 0, the Hopf algebra structures are exactly described by formal group laws of type ⊗-𝔈̂.

For the coordinate algebras of quantum groups defined by R-matrices, cf. [CP], like GL_{q} or O_{q} for 1 ≠ q ∈ K^{∗}, the theorem shows that their relations must induce commutator relations in the completion. The relations, written in variables x_{ij} ∈ ker ε, are for example q(1 + x_{11})x_{12} = x_{12}(1 + x_{11}). Since (q − 1)x_{12} occurs as linear term, x_{12} will be zero in the completion. Carrying this out one shows that the completions of all these Hopf algebras are commutative (see [Ho1]). This also holds for multi-parameter quantum deformations like the ones defined in [AST].

Remark 4.6. Let R ∈ ℕ^{∗}. Every F ∈ 𝔈[R]^{2} that fulfills condition (a) of Definition (4.1) is given by ∑_{r=1}^{R}(2^{r} − 2)m^{r+1} coefficients from K. Similarly, {F ∈ ⊗-𝔈[R]^{2} : F fulfills (a)} is a free K-module of rank ∑_{r=1}^{R}(r − 1)m^{r+1}. In the classical case this number is ∑_{r=1}^{R} m · (\binom{2m−1+r}{2m−1} − 2\binom{m−1+r}{m−1}) = m(\binom{2m+R}{2m} − 2\binom{m+R}{m} + 1). Let (c_{r})_{r} denote the recursively defined sequence with c_{1} = 1, c_{r} = ∑_{s=1}^{r−1} c_{s}c_{r−s} for r > 1 (Catalan numbers for binary trees). Then the corresponding number of coefficients in the noncommutative nonassociative case 𝔎[R]^{2} is ∑_{r=1}^{R}(2^{r} − 2)m^{r+1}c_{r}.

Condition (b) determines a system of equations for the coefficients, defining varieties (universal formal group law chunks) associated to the given type.

For a given F mod J^{r+s}, s ≥ 2, which is mod J^{r+1} an r-chunk, ΓF is of ord ≥ r +1 and we denote by Γ_{r+1}F the (degree r + 1)-component of ΓF. Similarly, for a given homomorphism of r-chunks that actually are s-chunks with s > r, we denote by Θ_{r+1}φ the (degree r + 1)-component of Θφ.

The following proposition, proven by Lazard for analyzers, also holds for pseudoanalyzers.

Proposition 4.7. Let F be a group law r-chunk in a pseudo-analyzer 𝔄.

(i) Γ_{r+1}F ∈ ker δ_{3}. Thus Γ_{r+1}F defines an element of H_{r+1}^{3}, called obstruction of F.

(ii) Let F be a group law r-chunk and F′ := F + L for a (homogeneous) L ∈ 𝔄_{r+1}^{2}. Then (Γ_{r+1}F − Γ_{r+1}F′) = δ_{2}(L). Thus there exists a (homogeneous) L ∈ 𝔄_{r+1}^{2} such that F +L is an (r+1)-chunk, if and only if the obstruction of F is 0. In this case, L is unique up to addition with an element of ker(δ_{2}).

(iii) If F and G are (r + 1)-chunks and φ : F → G is a homomorphism of the corresponding r-chunks, then Θ_{r+1}φ ∈ ker(δ_{2}). The residue class of Θ_{r+1}φ in H_{r+1}^{2} is called obstruction of φ.

(iv) Let φ, F and G be as in (iii) and φ′ := φ + ψ for a (homogeneous) ψ ∈ 𝔄_{r+1}^{1}. Then: (Θ_{r+1}φ − Θ_{r+1}φ′) = δ_{1}(ψ). Thus there exists a (homogeneous) ψ ∈ 𝔄_{r+1}^{1} such that φ+ψ is a homomorphism of (r+1)-chunks, if and only if the obstruction of φ is 0. In this case, ψ is unique up to addition with an element of ker(δ_{1}).

