Some successive quotients of group ring filtrations induced by N-series

Manfred Hartl

Institut de Recherche Mathématique Avancée, ULP et CNRS, 7 rue René-Descartes, 67084 Strasbourg Cedex, France

Abstract

For all groups G and N-series N of G a functorial description is given for the first three quotients of the filtration of the integral group ring which is induced by N. As a key tool N -polynomial maps on G are introduced and studied; an inductive characterization of such maps is given and used to identify an important class of nonlinear functions as N -polynomial maps.

1 Introduction

In this paper we attack with new, homological methods the following classical problem: A filtration of a group G by an N-series N = {N_{i}} gives rise to a descending filtration of the integral group ring ZZ(G) by ideals Δ_{N}^{n} (G) , see [9] or 2.1 below. For example, the lower central series γ of G induces the filtration by the powers of the augmentation ideal I(G) of ZZ(G). Now the problem consists in describing the filtration quotients Q_{n}^{N} (G) = Δ_{N}^{n} (G)/Δ_{N}^{n+1}(G) directly in terms of G.

While the identification Q_{1}^{N} (G) ≅ N_{1}/N_{2} is elementary a general result for n = 2 seems to be known only in case N = γ; and also here it has developed gradually in a sequence of papers; after results of Passi [7], Sandling [12] and Losey [4] for abelian, finite and finitely generated groups, respectively, an invariant and general solution was first given by Bachmann-Gruenenfelder [2], in the following setting (see [9] VIII): The groups Q_{n}^{N} (G) form the associated graded ring Gr^{N} (ZZ(G)) = ⨁_{n} Q_{n}^{N} (G). As an approximation of this ring consider the enveloping ring UL^{N} (G)of the graded Lie ring L^{N} (G) = ⨁_{i≥1} N_{i}/N_{i+1} where the Lie bracket is induced by the commutator pairing in G. The grading of L^{N} (G) induces a grading of UL^{N} (G), and a natural epimorphism of graded rings

Research supported by successive fellowships of the Sonderforschungsbereich 343, Bielefeld, and the Human Capital and Mobility Programme of the European Union.

θ^{N} = ⨁_{n} θ_{n}^{N} : UL^{N} (G) → →Gr^{N} (ZZ(G))  (1)

is induced by the map N_{i} → Δ_{N}^{i} (G) ⊂ ZZ(G), a ↦ a − 1.

The isomorphism Q_{1}^{N} (G) ≅ N_{1}/N_{2} now means that θ_{1}^{N} is isomorphic, and in [2] it is proven that θ_{2}^{γ} is isomorphic, too. Nevertheless, in general θ_{n}^{γ} for n > 2 is far from being isomorphic; e.g., a finite abelian group has isomorphic θ^{γ} if and only if it is cyclic [1]. At least one knows that the kernel of θ^{γ} is a torsion group since Q ⊗ θ^{γ} is an isomorphism [11]. For finite groups G and all N-series the group Q_{3}^{N} (G) was determined by Tahara [13] in terms of cyclic decompositions of various abelian subquotients of G. Pirashvili [10] determined the kernel of θ_{3}^{γ} for all abelian groups by use of Dold-Puppe derived functors.

In this paper we show that θ_{2}^{N} is isomorphic for all groups and N-series and give a functorial description of the kernel of θ_{3}^{N} . The latter one turns out to reduce to an interesting fundamental phenomenon, namely the difference of left and right connecting homomorphisms, as follows.

Since θ_{1}^{N} and θ_{2}^{N} are isomorphic one has a short exact sequence

U_{2}L^{N} (G) ⟶^{>} I(G)/Δ_{N}^{3} (G) ⟶ N_{1}/N_{2}

Tensoring with N_{1}/N_{2} from the left and from the right gives rise to connecting homomorphisms τ_{1}, τ_{2} in the following diagram, where we adopt the notation (N_{1}/N_{2}) ∗ (N_{1}/N_{2}) =^{def} Tor_{1}^{Z Z}(N_{1}/N_{2}, N_{1}/N_{2}) used in the book of Mac Lane [5].

\begin{matrix}(N_{1}/N_{2}) ∗ (N_{1}/N_{2}) & ⟶^{τ_{1}} & U_{2}L^{N} (G) ⊗ (N_{1}/N_{2}) \\ ↓^{τ_{2}} & & ↓^{q_{1}} \\ (N_{1}/N_{2}) ⊗ U_{2}L^{N} (G) & ⟶^{q_{2}} & U_{3}L^{N} (G) \end{matrix}  (2)

which can be compared to each other by composing with the maps q_{1} and q_{2} given by N_{1}/N_{2} ≅ U_{1}L^{N} (G) and multiplication in UL^{N} (G). Then we prove the following

Theorem 1.1 Let δ = q_{1}τ_{1} − q_{2}τ_{2}. Then the natural sequence of homomorphisms

(N_{1}/N_{2}) ∗ (N_{1}/N_{2}) ⟶^{δ} U_{3}L^{N} (G) ⟶^{θ_{3}^{N}} Q_{3}^{N} (G) → 0

is exact for all groups G and N-series N of G. More explicitely, let ⟨x_{1}, k, x_{2}⟩ be a typical generator of the torsion product N_{1}/N_{2} ∗ N_{1}/N_{2}, i.e. k ∈ ZZ and x_{i} = x′_{i}N_{2} ∈ L_{1}^{N} (G) = U_{1}L^{N} (G) for some x′_{i} ∈ G, such that (x′_{1})^{k}, (x′_{2})^{k} ∈ N_{2}, cf. [5] V.6. Then

δ⟨x_{1}, k, x_{2}⟩ = (x′_{1})^{k}x_{2} − x_{1}(x′_{2})^{k} + \binom{k}{2}(x_{1}x_{2}^{2} − x_{1}^{2}x_{2}) ,  (3)

where (x′_{i})^{k} ∈ N_{2} is considered as an element of U_{2}L^{N} (G) via the canonical maps N_{2} → → N_{2}/N_{3} = L_{2}^{N} (G) → U_{2}L^{N} (G) , and the products are taken in the ring UL^{N} (G) . □

Note that δ annihilates all diagonal elements ⟨x, k, x⟩ since [x^{k}, x] vanishes in γ_{3}(G) and thus also in U_{3}L^{N} (G) .

Actually the theorem is derived from a computation of the rings I(G)/Δ_{N}^{n} (G) , i = 3, 4, in terms of certain abelian subquotients of G, see section 2.

We also point out that the theorem determines the fourth integer dimension subgroup with respect to N, D_{4}^{N} (G) = G ∩ (1 + Δ_{N}^{4} (G)), of any group G and N-series N as

D_{4}^{N} (G)/N_{4} = L_{3}^{N} (G) ∩ Im(δ)

since the canonical map from a Lie ring to its enveloping ring is known to be injective. Evaluating this intersection for finitely generated groups one rediscovers Tahara’s description of D_{4}^{N} (G) for finite groups [13].

The method of proof relies on homological methods and an extensive use of N -polynomial maps generalizing polynomial maps in the sense of Passi. Therefore the first section is devoted to a study of such maps which might be of independent interest even for ordinary polynomial maps.

2 N -polynomial maps

In this section we introduce and study a notion of N -polynomial maps from groups to abelian groups. This generalizes Passi’s notion of polynomiality in now refering to a given N-series N of the group. We exhibit criteria for N -polynomiality of functions from groups to abelian groups. An inductive characterization of N - polynomiality is given which is inspired by the classical definition of quadratic forms. It applies easily, for example, to the diagonal maps from an abelian group G to its n-th symmetric or tensor power, where only the symmetric case seems to be treated in the literature, [14] or Sandling, cf. [9]. Moreover, we prove N - polynomiality of functions given by polynomials in certain quotients of the group ring. For example, all the power maps e_{i}: G → ZZ(G)/I(G)^{n+1}, a ↦ (a − 1)^{i} + I(G)^{n+1}, turn out to be polynomial of degree ≤ n.