Proof. Since all inserted elements are compatible, the proofs of (i)-(iv) are the same for pseudo-analyzers as for analyzers, thus compare [La1] for the proofs. We give the proof of (iv), the other proofs are longer but similar. We use the composition lemma. B′ := G − e_{2,1} − e_{2,2} is ≡ 0 mod J^{2}. φ′ ≡ φ mod J^{r+1}. Thus η_{φ′^{<},φ′^{>}}^{𝔄,2,2}(G)

= η_{e_{2,1}}^{𝔄,1,2}(φ)+η_{e_{2,1}}^{𝔄,1,2}(ψ)+η_{e_{2,2}}^{𝔄,1,2}(φ)+η_{e_{2,2}}^{𝔄,1,2}(ψ)+η_{φ′^{<},φ′^{>}}^{𝔄,2,2}(B′) ≡ η_{e_{2,1}}^{𝔄,1,2}(ψ)+η_{e_{2,2}}^{𝔄,1,2}(ψ)+

η_{φ^{<},φ^{>}}^{𝔄,2,2}(G) mod J^{r+2}. Similarly η_{F}^{𝔄,1,2}(φ′) = η_{F}^{𝔄,1,2}(φ) + η_{F}^{𝔄,1,2}(ψ) ≡ η_{F}^{𝔄,1,2}(φ) +

η_{e_{2,1}+e_{2,2}}^{𝔄,1,2}(ψ) mod J^{r+2}. Subtraction yields assertion (iv).

5. A Spectral Sequence

The torsion theorem (10.1’) of [La1] says that H_{r}^{n}(𝔄) = 0 for all n ≠ r and all analyzers 𝔄 that are rational, defined as follows:

Definition 5.1. For r ∈ N^{∗}, let ℚ_{r} := M^{−1}ℤ, where M is the multiplicative system {n ∈ ℕ^{∗} : if p prime with p|n then p ≤ r}. Then a complete or incomplete pseudoanalyzer over K is called rational, if 𝔄_{r}^{n} is a ℚ_{r}-module for all r ∈ ℕ^{∗}.

For 𝔄 a pseudo-analyzer, denote by 𝔄̃ the subcomplex of elements f ∈ 𝔄^{n} with η_{e_{n,1},...,e_{n,i−1},0,e_{n,i+1},...,e_{n,n}}^{n,n}(f) = 0 for all i.

Remark 5.2. To compute cohomology in the analyzer case, we can replace 𝔄 by 𝔄̃. Also in the pseudo-analyzer case, we will only be interested in the cohomology of 𝔄̃.

It is clear that H_{r}^{n}(𝔄̃) = 0 for all n > r (for all not necessarily rational pseudoanalyzers).

In [La1], p.356, it is shown for analyzers 𝔄: There is a canonical map φ_{r} from the submodule {f ∈ 𝔄̃_{r}^{r} : f anti-symmetric } of 𝔄_{r}^{r} to H_{r}^{r}(𝔄̃), r ∈ ℕ^{∗}. If f ∈ 𝔄̃_{r}^{r} is an element of im(δ_{r−1}) then ∑_{σ∈S_{r}}sign(σ)σf = 0. Therefore any anti-symmetric F is in the kernel of φ_{r} iff r!f = 0.

The cokernel of φ_{r} is similarly described as additive torsion group (0 in the rational case), see [La1] §9. (Even if ḡ ∈ H_{r}^{r}(𝔄̃) is not represented by an anti-symmetric cocycle, the cohomology class of r!g is.)

Proposition 5.3. Let 𝔄 ∈ {𝔈, ℌ, 𝔎}, F ∈ 𝔄̃_{3}^{3}.

For λ = (λ_{1}, λ_{2}, λ_{3}) a permutation of (1, 2, 3), i_{1}, i_{2}, i_{3}, n ∈ {1, ..., m}, we denote the coefficient of x_{i_{1}}^{(λ_{1})}(x_{i_{2}}^{(λ_{2})}x_{i_{3}}^{(λ_{3})}) in F_{n} by Φ(n; λ_{1}, λ_{2}, λ_{3}; i_{1}, i_{2}, i_{3}), where we furthermore assume λ_{2} < λ_{3} for 𝔄 = ℌ. The coefficient of (x_{i_{1}}^{(λ_{1})}x_{i_{2}}^{(λ_{2})})x_{i_{3}}^{(λ_{3})} in F_{n} for 𝔄 = 𝔎 is denoted by Φ′(n; λ_{1}, λ_{2}, λ_{3}; i_{1}, i_{2}, i_{3}).

For 𝔄 ∈ {𝔈, 𝔎}, F is a coboundary iff for all n, all ι = (ι_{1}, ι_{2}, ι_{3}) ∈ {1, ..., m}^{3}

∑_{λ} sign(λ)a(n; λ; ι) = 0,

and for 𝔄 = 𝔎 :  ∑_{λ} sign(λ)a′(n; λ; ι) = 0.

For 𝔄 = ℌ, the condition for F to be a coboundary is:

∑_{λ∈S_{3}:λ_{2}<λ_{3}} sign(λ)(a(n; λ; ι) − a(n; λ; ι_{1}, ι_{3}, ι_{2})) = 0, all n, ι (ι_{2} < ι_{3}).

Proof. We have to show that F is of the form δ_{2}(L) = (η_{e^{(1)},(e^{(2)}+e^{(3)})}^{𝔄,2,3} − η_{e^{(1)},e^{(2)}}^{𝔄,2,3} −

η_{e^{(1)},e^{(3)}}^{𝔄,2,3}) −(η_{(e^{(1)}+e^{(2)}),e^{(3)}}^{𝔄,2,3} − η_{e^{(2)},e^{(3)}}^{𝔄,2,3} − η_{e^{(1)},e^{(3)}}^{𝔄,2,3}) iff the asserted equations hold.

Here L ∈ 𝔄̃_{3}^{2} is given by 12m^{4} coefficients Φ(n; λ; ι), Φ′(n; λ; ι) in case 𝔄 = 𝔎, 6m^{4} in case 𝔄 = 𝔈 and 3m^{4} + m^{3} in case 𝔄 = ℌ, as in all cases we can assume 1 ≤ λ_{1}, λ_{2}, λ_{3} ≤ 2, not all equal, and for ℌ furthermore λ_{2} ≤ λ_{3} and i_{2} ≤ i_{3} if λ_{2} = λ_{3}.

Now F = δ_{2}(L) is equivalent to the following system of equations:

a(n; λ; ι) = Φ(n; ⊓λ; ι) − Φ(n; ⊔λ; ι),
and for 𝔎 : a′(n; λ; ι) = Φ′(n; ⊓λ; ι) − Φ′(n; ⊔λ; ι)

where ⊓ : {1, 2, 3}^{3} → {1, 2}^{3} fixes 1 and 2 and replaces 3 by 2, whereas ⊔ fixes 1 and replaces 2 by 1 and 3 by 2. Thus for example ⊓(3, 1, 2) = (2, 1, 2) and ⊓(3, 1, 2) = (2, 1, 1). For 𝔄 = ℌ, when (1, 2, 2) or (2, 1, 1) occur, we have to replace ι by (ι_{1}, ι_{3}, ι_{2}) if ι_{2} > ι_{3}.

The only-if direction of the assertion - already noted above - follows.

To complete the proof we regard the 2m^{4} coefficients a(n; 132; ι) and a′(n; 132; ι) (all n, ι) for 𝔎, m^{4} coefficients a(n; 123; ι) for 𝔈 and the \frac{1}{2}(m^{4} − m^{3}) coefficients a(n; 123; ι) with ι_{2} > ι_{3} for ℌ as determined (via the system of equations of (5.3)) by the other coefficients a, a′, which we assume to be freely chosen.

If we set all Φ^{(′)}(n; 122; ι) = 0 and thus only use 10m^{4}, 5m^{4} or 3m^{4} − \frac{1}{2}(m^{4} − m^{3}) coefficients describing L for 𝔎, 𝔈, ℌ respectively, it is easy to determine the coefficients of L such that δ(L) = F. In fact, for 𝔈, 𝔎 we start with −a^{(′)}(n; 123; ι) = Φ^{(′)}(n; 112; ι) and use a^{(′)}(n; λ; ι) = Φ^{(′)}(n; ⊓λ; ι)−Φ^{(′)}(n; ⊔λ; ι) successively except for a^{(′)}(n; 132; ι). For ℌ we use the same equations, but first for ι_{2} ≤ ι_{3}, then for ι_{2} > ι_{3}. For ℌ, we note also that the m^{3} equations with ι_{2} = ι_{3} give empty conditions.

Remark 5.4. Let us look at the proof for the torsion theorem given in [La1], § 10.

The induction hypothesis is the following: For r ≤ 2 and all rational analyzers, H_{r}^{n}(𝔄̃) = 0 holds for all n ≠ r. To get this, it is essential to use (D5b), and (D5b) is not fulfilled for general pseudo-analyzers.

After fixing the total degree r, we are able to define an exhausting regular filtration 0 ⊆ A_{1} ⊆ ... ⊆ A_{r} of A_{r} = ⊕_{n}𝔄̃_{r}^{n} for pseudo-analyzers as Lazard did for analyzers. For f ∈ 𝔄^{n}, f is an element of A_{i} iff the degree α_{n} with respect to (the last argument) e_{n,n} is ≤ i for every nonzero component of f.