Definitions 2.1 Let G be a group. Recall that an N-series N = {N_{i}}_{i} is a descending chain of subgroups

G = N_{1} ⊃ N_{2} ⊃ … ⊃ 1

such that [N_{i}, N_{j}] ⊂ N_{i+j} (with [a, b] = aba^{−1}b^{−1} for a, b ∈ G). An N-series N induces a descending chain of both-sided ideals

ZZ(G) ⊃ Δ_{N}^{1} (G) ⊃ Δ_{N}^{2} (G) ⊃ … ⊃ 0

by defining Δ_{N}^{n} (G) to be the subgroup of ZZ(G) generated by all elements

(a_{1} − 1) ⋯ (a_{r} − 1) , a_{i} ∈ N_{k_{i}}, such that k_{1} + … + k_{r} ≥ n.

Note that Δ_{N}^{1} (G) equals I(G), the augmentation ideal of ZZ(G).

Now let f : G → A be a function to some abelian group A. We call f (normalized) N -polynomial of degree ≤ n if its linear extension f̄: ZZ(G) → A annihilates 1 + Δ_{N}^{n+1}(G) . Clearly, the map

p_{n}^{N} : G → P_{n}^{N} (G) =^{def} I(G)/Δ_{N}^{n+1}(G) , p_{n}^{N} (a) = a − 1 + Δ_{N}^{n+1}(G) ,

is universal among normalized N -polynomial maps of degree ≤ n from G to abelian groups. We shall frequently write just p_{n} instead of p_{n}^{N} . Occasionally we shall also use the obvious functorial property of P_{n}^{N} (−) and naturality of the map p_{n}^{N} .

Other structures related to an N-series and used below are defined in the introduction.

Examples 2.2

(1) The lower central series γ, γ_{1}(G) = G and γ_{n+1}(G) = [G, γ_{n}(G)], is an N-series. By the inclusion I(γ_{n}(G)) ⊂ I(G)^{n} ⊂ Δ_{γ}^{n}(G) one has

Δ_{γ}^{n}(G) = I(G)^{n} .

Thus a function f as above is γ -polynomial of degree ≤ n iff it is polynomial of degree ≤ n as defined by Passi [6], and

P_{n}^{γ}(G) = P_{n}(G) =^{def} I(G)/I(G)^{n+1} .

(2) The isolators \sqrt{γ_{n}(G)} of γ_{n}(G) in G form an N-series \sqrt{γ}, see IV 1.3 in [9]. It is shown in [3] that

θ^{\sqrt{γ}}: UL^{\sqrt{γ}}(G) → Gr^{\sqrt{γ}}(ZZ(G))

is an isomorphism for all groups G.

(3) Let N be any N-series. Then a function f : G → A as above is normalized N - polynomial of degree 0 iff f(G) = 0 and of degree ≤ 1 iff it is linear and annihilates N_{2}. The latter statement follows from 2.8 below. Thus the universal property of p_{1}^{N} gives rise to a canonical isomorphism

P_{1}^{N} (G) →^{≅} N_{1}/N_{2} , (a − 1) + Δ_{N}^{2} (G) ↦ aN_{2} .  (4)

In other words, we have obtained the following elementary Corollary 2.3 For all groups G and N-series N the natural map

θ_{1}^{N} : L_{1}^{N} (G) = U_{1}L^{N} (G) → Δ_{N}^{1} (G)/Δ_{N}^{2} (G)

is an isomorphism.

For the rest of the paper we fix a group G and an N-series N of G. Lemma 2.4 Let f : G → A be a N -polynomial map of degree ≤ n. Then for k ≥ 1 the restriction g: N_{k} ↪^{i} G ⟶^{f} A of f is γ -polynomial of degree ≤ m, where m is the least integer ≥ \frac{n+1}{k} − 1. Proof: One has a factorization ḡ: ZZ(N_{k}) ⟶^{Z Z(i)} ZZ(G) ⟶^{f̄} A where ZZ(i) is an augmented ring homomorphism. It maps I(N_{k}) into Δ_{N}^{k} (G) , whence I(N_{k})^{m+1} into Δ_{N}^{k(m+1)}(G) which is annihilated by f̄ whenever k(m + 1) ≥ n + 1. □ Lemma 2.5 The map p_{n} satisfies the following relations for a, b, a_{i} ∈ G.

p_{n}(ab) = p_{n}(a) + p_{n}(b) + p_{n}(a)p_{n}(b)  (5)
p_{n}[a, b] = [p_{n}(a), p_{n}(b)]a^{−1}b^{−1}
= ∑_{i=0}^{n−2}(−1)^{i}[p_{n}(a), p_{n}(b)]p_{n}(a)^{i} ∑_{j=0}^{n−2}(−1)^{j}p_{n}(b)^{j}
= ∑_{k=0}^{n−2} ∑_{l=0}^{k}(−1)^{k}[p_{n}(a), p_{n}(b)]p_{n}(a)^{l}p_{n}(b)^{k−l}  (6)
p_{n}[… [a_{1}, a_{2}], …], a_{n}] = [… [p_{n}(a_{1}), p_{n}(a_{2})], …], p_{n}(a_{n})]  (7)
p_{n}γ_{n+1}(G) = 0  (8)

with [p_{n}(a), p_{n}(b)] = p_{n}(a)p_{n}(b) − p_{n}(b)p_{n}(a) and setting p_{n}(a)^{0} = 1. □

In order to characterize N -polynomiality of a map inductively we consider its deviation from linearity, as follows. Definition 2.6 Let f : G → A be a function between groups where A is abelian. Define the deviation function to be the coboundary of the 1-cochain f, namely

w_{f} : G × G → A , w_{f}(a, b) = f(ab) − f(a) − f(b) ,

and let

w̄_{f} : ZZ(G) ⊗ ZZ(G) ≅ ZZ(G × G) → A

be given by linear extension of w_{f}.

Note that for normalized f (i.e. f(1) = 0) one has

w̄_{f}((a − 1) ⊗ (b − 1)) = w_{f}(a, b) = f̄((a − 1)(b − 1)) .  (9)

For example, (5) now writes

w_{p_{n}}(a, b) = p_{n}(a)p_{n}(b) .

Lemma 2.7 The map f is normalized iff w_{f} is normalized in one of its variables, in fact,

w_{f}(a, 1) = −f(1) = w_{f}(1, b) .  (10)

Moreover, if f is normalized then the following relations hold for x, y, z ∈ Δ_{Z Z}(G) :

w̄_{f}(x ⊗ y) = f̄(xy)  (11)
w̄_{f}(xy ⊗ z) = w̄_{f}(x ⊗ yz) .  (12)

Proof: It suffices to check (11) on generators (a−1)⊗(b−1) of Δ_{Z Z}(G)⊗Δ_{Z Z}(G) , a, b ∈ G, where it is equation (9). Then (11) implies (12). □

Proposition 2.8 Let n ≥ 1. A function f : G → A as in 2.1 is a normalized N -polynomial map of degree ≤ n if and only if f(N_{n+1}) = 0 and for every a ∈ N_{i} (1 ≤ i ≤ n) the function w_{f}(a, −) : G → A is normalized N -polynomial of degree ≤ n − i (or if, equivalently, the same holds after replacing w_{f}(a, −) by w_{f}(−, a)).

Proof: With regard to the normalization properties use (10). Assuming f(1) = 0 relation (11) yields for k ≥ 1

f̄((a − 1)Δ_{N}^{k} (G) = w̄_{f}((a − 1) ⊗ Δ_{N}^{k} (G)) = w̄_{f}(a ⊗ Δ_{N}^{k} (G)) = \overline{w_{f}(a, −)}(Δ_{N}^{k} (G))

The assertion follows immediately from this equation by putting k = n − i + 1. □

We apply this criterion to the following ‘reduced diagonal’ function.

2.9 For integers k, n ≥ 1 consider the map

δ_{N} (k, n) : G → I(G)^{⊗k}/F (k, n + 1)

with

F (k, n) = F (G, N , k, n) = ∑_{l_{1}+...+l_{k}≥n} Δ_{N}^{l_{1}} (G) ⊗ ⋯ ⊗ Δ_{N}^{l_{k}} (G)

and δ_{N} (k, n)(a) = (a − 1)^{⊗k} + F (k, n + 1).