We get a spectral sequence (E^{p,n}, d) with E_{0}^{p,n} = A_{p}^{n}/A_{p−1}^{n} (zero if not p+n−1 ≤ r, n, p ∈ ℕ^{∗}) and d induced by δ as in Lazard’s proof.

We note that A_{p}^{1} = 0 if p ≠ r and that E_{0}^{p,1} = A_{p}^{1} = E_{1}^{p,1}. For n ≥ 2 and all p, the complex (E_{0}^{p,n}, d_{0} : E_{0}^{p,n} → E_{0}^{p,n+1}) and its cohomology E_{1}^{p,n} is determined by the the complex for total degree r − p and n − 1 arguments: d_{0} fixes all variables with superindex (n) and acts as δ_{n−1} with respect to the other n − 1 arguments.

Let us look at the case 𝔄 = ⊗-^{m}𝔈 for small total degrees r. The case r = 1 is trivial. Here E_{0}^{1,1} = A_{1}^{1} = H_{1}^{1} is given by all m-tuples with entries from V, where V is the vector space generated by X.

By definition H_{r}^{1} = ker δ_{1}|𝔄_{r}^{1} is given by the pseudolinear elements of degree r.

Since pseudolinearity can be checked componentwise, let us call f ∈ ˆ|𝔄|^{1} pseudolinear, iff (f, 0, ..., 0) is.

Lemma 5.5. For { \atop | ⊗ -^{m}𝔈|}^{1}, the following elements (and their K-linear combinations) are pseudolinear:

(i) variables x_{i} ∈ X (ii) the commutators [g, h] := gh − hg of pseudolinear g and h

Proof. We can restrict to the case where f is homogeneous of degree r. To be pseudolinear, f has to fulfill P_{(s,r−s)}η_{e^{(1)}+e^{(2)}}(f) = 0 for all 0 < s < \frac{r+1}{2} (this is equivalent to the definition). (i) is clear. For (ii), by assumption we only have to look at the case where s is the degree of g (and r − s the degree of h) and η_{e^{(1)}} is applied to the factor g, η_{e^{(2)}} to the factor h. Now clearly [η_{e^{(1)}}(g), η_{e^{(2)}}(h)] = 0.

Corollary 5.6. The torsion theorem (10.1’) of [La1] is not true for rational pseudoanalyzers.

Proof. For 𝔄 = ⊗-^{m}𝔈, m > 1, r > 1, H_{r}^{1}(𝔄̃) contains m-tuples of commutators given by (5.5).

Theorem 5.7. For r ∈ {2, 3} and the rational pseudo-analyzer given by 𝔄 = ⊗-^{m}𝔈, the nonzero modules of the cohomology in total degree r, H_{r}^{n}(𝔄̃), are given by:

(i) For r = 2, n = 1, all m-tuples of linear combinations of {[x_{h}^{(1)}, x_{i}^{(1)}] : h < i}; for n = 2, all m-tuples of linear combinations of {[x_{h}^{(1)}, x_{i}^{(2)}] : h < i}.

(ii) For r = 3, n = 1, spanned by all m-tuples with entries [x_{h}^{(1)}, [x_{i}^{(1)}, x_{j}^{(1)}]]; for n = 2, all m-tuples of linear combinations of {[x_{h}^{(1)}, x_{i}^{(1)}]x_{j}^{(2)} : h < i} (or equivalently {x_{h}^{(1)}[x_{i}^{(2)}, x_{j}^{(2)}] : i < j});

for n = 3, all anti-symmetric elements of ℭ̃_{3}^{3}.

Proof. 1) We saw above that, for r = 2, only (p, n) ∈ {(2, 1), (1, 2)} have to be considered (where p is the filtration degree). The only nontrivial map is d_{1} : A_{2}^{1} = E_{1}^{2,1} → E_{1}^{1,2} = A_{1}^{2}. Its kernel E_{2}^{2,1} = E_{∞}^{2,1} is given by the pseudolinear elements of degree 2, i.e. tuples of commutators and their linear combinations. Since all elements of H_{2}^{1} have degree 2 with respect to their last (and only) argument, E_{∞}^{2,1} = H_{2}^{1}. Similarly the cokernel E_{2}^{1,2} of this map, spanned by anti-symmetric f ∈ 𝔄̃_{2}^{2}, is H_{2}^{1}.