2.10 For example, by (4) the map δ_{N} (k, k) is equivalent to the diagonal

δ_{k,N} : G → (N_{1}/N_{2})^{⊗k} , a ↦ (aN_{2})^{⊗k} .

2.11 Clearly, δ_{N} (k, n) = 0 whenever k > n.

Lemma 2.12 The map δ_{N} (k, n) is N -polynomial of degree ≤ n.

Proof: We use induction on n. For n = 1 the assertion is trivial by 2.11, 2.10 and 2.2 (3). Now fix b ∈ N_{i} and expand

w_{δ(k,n)}(a, b) = ((a − 1) + (b − 1) + (a − 1)(b − 1))^{⊗k} − (a − 1)^{⊗k} − (b − 1)^{⊗k}

as a sum of monomial tensor product elements v_{m}(a, b) and consider these as maps v_{m}(−, b) : G → Δ_{Z Z}(G)^{⊗k}/F (G, k, n + 1). Let n_{a}, n_{b}, n_{ab} be the number of tensor factors (a − 1), (b − 1), (a − 1)(b − 1) occuring in v_{m}(a, b), respectively. Then we have an obvious factorization of v_{m}(−, b),

\begin{aligned}G & ⟶^{δ′} & Δ_{Z Z}(G)^{⊗n_{a}+n_{ab}}/F (G, n_{a} + n_{ab} , n + 1 − n_{b}i − n_{ab}i)) \\ ↓ & fm \\ & & Δ_{Z Z}(G)^{⊗k}/F (G, k, n + 1) \end{aligned}  (13)

with δ′ = δ(n_{a} + n_{ab} , n + 1 − n_{b}i − n_{ab}i) and where f_{m} is the homomorphism defined by inserting the missing factors (b − 1). (∗) Since n_{b} + n_{ab} ≥ 1 holds the map δ′ – and thus v_{m}(−, b) – is N -polynomial of degree ≤ n − n_{b}i − n_{ab}i ≤ n − i by induction, whence the assertion follows from 2.8. □

Corollary 2.13 Let G be an abelian group. Then the n-fold diagonal map δ_{n}: G → G^{⊗n} is a polynomial map of degree ≤ n. Moreover, it satisfies

δ̄_{n}((a_{1} − 1) ⋯ (a_{n} − 1)) = ∑_{π∈Σ_{n}} a_{π(1)} ⊗ ⋯ ⊗ a_{π(n)} .

In particular, the symmetrized diagonal map into the n-fold symmetric tensor product SP^{n}(G),

λ_{n}δ_{n}: G → SP^{n}(G) ,

λ_{n} : G^{⊗n} → →SP^{n}(G) , is a N -polynomial map of degree ≤ n and satisfies

λ_{n}δ̄_{n}((a_{1} − 1) ⋯ (a_{n} − 1)) = n! λ_{n}((a_{1} − 1) ⋯ (a_{n} − 1)) .

The symmetrized case was treated by more complicated combinatorial methods by Vermani [14] and Sandling [9].

Proof: This is the case k = n of 2.12, see 2.10. To evaluate δ_{n} on I(G)^{n} observe that by (11) and (13)

δ̄_{n}(∏_{1}^{n}(a_{j} − 1)) = w̄_{δ_{n}}( ∏_{1}^{n−1}(a_{j} − 1) ⊗ (a_{n} − 1))
= ∑_{m} v̄_{m}( ∏_{1}^{n−1}(a_{j} − 1) , a_{n})  (14)

The m-th summand vanishes whenever v_{m}(−, b) is of degree < n−1, in particular when n − n_{b} − n_{ab} < n − 1 by (∗) in the proof of 2.9 above. The same holds when n_{ab} > 0 since f_{i} maps into F (n, n_{a} + n_{ab} + n_{b} + n_{ab}) = F (n, n + n_{ab}). So only those n summands v̄_{i}(∏_{1}^{n−1}(a_{j} − 1) , a_{n}) are left for which v_{m}(a, b) contains no factor (a − 1)(b − 1), exactly one factor (b − 1) and thus n − 1 factors (a − 1). By induction and (14) we obtain

δ̄_{n}(∏_{1}^{n}(a_{j} − 1)) = a_{n} ⊗ ∑_{π∈Σ_{n−1}} a_{π(1)} ⊗ ⋯ ⊗ a_{π(n−1)}

+ ∑_{i=2}^{n−1} ∑_{π∈Σ_{n−1}} a_{π(1)} ⊗ ⋯ ⊗ a_{π(i−1)} ⊗ a_{n} ⊗ a_{π(i)} ⊗ ⋯ ⊗ a_{π(n−1)}

+ ∑_{π∈Σ_{n−1}} a_{π(1)} ⊗ ⋯ ⊗ a_{π(n−1)} ⊗ a_{n}

= ∑_{π∈Σ_{n}} a_{π(1)} ⊗ ⋯ ⊗ a_{π(n)} ,

as asserted. □

Now we are ready to consider functions given by polynomials in the group elements.

Recall that the N -weight of an element x ∈ ZZ(G) is the maximal number w such that x ∈ Δ_{N}^{w} (G) (with Δ_{N}^{0} (G) =^{def} ZZ(G)).

Proposition 2.14 Let G be a group, r > 1, x_{1}, …, x_{r} ∈ ZZ(G) and e_{1}, …, e_{r−1} integers ≥ 1. Then for n ≥ 1 the map

f : G → P_{n}^{N} (G) , a ↦ x_{1}(a − 1)^{e_{1}} ⋯ x_{r−1}(a − 1)^{e_{r−1}}x_{r} + Δ_{N}^{n+1}(G)

is N -polynomial of degree ≤ n − ∑_{j=1}^{r} w(x_{j}).

Proof: Let e = e_{1} + … + e_{r−1} and w = w(x_{1}) + … + w(x_{r}). Then there is an obvious factorization

f : G ⟶^{δ_{N} (e,n+1−w)} I(G)^{⊗e}/F (G, N , e, n + 1 − w) ⟶^{f ′} P_{n}^{N} (G)

where f′ is the homomorphism defined by inserting the factors x_{j} and multiplying together in P_{n}^{N} (G) . Now the assertion follows from 2.12. □

By an obvious adaptation of the proof of 2.14 one shows

Proposition 2.15 Let G be a group, r, s > 1, x_{1}, …, x_{r}, y_{1}, …, y_{s} ∈ ZZ(G) and e_{1}, …, e_{r−1}, f_{1}, …, f_{s−1} ≥ 1. For n ≥ 1 define the map

g: G → P_{n}^{N} (G) ⊗ P_{n}^{N} (G)/(∑_{i+j>n+1} Δ_{N}^{i} (G) ⊗ Δ_{N}^{j} (G)) ,

a ↦ (ρ ⊗ ρ)(x_{1}(a − 1)^{e_{1}} ⋯ x_{r−1}(a − 1)^{e_{r−1}}x_{r}) ⊗ (y_{1}(a − 1)^{f_{1}} ⋯ y_{s−1}(a − 1)^{f_{s−1}}y_{s}) ,

with ρ : Δ_{Z Z}(G) → → P_{n}^{N} (G) being the canonical quotient map. Then g is N - polynomial of degree ≤ max(0, n + 1 − ∑_{i=1}^{r} w(x_{i}) − ∑_{j=1}^{s} w(y_{j})). □

A fortiori, the target of the map g can be replaced by its quotient

P_{n}^{N} (G) ⊗_{N} P_{n}^{N} (G) =^{def} P_{n}^{N} (G) ⊗_{G} P_{n}^{N} (G)/( ∑_{i+j>n+1} Δ_{N}^{i} (G) ⊗ Δ_{N}^{j} (G))

which is used below.

3 The structure of P_{2}^{N} (G) and P_{3}^{N} (G)

In this section we determine the ring structure of P_{2}^{N} (G) and P_{3}^{N} (G) in terms of certain abelian subquotients of G obtained from N.