2) Let now r = 3. For n ≥ 2 the modules E_{1}^{p,n} are determined by the results for lower total degrees (see above). Nonzero are E_{1}^{2,2}, spanned by the tuples with entries x_{h}^{(1)}x_{i}^{(2)}x_{j}^{(2)}, E_{1}^{1,2}, spanned by tuples with entries x_{h}^{(1)}[x_{i}^{(2)}, x_{j}^{(2)}] and E_{1}^{1,3}, given by multilinear elements with entries [x_{h}^{(1)}, x_{i}^{(2)}]x_{j}^{(3)}. For n = 1, only E_{1}^{3,1} = A_{3}^{1} occurs.

3) On E_{1}^{3,1}, d_{1} is the map P_{(1,2)}η_{e^{(1)}+e^{(2)}} : A_{3}^{1} → A_{2}^{2}/A_{1}^{2}. Thus its kernel E_{2}^{3,1} is given by the pseudolinear elements of degree 3. We claim that for |⊗-^{m}𝔈|_{3}^{1} the elements given in (5.5) are the only pseudolinear elements. Assume that f is pseudolinear and not a linear combination of elements [x_{h}^{(1)}, [x_{i}^{(1)}, x_{j}^{(1)}]]. Since there are no pseudolinear elements of degree 3 for type ℭ, f is contained in the ideal generated by commutators. It suffices to look at linear combinations of [x_{h}^{(1)}x_{i}^{(1)}, x_{j}^{(1)}], x_{h}^{(1)}[x_{i}^{(1)}, x_{j}^{(1)}] and [x_{i}^{(1)}, x_{j}^{(1)}]x_{h}^{(1)}. Aside from forming the commutator, we cannot combine the last two to have zero image under P_{(1,2)}η_{e^{(1)}+e^{(2)}}, as their images are x_{h}^{(1)}[x_{i}^{(2)}, x_{j}^{(2)}]. The image of [x_{h}^{(1)}x_{i}^{(1)}, x_{j}^{(1)}] is x_{h}^{(1)}[x_{i}^{(2)}, x_{j}^{(2)}] + x_{i}^{(1)}[x_{h}^{(2)}, x_{j}^{(2)}] and the only way to cancel this out is to form a combination like [x_{h}^{(1)}x_{i}^{(1)}, x_{j}^{(1)}] − x_{h}^{(1)}[x_{i}^{(1)}, x_{j}^{(1)}] − x_{i}^{(1)}[x_{h}^{(1)}, x_{j}^{(1)}] which is [x_{i}^{(1)}, [x_{j}^{(1)}, x_{h}^{(1)}]].

4) The map d_{1} : E_{1}^{2,2} → E_{1}^{1,3} induced by δ_{2} (i.e. induced by η_{e^{(1)},e^{(2)}+e^{(3)}}) has

the cokernel E_{2}^{1,3} = E_{∞}^{1,3} = H_{3}^{3} = { anti-symmetric elements of ℭ̃_{3}^{3}}, because the quotient in question is the same quotient that occurs for type ℭ. Note that x_{h}^{(1)}x_{i}^{(2)}x_{j}^{(2)} and x_{h}^{(1)}x_{j}^{(2)}x_{i}^{(2)} have the same image. Analogously, we get that E_{2}^{2,2} = 0: In comparison to the type ℭ case (where E_{2}^{2,2} = 0) the only new cocycles in E_{1}^{2,2} are given by tuples with entries x_{h}^{(1)}[x_{i}^{(2)}, x_{j}^{(2)}], which are in the image of P_{(1,2)}η_{e^{(1)}+e^{(2)}}.

5) At last we look at the only possibly nonzero map d_{2} : E_{2}^{3,1} → E_{2}^{1,2}. Since E_{2}^{3,1} consists of pseudolinear elements, this is also zero. Thus E_{2}^{3,1} = E_{∞}^{3,1} gives H_{3}^{1} and E_{2}^{1,2} = E_{∞}^{1,2} gives the elements of H_{3}^{2} with degree 1 with respect to the second argument. Since E_{2}^{2,2} = 0, they represent all elements of H_{3}^{2}.

Remark 5.8. There is no reason, why the case r = 3 of theorem (5.7)(ii) should not be typical. It is not clear though, if the description of pseudolinear elements is as trivial for higher r as it is for r ≤ 3.

We remark, that for all n one can find cocycles L ∈ ⊗-^{m}𝔈̃^{n} that are not coboundaries, like the one given by L_{1} = x_{1}^{(1)} · ... · x_{1}^{(n−1)}(x_{2}^{(n)}x_{1}^{(n)} − x_{1}^{(n)}x_{2}^{(n)}) and L_{i} = 0 (i ≥ 2).

We noted in part 4) of the proof above, that the computation of H_{r}^{r}(⊔ℭ-^{m}𝔈̃) is exactly the same as in type ℭ, because we look at multilinear elements and multilinear elements obey the relations of ℭ. It is easy to see, that a similar result holds for the other mixed types of (3.5), and for all r. Thus we get:

Proposition 5.9. For the pseudo-analyzers ⊔𝔅-^{m}𝔄 given in (3.5) in the rational case, it holds (for all r):

H_{r}^{r}(⊔𝔅-^{m}𝔄̃) = H_{r}^{r}( ^{m}𝔅) = {f ∈ ^{m}𝔅̃_{r}^{r} : f anti-symmetric }.

6. Interactions between different Group Law Types

Our aim in this last section is to find solutions for the system of equations, given in (4.6), associated to universal formal group laws (or chunks) for a given type.

Proposition (4.7) and the torsion theorem (for the cases where it holds) will be our main tools. We can restrict to the subcomplex 𝔄̃ of 𝔄, as ΓF, Θφ ∈ 𝔄̃.

The first nontrivial extension problem is the extension from 2-chunks to 3-chunks (here the obstructions are in H_{3}^{3}(𝔄̃)). We denote the coefficient of x_{i_{1}}^{(λ_{1})}x_{i_{2}}^{(λ_{2})} in F_{n} by Φ(n; λ_{1}, λ_{2}; i_{1}, i_{2}) with i_{1}, i_{2}, n ∈ {1, ..., m} and λ_{1} ≠ λ_{2} ∈ {1, 2}, where 𝔄 is any of the pseudo-analyzers given in (3.5).

Remark 6.1. The group laws of types ℭ, 𝔈, 𝔎 and ℌ, are group laws over operads. For the rational analyzer given by ^{m}ℭ̂ it is well-known that the extension of a 2chunk to 3-chunk and group law is possible (and unique up to strict isomorphism) iff the elements Ψ(n; i_{1}, i_{2}) := Φ(n; 1, 2; i_{1}, i_{2}) − Φ(n; 1, 2; i_{2}, i_{1}) of K are structure constants of an m-dimensional Lie algebra over K. This correspondence between classical group laws and Lie algebras defines an equivalence of categories.

The notion structure constants means that a structure of a (not necessarily unitary or associative) algebra is defined on the free K-module with basis e_{1}, ..., e_{m} by e_{i_{1}}e_{i_{2}} := ∑_{n=1}^{m} Ψ(n; i_{1}, i_{2})e_{n}.

We have seen, that H_{3}^{3}(⊗-𝔈̃) = H_{3}^{3}(ℭ). This leads to the following proposition (of [Ho1], where a direct computation is given). For the nonuniqueness up to strict isomorphisms (see [Ho1] §6) we only refer to the nontriviality of H_{3}^{2}(⊗-𝔈̃).

Proposition 6.2. A 2-chunk F in ⊗-𝔈 is nonuniquely extendable to a 3-chunk, iff the Ψ_{h,i}^{(n)} := (Φ(n; 1, 2; h, i) − Φ(n; 1, 2; i, h)), h < i, are structure constants of a Lie algebra.

Theorem 6.3. Let F be a group law chunk of degree 2 in 𝔎, ℌ or 𝔈 given by its coefficients Φ(n; λ_{1}, λ_{2}; i_{1}, i_{2}) with i_{1}, i_{2}, n ∈ {1, ..., m} and λ_{1}, λ_{2} ∈ {1, 2}, and let us denote Φ(n; 1, 2; i_{1}, i_{2}) − Φ(n; 2, 1; i_{1}, i_{2}) by Ψ(n; i_{1}, i_{2}), where Φ(n; 2, 1; i_{1}, i_{2}) = Φ(n; 1, 2; i_{2}, i_{1}) for 𝔄 = ℌ.

(i) For 𝔄 = 𝔈, the extension of F to a group law 3-chunk is possible if and only if (for all n, ι_{1}, ι_{2}, ι_{3}):

∑_{l=1}^{m} Ψ(n; l, ι_{3})Ψ(l; ι_{1}, ι_{2}) = ∑_{l=1}^{m} Ψ(n; ι_{1}, l)Ψ(l; ι_{2}, ι_{3}).

In the rational case the condition means that the elements Ψ(n; ι_{1}, ι_{2}) form structure constants of an algebra (without unit) with associative multiplication.