Recall that the exterior square A ∧ A of an abelian group A is defined to be the quotient of A ⊗ A modulo the subgroup generated by the diagonal elements.

For a group G with N-series N one has natural homomorphisms

γ_{2}(G)/γ_{3}(G) \overleftarrow{⟵}^{c_{2}} (N_{1}/N_{2})∧(N_{1}/N_{2}) ⟶^{>}^{l_{2}} (N_{1}/N_{2})⊗(N_{1}/N_{2})⟶^{μ_{2}} P_{2}^{N} (G) ⟶^{ρ} N_{1}/N_{2}

defined for a, b ∈ G by c_{2}(aN_{2} ∧ bN_{2}) = [a, b]N_{3} with [a, b] = aba^{−1}b^{−1},

l_{2}(aN_{2} ∧ bN_{2}) = aN_{2} ⊗ bN_{2} − bN_{2} ⊗ aN_{2}
μ_{2}(aN_{2} ∧ bN_{2}) = (a − 1)(b − 1) + Δ_{N}^{3} (G)

and ρ(a − 1 + Δ_{N}^{3} (G)) = aN_{2}. By (5) μ_{2} is welldefined. Furthermore, choose a transversal, s, for the quotient map N_{1}/N_{4} → → N_{1}/N_{2}. One gets a 2-cocycle

d: (N_{1}/N_{2}) × (N_{1}/N_{2}) → γ_{2}(G)/γ_{4}(G) , sx · sy = d(x, y)s(xy) .

Let π : N_{2}/N_{4} → →N_{2}/N_{3} be the canonical quotient map and p: N_{2}/N_{3} → P_{2}^{N} (G) be the homomorphism (see (5)) given by restriction of p_{2} to N_{2}/N_{3}.

Theorem 3.1 Let G be a group and N an N-series of G. Then the following sequence of homomorphisms is exact:

(N_{1}/N_{2})∧(N_{1}/N_{2}) ⟶^{(c_{2},−l_{2})^{t}} N_{2}/N_{3} ⊕(N_{1}/N_{2})⊗(N_{1}/N_{2}) ⟶^{(p,μ_{2})} P_{2}^{N} (G) ⟶^{ρ} N_{1}/N_{2} ⟶ 0

Moreover, the congruence class of the corresponding extension

cok(c_{2}, −l_{2})^{t} ⟶^{>}^{μ̄_{2}} P_{2}^{N} (G) ⟶^{ρ} N_{1}/N_{2}

(with μ̄_{2} induced by (p, μ_{2})) is determined by the symmetric cocycle

D_{2} : (N_{1}/N_{2}) × (N_{1}/N_{2}) → cok(c_{2}, −l_{2})^{t} ,
D_{2}(x, y) = (πd(x, y) , −x ⊗ y) + Im(c_{2}, −l_{2})^{t} .
□

Theorem 3.1 actually follows from an analogous result for P_{3}^{N} (G) which we describe after noting some corollaries.

Since the composition U_{2}L^{N} (G) ≅ cok(c_{2}, −l_{2})^{t} ≅^{μ̄_{2}} Δ_{N}^{2} (G)/Δ_{N}^{3} (G) coincides with θ_{2}^{N} we obtain

Corollary 3.2 For all groups G and N-series N of G the natural map

θ_{2}^{N} : U_{2}L^{N} (G) → Δ_{N}^{2} (G)/Δ_{N}^{3} (G)

is an isomorphism. □

In case N = γ this result is due to [2]; for finite groups it follows from Tahara’s result in [13].

Now observe that the canonical map N_{2}/N_{3} → cok(c_{2}, −l_{2})^{t} is injective since the map l_{2} is injective; this wellknown general fact is easily checked first in case N_{1}/N_{2} is finitely generated (by cyclic decomposition) and then extended to the general case by a colimit argument. Thus we get

Corollary 3.3 For all groups G and N-series N of G there is a natural exact sequence of homomorphisms

0 → N_{2}/N_{3} ⟶^{p_{2}|N_{2}} P_{2}^{N} (G) ⟶^{P_{2}(α)} P_{2}(N_{1}/N_{2}) → 0

with the anonical quotient map α : N_{1}/N_{3} → →N_{1}/N_{2} .

This implies the following result on dimension subgroups with respect to Nseries:

Corollary 3.4 For all groups G and N-series N of G one has

G ∩ (1 + Δ_{N}^{3} (G)) = N_{3} .

We remark that this is equivalent to Passi’s result in [8] that

G ∩ (1 + I(ζ_{1}(G))I(G) + I(G)^{3}) = γ_{3}(G)

where ζ_{1}(G) denotes the center of G. In fact, the lower central series of G remains an N-series when one replaces γ_{2}(G) by N_{2} =^{def} ζ_{1}(G)γ_{2}(G); but then Δ_{N}^{3} (G) = I(ζ_{1}(G))I(G) + I(G)^{3}.

Now we turn to a study of P_{3}^{N} (G) . First we construct homomorphisms

γ_{2}(G)/γ_{4}(G) \overleftarrow{⟵}^{c_{3}} P_{2}^{N} (G) ∧ P_{2}^{N} (G) ⟶^{l_{3}} P_{2}(G) ⊗_{G} P_{2}(G) ⟶^{μ_{3}} P_{3}^{N} (G) (15)

which will play the same role for P_{3}^{N} (G)as the maps c_{2}, l_{2} and μ_{2} for P_{2}^{N} (G)above. Clearly μ_{3} is well defined by sending the coset of (a − 1) ⊗ (b − 1) to the coset of (a − 1)(b − 1). Consider the functions

γ_{2}(G)/γ_{4}(G) ⟵^{c} G × G ⟶^{l} P_{2}(G) ⊗_{G} P_{2}(G)

defined by c(a, b) = [a, b]N_{4} and

l(a, b) = p_{2}(a) ⊗ p_{2}(b) − p_{2}(b) ⊗ p_{2}(a) − [p_{2}(a), p_{2}(b)] ⊗ (p_{2}(a) + p_{2}(b)) (16)

By the Witt-Hall identities and the relation γ_{4}(G) ⊂ N_{4} we have for a, g, h ∈ G

w_{c(−,a)}(g, h) = [g, [h, a]] γ_{4}(G)  (17)
w_{c(a,−)}(g, h) = [g, [a, h]] γ_{4}(G)

The right hand term is trilinear and annihilates N_{2} in all variables. So we conclude from 2.8 that c is N -polynomial of degree ≤ 2 in both variables; the same holds for l by 2.15. Thus c and l induce homomorphisms

γ_{2}(G)/γ_{4}(G) ⟵ P_{2}^{N} (G) ⊗ P_{2}^{N} (G) ⟶ P_{2}(G) ⊗_{G} P_{2}(G) .

Both factor through P_{2}(G)∧P_{2}(G) since they annihilate the elements p_{2}(a)⊗ p_{2}(a) and p_{2}(a) ⊗ p_{2}(b) + p_{2}(b) ⊗ p_{2}(a), a, b ∈ G. The induced maps are the desired homomorphisms c_{3} and l_{3} in (15).

Furthermore, by the isomorphisms

P_{2}^{N} (G)/(Δ_{N}^{2} (G)/Δ_{N}^{3} (G)) ≅ Δ_{N}^{1} (G)/Δ_{N}^{2} (G) ≅ N_{1}/N_{2}

one has natural homomorphisms

(Δ_{N}^{2} (G)/Δ_{N}^{3} (G))⊗(N_{1}/N_{2}) ⟶^{j_{1}}P_{2}(G)⊗_{G}P_{2}(G) ⟵^{j_{2}} (N_{1}/N_{2})⊗(Δ_{N}^{2} (G)/Δ_{N}^{3} (G))

well defined by

j_{1} = ι ⊗ p_{2}s and j_{2} = p_{2}s ⊗ ι , ι: Δ_{N}^{2} (G)/Δ_{N}^{3} (G) ⊂ P_{2}^{N} (G) .  (18)

Theorem 3.5 The following sequence of homomorphisms is exact:

P_{2}^{N} (G)∧P_{2}^{N} (G) ⟶^{(c_{3},−l_{3})^{t}}N_{2}/N_{4} ⊕ P_{2}(G)⊗_{G}P_{2}(G) ⟶^{(p_{3}|N_{2},μ_{3})} P_{3}^{N} (G) ⟶^{ρ}N_{1}/N_{2}⟶0

Moreover, the congruence class of the corresponding extension

cok(c_{3}, −l_{3})^{t} ⟶^{>}^{μ̄_{3}} P_{3}^{N} (G) ⟶^{ρ} N_{1}/N_{2}

(with μ̄_{3} induced by (p_{3}|N_{2}, μ_{3})) is determined by the symmetric cocycle

D_{3} : (N_{1}/N_{2}) × (N_{1}/N_{2}) → cok(c_{3}, −l_{3})^{t} ,

D_{3}(x, y) = (d(x, y) , −p_{2}(sx) ⊗ p_{2}(sy) − j_{1}(p_{2}d(x, y) ⊗ (x + y))) + Im(c_{3}, −l_{3})^{t} .

Proof: From now on we write the abelian group A =^{def} N_{1}/N_{2} additively. Write B = cok(c_{3}, −l_{3})^{t} and let

N_{2}/N_{4} ⟶^{l̄} B ⟵^{c̄} P_{2}(G) ⊗_{G} P_{2}(G) = \frac{P_{2}^{N} (G) ⊗_{G} P_{2}^{N} (G)}{Im(μ̄_{2}) ⊗ Im(μ̄_{2})}

be the canonical maps. From (6) one derives the relation (p_{3}|N_{2}, μ_{3})(c_{3}, −l_{3}) = 0, so the homomorphism μ̄_{3} : B → Δ_{N}^{2} (G)/Δ_{N}^{4} (G) is well defined and clearly surjective. We wish to construct a left inverse of μ̄_{3} by the universal property of P_{3}^{N} (G). For this we extend B to an abelian group C ⊃ B which admits an extension of the map N_{2} → → N_{2}/N_{4} ⟶^{l̄} B ↪ C to an N -polynomial map f : G → C of degree ≤ 3. For doing this we check below the following claim:

The map D_{3} is a symmetric 2-cocycle with respect to the trivial A-action on B. Thus an abelian group C is defined by taking the set C = B ×A with addition

(α, x) + (β, y) = (α + β + D_{3}(x, y) , x + y) .

Consider the functions

N_{2}/N_{4} ⟵^{t} G ⟶^{f}B × A ,

t(a) = as(ā)^{−1}N_{4} , ā = aγ_{2}(G) ∈ A ,
f(a) = (l̄t(a) + c̄j_{1}(p_{2}t(a) ⊗ ā) , ā) .

Note that f(N_{4}) = 0. Moreover, we check below that for a, b ∈ G we have

w_{f}(a, b) = (c̄(p_{2}(a) ⊗ p_{2}(b)) , 0) .  (19)

Assuming this for a moment, we conclude from 2.8 that f is a N -polynomial map of degree ≤ 3, hence induces a homomorphism f̄ which fits into the following commutative diagram with short exact rows:

p_{3}(N_{2}) + P_{3}^{N} (G)^{2} ↪ P_{3}^{N} (G) → → P_{1}^{N} (G)
↓f̄^{res}  ↓f̄  ↓≅
B  ⟶^{>}^{(1,0)^{t}}  C  ⟶^{(0,1)}  A

But f̄^{res} is clearly a left inverse of the surjection μ̄_{3} by the following equations for a, b ∈ G, d ∈ N_{2}:

(1, 0)^{t}f̄^{res}μ̄_{3}c̄(p_{2}a ⊗ p_{2}b) = f̄((p_{2}a ⊗ p_{2}b) = w_{f}(a, b) = (1, 0)^{t}c̄(p_{2}a ⊗ p_{2}b)

(1, 0)^{t}f̄^{res}μ̄_{3}l̄(d) = f̄p_{2}d = f(d) = (1, 0)^{t}l̄(d)

Thus μ̄_{3}, f̄^{res} and f̄ are isomorphisms which proves the theorem. It remains to verify relation (19) and the symmetric cocycle property of D_{3}.

Let x, y, z ∈ N_{1}/N_{2} and a ∈ γ_{2}(G). The following relations are needed below:

p_{2}s(x + y) = − p_{2}d(x, y) + p_{2}(sx) + p_{2}(sy) + p_{2}(sx)p_{2}(sy) by (5)  (20)
p_{2}(^{x}a) = p_{2}([sx, a]a) = p_{2}(a) by (5) and (8)  (21)
l̄(^{x}a) = l̄[sx, a] + l̄(a)
= l̄(a) + c̄l(p_{2}(sx) ⊗ p_{2}(a))
= l̄(a) + c̄j_{2}(x ⊗ p_{2}(a)) − c̄j_{1}(p_{2}(a) ⊗ x) by (5) and (6) (22)
[sx, sy] = d(x, y) − d(y, x)  (23)

To check symmetry of D_{3} we calculate

l̄(d(x, y) − d(y, x)) =^{(23)} l̄[sx, sy] = c̄l(sx, sy)
= c̄(p_{2}(sx) ⊗ p_{2}(sy) − p_{2}(sy) ⊗ p_{2}(sx) +
− j_{1}([p_{2}(sx), p_{2}(sy)] ⊗ (x + y)))
= c̄(p_{2}(sx) ⊗ p_{2}(sy) − p_{2}(sy) ⊗ p_{2}(sx) +
− j_{1}(p_{2}[sx, sy] ⊗ (x + y)))
= c̄(p_{2}(sx) ⊗ p_{2}(sy) − p_{2}(sy) ⊗ p_{2}(sx) +
− j_{1}((p_{2}d(x, y) − p_{2}d(y, x)) ⊗ (x + y)))

Equality of the first and last term in this calculation is equivalent with the identity D_{3}(x, y) − D_{3}(y, x) = 0.

Next we check the cocycle condition for D_{3}:

δD_{3}(x, y, z) = D_{3}(y, z) − D_{3}(x + y, z) + D_{3}(x, y + z) − D_{3}(x, y)

=^{(20)} l̄d(y, z) − c̄(p_{2}(sy) ⊗ p_{2}(sz)) + c̄j_{1}(p_{2}d(y, z) ⊗ (y + z)) +

− l̄d(x + y, z) − c̄j_{1}(p_{2}d(x, y) ⊗ z) + c̄(p_{2}(sx) ⊗ p_{2}(sz)) + c̄(p_{2}(sy) ⊗ p_{2}(sz)) +

+ c̄(p_{2}(sx)p_{2}(sy)) ⊗ p_{2}(sz)) − c̄j_{1}(p_{2}d(x + y, z) ⊗ (x + y + z)) +

+ l̄d(x, y + z) + c̄j_{2}(x ⊗ p_{2}d(y, z)) − c̄(p_{2}(sx) ⊗ p_{2}(sy)) +

+ c̄( − p_{2}(sx) ⊗ p_{2}(sz) − p_{2}(sx) ⊗ (p_{2}(sy)p_{2}(sz))) +

+ c̄j_{1}(p_{2}d(x, y + z) ⊗ (x + y + z)) +

− l̄d(x, y) + c̄(p_{2}(sx) ⊗ p_{2}(sy)) − c̄j_{1}(p_{2}d(x, y) ⊗ (x + y))

=^{(22)} l̄(^{x}d(y, z)) + c̄j_{1}(p_{2}d(y, z) ⊗ x) − c̄j_{2}(x ⊗ p_{2}d(y, z)) +

+ l̄( − d(x + y, z) + d(x, y + z) − d(x, y)) + c̄j_{1}(p_{2}d(y, z) ⊗ (y + z)) +

+ c̄j_{1}(p_{2}(−d(x + y, z) + d(x, y + z) − d(x, y)) ⊗ (x + y + z)) +

+ c̄j_{2}(x ⊗ p_{2}d(y, z))
= l̄δd(x, y, z) + c̄j_{1}(p_{2}δd(x, y, z) ⊗ (x + y + z))
= 0

The last two equations are implied by the cocycle condition for d.