(ii) For 𝔄 = 𝔎, the extension of F to a group law 3-chunk is possible if and only if (for all n, ι_{1}, ι_{2}, ι_{3}) both sides of the equation above equal zero. In the rational case the condition means that the elements Ψ(n; ι_{1}, ι_{2}) form structure constants of an algebra (without unit) with cube-zero multiplication, i.e. (ab)c = 0 = a(bc) for all a, b, c.

(iii) For 𝔄 = ℌ, Ψ(n; i_{2}, i_{1}) = −Ψ(n; i_{1}, i_{2}) and the condition F to be exentable to a group law 3-chunk is given by

0 = ∑_{l=1}^{m} Ψ(n; ι_{1}, l)Ψ(l; ι_{2}, ι_{3}) (ι_{2} < ι_{3})

In the rational case the condition means that the elements Ψ(n; ι_{1}, ι_{2}) form structure constants of a 3-nilpotent Lie algebra (cube-zero bracket).

Proof. We use the criterion (5.3) for Γ_{3}F to be a coboundary and collect summands associated to (1, 2, 3), (3, 1, 2), (2, 3, 1) on the lefthand side and to (1, 3, 2), (2, 1, 3), (3, 2, 1) on the righthand side (to omit negative signs). For 𝔎 we get

− ∑_{l=1}^{m} Φ(n; 1, 2; ι_{1}, l)Φ(l; 1, 2; ι_{2}, ι_{3}) + ∑_{l=1}^{m} Φ(n; 2, 1; ι_{1}, l)Φ(l; 1, 2; ι_{2}, ι_{3}) + 0

= − ∑_{l=1}^{m} Φ(n; 1, 2; ι_{1}, l)Φ(l; 2, 1; ι_{2}, ι_{3}) + 0 + ∑_{l=1}^{m} Φ(n; 2, 1; ι_{1}, l)Φ(l; 2, 1; ι_{2}, ι_{3})

and ∑_{l=1}^{m} Φ(n; 1, 2; l, ι_{3})Φ(l; 1, 2; ι_{1}, ι_{2}) + 0 − ∑_{l=1}^{m} Φ(n; 2, 1; l, ι_{3})Φ(l; 1, 2; ι_{1}, ι_{2})

=0 + ∑_{l=1}^{m} Φ(n; 1, 2; l, ι_{3})Φ(l; 2, 1; ι_{1}, ι_{2}) − ∑_{l=1}^{m} Φ(n; 2, 1; l, ι_{3})Φ(l; 2, 1; ι_{1}, ι_{2}).

For 𝔈 (adding the equations) we get a similar system, or equivalently the system given in (i). For ℌ we get

\begin{matrix}− ∑_{l=1}^{m} Φ(n; 1, 2; ι_{1}, l)(Φ(l; 1, 2; ι_{2}, ι_{3}) − Φ(l; 1, 2; ι_{3}, ι_{2})) \\ + ∑_{l=1}^{m} Φ(n; 1, 2; l, ι_{1})(Φ(l; 1, 2; ι_{2}, ι_{3}) − Φ(l; 1, 2; ι_{3}, ι_{2})) = 0. \end{matrix}

Remark 6.4. Over a field K of characteristic 0, theorem (6.3) is a special example of the Koszul duality for quadratic operads given in [GK]. In fact, if we assume that the coefficients of the given 2-chunk can be divided by 2, then we can restrict to 2-chunks fulfilling the (canonical, cf. [La1]) property Φ(n; 2, 1; i_{1}, i_{2}) = −Φ(n; 1, 2; i_{1}, i_{2}) by applying a strict isomorphism. Now we can replace the Ψ(n; i_{1}, i_{2}) by 2Φ(n; 1, 2; i_{1}, i_{2}) or just by Φ(n; 1, 2; i_{1}, i_{2}) in the equations of (6.3). The classification of formal group laws in ^{m}𝔈̂ over a field K of characteristic 0, given in [GK], §2, follows from the torsion theorem (used both for existence and uniqueness of the extensions). A classification of formal group laws in ^{m}ℌ̂ and ^{m}𝔎̂ in the rational case is obtained similarly.

Example 6.5. For ^{1}𝔎 there are no group law chunks which are not abelian (F is called abelian if η_{e^{(2)},e^{(1)}}F = F).