In order to check (19) we also use the relation

t(ab) = ^{a}t(b)t(a)d(ā, b̄)

with x̄ = xγ_{2}(G) ∈ N_{1}/N_{2}. Now calculate

pr_{1}f(ab) =^{(22),(21)} l̄(tb) + c̄j_{2}(ā ⊗ p_{2}(tb)) − c̄j_{1}(p_{2}(tb) ⊗ ā) +

+ l̄(ta) + l̄d(ā, b̄) + c̄j_{1}((p_{2}(tb) + p_{2}(ta) + p_{2}d(ā, b̄)) ⊗ (ā + b̄))

= pr_{1}f(a) + pr_{1}f(b) + c̄j_{1}(p_{2}(ta) ⊗ b̄ + p_{2}d(ā, b̄) ⊗ (ā + b̄)) +
+ c̄j_{2}(ā ⊗ p_{2}(tb)) + l̄d(ā, b̄)
=^{(18)} pr_{1}(fa + fb) + c̄(p_{2}(sā) ⊗ p_{2}(sb̄)) + c̄(p_{2}(ta) ⊗ p_{2}(sb̄)) +
+ c̄(p_{2}(sā) ⊗ p_{2}(tb))
=^{(5)} pr_{1}(fa + fb) + c̄(p_{2}((ta)(sā)) ⊗ p_{2}(sb̄)) + c̄(p_{2}((ta)(sā)) ⊗ p_{2}(tb)) +
− c̄(p_{2}(ta) ⊗ p_{2}(tb))
= pr_{1}(fa + fb) + c̄(p_{2}(a) ⊗ p_{2}(b))

since p_{2}(ta) ⊗ p_{2}(tb) = 0 in P_{2}(G) ⊗_{G} P_{2}(G). Thus the theorem is proven. □

Proof of 3.1: The isomorphism μ̄: cok(c_{3}, −l_{3}) ≅ Δ_{N}^{2} (G)/Δ_{N}^{4} (G) induces an isomorphism between

N_{2}/N_{3}⊕N_{1}/N_{2}⊗N_{1}/N_{2} ≅ B/(l̄(N_{3}/N_{4})+c̄(Im(Δ_{N}^{2} (G) ⊗ Δ_{N}^{1} (G) + Δ_{N}^{1} (G) ⊗ Δ_{N}^{2} (G))))

and (Δ_{N}^{2} (G)/Δ_{N}^{4} (G))/(Δ_{N}^{3} (G)/Δ_{N}^{4} (G)) ≅ (Δ_{N}^{2} (G)/Δ_{N}^{3} (G)). This coincides with μ̄_{2} by definition. □

4 Proof of theorem 1.1

The kernel of θ_{3}^{N} is determined as kernel of the composition

U_{3}L^{N} (G) ⟶^{θ_{3}^{N}} Δ_{N}^{3} (G)/Δ_{N}^{4} (G) ↪^{i} Δ_{N}^{2} (G)/Δ_{N}^{4} (G) ≅ cok(c_{3}, −l_{3})^{t})

where the latter isomorphism was obtained in 3.5.

We abbreviate U_{i} = U_{i}L^{N} (G) . Consider the subgroups V = U_{2}U_{1} + U_{1}U_{2} ⊂ U_{3}L^{N} (G) and W = Im(Δ_{N}^{2} (G) ∩ I(G)^{2} ↪ Δ_{N}^{2} (G) → → Δ_{N}^{2} (G)/Δ_{N}^{4} (G)) of decomposables. Note that restricting the isomorphism μ̄_{2} : B ≅ Δ_{N}^{2} (G)/Δ_{N}^{4} (G) in 3.5 yields an isomorphism Im(c̄) ≅ W. Thus we obtain a commutative diagram

Im(c̄) ⟶^{>}^{μ̄_{3}} Δ_{N}^{2} (G)/Δ_{N}^{4} (G) ⟶  cok(c̄)  ≅  N_{2}/(N_{4}γ_{2}(N_{1}))
↑^{g}  ↑^{iθ_{3}^{N}}  ↑^{h}  ↑^{j}
V  ⟶^{>}  U_{3}L^{N} (G)  ⟶ L_{3}^{N} (G)/(L_{3}^{N} (G) ∩ V) \overleftarrow{⟵}^{π} N_{3}/(N_{4}(N_{3} ∩ γ_{2}(N_{1})))

Here g and h are induced by iθ_{3}^{N} , j is induced by the inclusion N_{3} ↪ N_{2} and π by the quotient map N_{3} → → L_{3}^{N} (G) . Since j is injective so are π an h. From exactness of the rows in the second column it now follows that

Ker(θ_{3}^{N}) = Ker(iθ_{3}^{N}) = Ker(g) .  (24)

Let B_{2} = cok(c_{2}, −l_{2})^{t}, see 3.1, with the canonical maps

N_{2}/N_{3} ⟶^{l̄_{2}} B_{2} ⟵^{c̄_{2}} A ⊗ A

Consider the following homomorphisms.

V \overleftarrow{⟵}^{m} N_{2}/N_{3} ⊗ A ⊕ A ⊗ B_{2} ⟶^{ϕ} \frac{P_{2}^{N} (G) ⊗ P_{2}^{N} (G)}{Im(μ̄_{2}) ⊗ Im(μ̄_{2})} ⟶^{q} \frac{P_{2}^{N} (G) ⊗_{G} P_{2}^{N} (G)}{Im(μ̄_{2}) ⊗ Im(μ̄_{2})}

Here m is defined as the composition

N_{2}/N_{3} ⊗ A ⊕ A ⊗ B_{2} ⟶^{l̄_{2}⊗A⊕id} B_{2} ⊗ A ⊕ A ⊗ B_{2} ⟶^{ϵ} U_{2} ⊗ U_{1} ⊕ U_{1} ⊗ U_{2} ⟶^{(q_{1},q_{2})} V

with ϵ = (θ_{2}^{N})^{−1}μ̄_{2} ⊗ (θ_{1}^{N})^{−1} ⊕ (θ_{1}^{N})^{−1} ⊗ (θ_{2}^{N})^{−1}μ̄_{2}.

The map ϕ is well defined by ϕ = (ϕ_{1}, ϕ_{2}) with ϕ_{1} = p ⊗ p_{2}s, ϕ_{2} = p_{2}s ⊗ μ̄_{2}, and q is the canonical quotient map. Since gm = c̄qϕ we have

Ker(g) = m Ker(c̄qϕ)  (25)

The kernel of c̄qϕ is now determined in three steps according to the three factors.

For this we need the following exact sequences with appropriate connecting homomorphisms τ_{i} which are deduced from the exact sequences in 3.1 and 3.3. Here let SP^{2}(A) = cok(l_{2}) with quotient map σ : A ⊗ A→ →S^{2}A .

P_{2}^{N} (G) ∗ A ⟶^{τ_{3}} P_{2}^{N} (G) ⊗ B_{2} ⟶^{id⊗μ̄_{2}} P_{2}^{N} (G) ⊗ P_{2}^{N} (G) ⟶^{id⊗ρ} P_{2}^{N} (G) ⊗ A → 0 (26)
P_{2}(A) ∗ A ⟶^{τ_{4}} P_{2}(A) ⊗ B_{2} ⟶^{id⊗μ̄_{2}} P_{2}(A) ⊗ P_{2}^{N} (G) ⟶^{id⊗ρ} P_{2}(A) ⊗ A → 0 (27)
P_{2}(A) ∗ A ⟶^{τ_{5}} N_{2}/N_{3} ⊗ A ⟶^{p⊗A} P_{2}^{N} (G) ⊗ A ⟶^{P_{2}(α)⊗A} P_{2}(A) ⊗ A → 0  (28)
A ∗ A ⟶^{τ_{6}} S^{2}(A) ⊗ A ⟶^{μ_{2}⊗A} P_{2}(A) ⊗ A ⟶^{ρ⊗A} A ⊗ A → 0  (29)

We claim that the following sequence is exact.

P_{2}^{N} (G) ∗ A ⊕ P_{2}(A) ∗ A ⟶^{τ} N_{2}/N_{3} ⊗ A ⊕ A ⊗ B_{2} ⟶^{ϕ} \frac{P_{2}^{N} (G) ⊗ P_{2}^{N} (G)}{Im(μ̄_{2}) ⊗ Im(μ̄_{2})} (30)

with

τ = ( \begin{matrix}0 & − τ_{5} \\ (ρ ⊗ B_{2})τ_{3} & (ρ ⊗ B_{2})τ_{4} \end{matrix} )

To see this observe first that sequence (26) induces another exact sequence

P_{2}^{N} (G) ∗ A ⟶^{(ρ⊗B_{2})τ_{3}} A ⊗ B_{2} ⟶^{ϕ_{2}} \frac{P_{2}^{N} (G) ⊗ P_{2}^{N} (G)}{Im(μ̄_{2}) ⊗ Im(μ̄_{2})} ⟶^{id⊗ρ} P_{2}^{N} (G) ⊗ A  (31)
Hence ϕ_{1}^{−1}Im(ϕ_{2}) = Ker((id ⊗ ρ)ϕ_{1}) = Ker(p ⊗ A) = Im(τ_{5}) .  (32)

Now let ⟨u, k, x⟩ be a typical generator of P_{2}(A)∗A, i.e. u ∈ P_{2}(A), k ∈ ZZ, x ∈ A such that ku = 0 = kx, see [5] V.6. Choose u′ ∈ P_{2}^{N} (G) with P_{2}(α)(u′) = u. Then

ϕ_{1}τ_{5}⟨u, k, x⟩ = ku′ ⊗ p_{2}sx
= u′ ⊗ kp_{2}sx
≡^{3.1} p_{2}(sρu′) ⊗ kp_{2}sx mod Im(μ̄_{2}) ⊗ Im(μ̄_{2})
= ϕ_{2}(ρu′ ⊗ μ̄_{2}^{−1}(kp_{2}sx))
= ϕ_{2}(ρ ⊗ B_{2})τ_{4}⟨u, k, x⟩  (33)

since ρu′ = ρu. Together with (31) and (32) this shows exactness of sequence (30).

Next we study the group Ker(qϕ) = ϕ^{−1}Ker(q) . Consider the following commutative diagram of homomorphisms with exact rows

A ∧ A ⊗ A  ⟶^{l_{2}⊗A}  A ⊗ A ⊗ A  ⟶^{σ⊗A} SP^{2}(A) ⊗ A
↓^{ψ}  ↓^{β}  ↓^{μ_{2}⊗A}
N_{2}/N_{3} ⊗ A ⊕ A ⊗ B_{2} ⟶^{ϕ} \frac{P_{2}^{N} (G) ⊗ P_{2}^{N} (G)}{Im(μ̄_{2}) ⊗ Im(μ̄_{2})} ⟶^{P_{2}(α)⊗ρ} P_{2}(A) ⊗ A

Here ψ = (−c_{2} ⊗ A , (A ⊗ c̄_{2})(l_{2} ⊗ A))^{t} and β = p_{2}s ⊗ μ_{2} − μ_{2} ⊗ p_{2}s. Thus Ker(q) = Im(β) , and

Im(ϕ) ∩ Ker(q) = β Ker((μ_{2} ⊗ A)(σ ⊗ A))
= β(Im(l_{2}) ⊗ A) + β(σ ⊗ A)^{−1}Im(τ_{6})
= ϕ(Im(ψ)) + β(σ ⊗ A)^{−1}Im(τ_{6})  (34)

To make the maps τ_{i} explicit note that for x ∈ A with kx = 0 we have

kp_{2}(sx) = μ̄_{2}(l̄_{2}(s(x)^{k}) − \binom{k}{2}c̄_{2}(x ⊗ x))  (35)

by (5). Let ⟨x, k, y⟩ be a typical generator of A ∗ A, with x, y ∈ A, k ∈ ZZ such that kx = 0 = ky. Then

β(σ ⊗ A)^{−1}τ_{6}⟨x, k, y⟩
= β(σ ⊗ A)^{−1}(μ̄_{2}^{−1}(kp_{2}(x)) ⊗ y)
=^{(35)} β( −\binom{k}{2}x ⊗ x ⊗ y)
= \binom{k}{2}μ_{2}(x ⊗ x) ⊗ p_{2}(sy) − \binom{k}{2}p_{2}(sx) ⊗ μ_{2}(x ⊗ y) +
+ k p_{2}(sx) ⊗ p_{2}(sy) − k p_{2}(sx) ⊗ p_{2}(sy)
=^{(35)} p_{2}((sx)^{k}) ⊗ p_{2}(sy) − \binom{k}{2}p_{2}(sx) ⊗ μ_{2}(x ⊗ y) +
− p_{2}(sx) ⊗ (kp_{2}(sy))
= ϕ_{1}((sx)^{k} ⊗ y) − ϕ_{2}(\binom{k}{2}x ⊗ c̄_{2}(x ⊗ y) + τ_{2}⟨x, k, y⟩)  (36)

Now we are ready to compute mKer(qϕ) . By definition of the enveloping algebra one has m Im(ψ) = 0. For elements x, k, y as above we have

m((sx)^{k} ⊗ y , −\binom{k}{2}x ⊗ c̄_{2}(x ⊗ y) − τ_{2}⟨x, k, y⟩)
= q_{1}((sx)^{k} ⊗ y − \binom{k}{2}x^{2} ⊗ y) − q_{2}τ_{2}⟨x, k, y⟩)
=^{(35)} (q_{1}τ_{1} − q_{2}τ_{2})⟨x, k, y⟩ .

Thus by (34) and (36) one has

m Ker(qϕ) = Im(q_{1}τ_{1} − q_{2}τ_{2}) + mKer(ϕ) .  (37)

We assert that actually mKer(ϕ) ⊂ Im(q_{1}τ_{1} − q_{2}τ_{2}) . Recall that mKer(ϕ) = mIm(τ) by (30). Let ⟨w, k, x⟩ be a typical generator of P_{2}^{N} (G) ∗ A, with w ∈ P_{2}^{N} (G) ,k ∈ ZZ, x ∈ A, such that kw = 0 = kx. Then

m(0, (ρ ⊗ B_{2})τ_{3})^{t}⟨w, k, x⟩ = q_{2}τ_{2}⟨ρw, k, x⟩
= − (q_{1}τ_{1} − q_{2}τ_{2})⟨ρw, k, x⟩

since τ_{1}⟨ρw, k, x⟩ = ι^{−1}(kw) ⊗ x = 0. Now let u, k, x, u′ as in (33). Then ku′ ≡ kp_{2}(sρu) mod k Im(μ̄_{2}) by 3.1, whence

m(τ_{5}, −(ρ ⊗ B_{2})τ_{4})⟨u, k, x⟩ = m(p^{−1}(ku′) ⊗ x , −ρu ⊗ μ̄_{2}^{−1}(kp_{2}(sx)))
= q_{1}((θ_{2}^{N})^{−1}μ̄_{2} ⊗ (θ_{1}^{N})^{−1})(μ̄_{2}^{−1}(kp_{2}(sρu)) ⊗ x) +

− q_{2}τ_{2}⟨ρu, k, x⟩

= (q_{1}τ_{1} − q_{2}τ_{2})⟨ρu, k, x⟩

Thus m Ker(qϕ) = Im(q_{1}τ_{1} − q_{2}τ_{2}) .  (38)

In a last step we shall show that

m Ker(c̄qϕ) = m Ker(qϕ)  (39)

which finishes the proof by (24), (25) and (38).