Consider ^{2}𝔎 over K = ℚ, and let Φ(2; 1, 2; 1, 1) = −Φ(2; 2, 1; 1, 1) = \frac{1}{2}, all other coefficients 0. It is easy to show, that a solution of the system of equations (ii) is given. Thus the corresponding 2-chunk (F_{1}, F_{2}) = (x_{1}^{(1)}+x_{1}^{(2)}, x_{2}^{(1)}+x_{2}^{(2)}+ \frac{1}{2}[x_{1}^{(1)}, x_{1}^{(2)}]) is extendable to a 3-chunk and consequently to a group law. It corresponds to the 2-dimensional algebra A = xK[x]/(x^{3}K[x]) with basis x̄, x̄^{2}.

Corollary 6.6. Let K be a field of characteristic 0 and V an m-dimensional vector space. Assume that elements Φ(n; λ_{1}, λ_{2}; i_{1}, i_{2}) of K for i_{1}, i_{2}, n ∈ {1, ..., m} and λ_{1} ≠ λ_{2} ∈ {1, 2} are given such that

(i) Φ(n; 1, 2; i_{1}, i_{2})−Φ(n; 2, 1; i_{1}, i_{2}) are structure constants of a non-trivial associative algebra structure defined on V and

(ii) Φ(n; 1, 2; i_{1}, i_{2}) + Φ(n; 2, 1; i_{2}, i_{1}) − Φ(n; 1, 2; i_{1}, i_{2}) − Φ(n; 2, 1; i_{2}, i_{1}) are structure constants of a nonabelian Lie algebra structure defined on V.

Then there exist formal group laws of type ^{m}𝔈, ^{m}ℭ and ⊗-^{m}𝔈, with 2-chunk defined by the coefficients Φ(n; λ_{1}, λ_{2}; i_{1}, i_{2}) (for ^{m}𝔈) or Φ(n; λ_{1}, λ_{2}; i_{1}, i_{2})+Φ(n; λ_{2}, λ_{1}; i_{2}, i_{1}) (for ^{m}ℭ and ⊗-^{m}𝔈). All are non-trivial (i.e. not isomorphic to the trivial group law).

Proof. Theorem (6.3) shows that there exists the asserted formal group law in ^{m}𝔈. It is given by an m-tuple F with components from K⟨⟨X^{(2)}⟩⟩ and 0 = ΓF := η_{F^{<},e_{3,3}}^{𝔄,2,3}(F) − η_{e_{3,1},F^{>}}^{𝔄,2,3}(F) holds. Viewing the same m-tuple as elements of ^{m}ℭ and ⊗-^{m}𝔈, i.e. applying the module homomorphisms of (3.8), we get group laws, because the condition 0 = ΓF still holds. The new coefficients are sums Φ(n; i_{1}, i_{2}) := Φ(n; λ_{1}, λ_{2}; i_{1}, i_{2}) + Φ(n; λ_{2}, λ_{1}; i_{2}, i_{1}) of the old coefficients. We have to verify the assertion on non-triviality. Since the Lie algebra associated to the group law F in mℭ is nonabelian by assumption (ii), F itself is not isomorphic to the trivial group law, cf. (6.1). Now if the group law in ⊗-^{m}𝔈 were trivial, then the group law in ^{m}ℭ would be trivial, too. Thus all are non-trivial.

Example 6.7. The example corresponding to the completed general linear quantum group (4.3) is of this form.

The matrices e_{1} = \binom{1 0}{0 0} and e_{2} = \binom{0 1}{0 0} together with [,] form a 2-dimensional Lie algebra and it is closed under multiplication (of matrices). The theorem applies, and we get a non-abelian formal group law in ⊗-^{2}𝔈 with 2-chunk defined by the Φ(1; 1, 2) = 1, Φ(n; h, i) = 0 otherwise.

Remark 6.8. It is easy to generalize (6.6) for other group laws of mixed type. It is not clear if the group laws constructed that way are the only examples. For R-chunks with R < ∞, there are much more.

References

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[Ha] Hazewinkel, M. Formal Groups and Applications. New York 1978

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[HL] Huang, Y. and J. Lepowsky Vertex operator algebras and operads hep-th/9301009(1993)

[La1] Lazard, M. Lois de groupes et analyseurs. Ann. École Norm.Sup. 72(1955), pp. 299-400 [La2] Lazard, M. Analyseurs. Bollettino U.M.I.(4)9, Suppl.fasc.2(1974), pp. 49-59

[ML1] MacLane, S. Categorical algebra. Bull. Amer. Math. Soc. 71(1965), pp. 40-106

[ML2] MacLane, S. Categories for the working mathematician. Springer, Berlin, 1971

Ralf Holtkamp, Mathematical Sciences Research Institute, Berkeley, CA 94720

  E-mail address: ralf.holtkamp@ruhr-uni-bochum.de