First observe that

Ker(c̄qϕ) = (qϕ)^{−1}Ker(c̄) = (qϕ)^{−1}l_{3}Ker(c_{3}) .  (40)

Consider the following diagram with exact rows from the second column onwards. Here the map qϕ is welldefined by (34).

V  ⟵^{m} \frac{N_{2}/N_{3} ⊗ A ⊕ A ⊗ B_{2}}{Im(ψ)} ⟶^{qϕ} \frac{P_{2}^{N} (G) ⊗_{G} P_{2}^{N} (G)}{Im(μ̄_{2}) ⊗ Im(μ̄_{2})} ⟶^{ρ⊗ρ} A ⊗ A
↓  ↑_{l′}  ↑^{l_{3}}  ↑^{l_{2}}
U_{3}L^{N} (G)  B_{2} ⊗ P_{2}^{N} (G)  ⟶^{μ̄_{2}⊗id} P_{2}^{N} (G) ∧ P_{2}^{N} (G) ⟶^{ρ ∧ ρ} A ∧ A
↑  ↓_{c′}  ↓^{c_{3}}  ↓^{c_{2}}
L_{3}^{N} (G) =  N_{3}/N_{4}  ↪  N_{2}/N_{4}  ⟶ N_{2}/N_{3}

Here l′ and c′ are defined as follows. Let b ∈ N_{2}/N_{3}, x, y, z ∈ A, a ∈ G and ā = aN_{2} ∈ A. Write [x, [y, z]] = x ⊗ (y ⊗ z − z ⊗ y) − (y ⊗ z − z ⊗ y) ⊗ x in A ⊗ A ⊗ A. Then let

l′((l̄_{2}(b) + c̄_{2}(x ⊗ y)) ⊗ p_{2}(a)) = (b ⊗ ā , − ā ⊗ l̄_{2}(b) + (A ⊗ c̄_{2})[x, [y, ā]]) + Im(ψ)

c′((l̄_{2}(b) + c̄_{2}(x ⊗ y)) ⊗ p_{2}(a)) = [b, a][sx, [sy, a]]N_{4}

To check that l′ and c′ are well defined homomorphisms first observe that the triple commutator [−, [−, −]]N_{4} is trilinear and annihilates N_{2} in all variables. Now calculate

l′(c̄_{2}l_{2}(x ∧ y) ⊗ p_{2}(a)) = (0, − (A ⊗ c̄_{2})[ā, [x, y]]) by the Jacobian identity
= (0, − ā ⊗ c̄_{2}l_{2}(x ∧ y)) + ψ(x ∧ y ⊗ ā) + (c_{2}(x ∧ y) ⊗ ā , 0)
≡ (c_{2}(x ∧ y) ⊗ ā , − ā ⊗ l̄_{2}c_{2}(x ∧ y)) mod Im(ψ)
= l′(l̄_{2}c_{2}(x ∧ y) ⊗ p_{2}(a))

c′(c̄_{2}l_{2}(x ∧ y) ⊗ p_{2}(a)) = [sx, [sy, a]][sy, [sx, a]]^{−1}N_{4}
= [a, [sx, sy]]^{−1}N_{4} by the Jacobian identity
= [[sx, sy], a]N_{4}
= c′(l̄_{2}c_{2}(x ∧ y) ⊗ p_{2}(a))

We shall check below that the diagram commutes. Assuming this for the moment, we conclude from exactness of rows and from injectivity of l_{2} (cf. the remark preceding 3.3) that

Im(qϕ) ∩ Im(l_{3}) = Im(qϕl′)

whence mKer(c̄qϕ) =^{(40)} m(qϕ)^{−1}l_{3}Ker(c_{3})
= ml′Ker(c′) + mKer(qϕ)
= mKer(qϕ)

by commutativity of the left hand part of the diagram. Thus (39) and hence the theorem are proven when we have checked commutativity of the diagram. For the left and right hand part this is immediate from the definitions. For the bottom middle square recall from (9) and (17) that

c_{3}(μ̄_{2} ⊗ id)((l̄_{2}(b) + c̄_{2}(x ⊗ y)) ⊗ p_{2}(a)) = c_{3}((p_{2}(b) + p_{2}(sx)p_{2}(sy)) ⊗ p_{2}(a))
= [b, a] w_{c(−,a)}(sx, sy)
= [b, a] [sx, [sy, a]]N_{4}
= c′((l̄_{2}(b) + c̄_{2}(x ⊗ y)) ⊗ p_{2}(a))

Commutativity of the upper middle square is checked on the generating subgroups Im(l̄_{2}) ⊗ P_{2}^{N} (G) and Im(c̄_{2}) ⊗ P_{2}^{N} (G) separately.

qϕl′(l̄_{2}(b) ⊗ p_{2}(a)) = p_{2}(b) ⊗ p_{2}(sā) − p_{2}(sā) ⊗ p_{2}(b)
=^{3.1} p_{2}(b) ⊗ p_{2}(a) − p_{2}(a) ⊗ p_{2}(b)
= l(b, a) since [p_{2}(b), p_{2}(a)] ∈ Δ_{N}^{3} (G)
= l_{3}(p_{2}(b) ∧ p_{2}(a))
= l_{3}(μ̄_{2} ⊗ id)(l̄_{2}(b) ⊗ p_{2}(a))

Write g = sx and h = sy.

l_{3}(μ_{2}q ⊗ id)(c̄_{2}(x ⊗ y) ⊗ p_{2}(a)) =^{(5)} l((p_{2}(gh) − p_{2}(g) − p_{2}(h)) ⊗ p_{2}(a))

= l(gh, a) − l(g, a) − l(h, a)
=^{(16),(5)} p_{2}(g)p_{2}(h) ⊗ p_{2}(a) − p_{2}(a) ⊗ p_{2}(g)p_{2}(h) +
− [p_{2}(g) + p_{2}(h) , p_{2}(a)] ⊗ (p_{2}(g) + p_{2}(h) + p_{2}(a)) +

+ [p_{2}(g) , p_{2}(a)] ⊗ (p_{2}(g) + p_{2}(a)) + [p_{2}(h) , p_{2}(a)] ⊗ (p_{2}(h) + p_{2}(a))

= p_{2}(g)p_{2}(h) ⊗ p_{2}(a) − p_{2}(a) ⊗ p_{2}(g)p_{2}(h) +
− [p_{2}(g) , p_{2}(a)] ⊗ p_{2}(h) − [p_{2}(h) , p_{2}(a)] ⊗ p_{2}(g)
= p_{2}(g)p_{2}(h) ⊗ p_{2}(a) − p_{2}(g)p_{2}(a) ⊗ p_{2}(h) − [p_{2}(h) , p_{2}(a)] ⊗ p_{2}(g)
=^{(∗)} p_{2}(g) ⊗ [p_{2}(h) , p_{2}(sā)] − [p_{2}(h) , p_{2}(sā)] ⊗ p_{2}(g)

= p_{2}(g) ⊗ [p_{2}(h) , p_{2}(sā)] − p_{2}(h) ⊗ p_{2}(sā)p_{2}(g) + p_{2}(sā) ⊗ p_{2}(h)p_{2}(g)
= p_{2}(g) ⊗ μ_{2}l_{2}(y ∧ ā) − p_{2}(h) ⊗ μ̄_{2}(ā ⊗ x) + p_{2}(sā) ⊗ μ_{2}(y ⊗ x)
= qϕ_{2}(A ⊗ c̄_{2})[x, [y, ā]]
= qϕl′(c̄_{2}(x ⊗ y) ⊗ p_{2}(a))

where equation (∗) is implied by the relation p_{2}(a) − p_{2}(sā) ∈ Im(μ̄_{2}) as follows from 3.1. Thus the theorem is proven. □

Acknowledgments

I would like to express my gratitude to the Sonderforschungsbereich 343 in Bielefeld and the Human Capital and Mobility programme of the European Union for awarding researchs positions to me which enables me to persue freely my work programme. Moreover, I am grateful to Professor H. Abels in Bielefeld and Professors J.-L. Loday and C. Kassel in Strasbourg for creating a friendly and stimulating atmosphere and for many interesting discussions.

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