Homology, Homotopy and Applications, vol.4(1), 2002, pp.43–57

SIMPLICIAL AND CROSSED LIE ALGEBRAS

İ. AKÇA and Z. ARVASI

(communicated by Ronald Brown)

Abstract In this paper we examine higher order Peiffer elements in simplicial Lie algebras and apply them to the Lie 2-crossed module and Lie crossed squares introduced by Ellis.

Introduction

Simplicial Lie algebras arise in simplicial homotopical algebra, [6], [13], and their homotopy theory is of considerable interest. Kassel and Loday [10] (see also [11] and [12]) introduced crossed modules of Lie algebras as computational algebraic objects equivalent to simplicial Lie algebras with associated Moore complex of length 1. Following work of Conduché in a group-theoretic setting, Ellis [9] captured the algebraic structure of a Moore complex of length 2 in his definition of a 2-crossed module of Lie algebras. Within the homotopy theory of simplicial Lie algebras, analogues of Samelson and Whitehead products are given by sums over shuffles (a; b) of Lie products of the form [s_{b}x, s_{a}y]. In this paper we explain the relationship of these shuffles to crossed modules and crossed 2-modules. More precisely, let L be a simplicial Lie algebra with Moore complex NL. Let ∂ denote the boundary homomorphism of the Moore complex. For n > 1 let D_{n} be the ideal in L_{n} generated by the degenerate elements. We show in Proposition 2.3 that if L_{n} = D_{n}, then

NL_{n} = I_{n}

where I_{n} is an ideal in L_{n} generated by certain shuffles. We use this equality to prove the following theorem (in which the face homomorphisms of the simplicial Lie algebra L are denoted by d_{i}, and for I ⊆ {1, …, n} the intersection ∩_{i∈I}Kerd_{i} is denoted by K_{I}). Theorem 1. Let L be a simplicial Lie algebra.

(i) L_{2} = D_{2} then ∂_{2}(NL_{2}) = [Kerd_{0}, Kerd_{1}]. (ii) L_{3} = D_{3} then

∂_{3}(NL_{3}) = [K_{{0,1}}, K_{{0,2}}] + [K_{{0,2}}, K_{{1,2}}] + [K_{{0,1}}, K_{{1,2}}] + ∑_{I,J}[K_{I}, K_{J}].

Received October 11, 2001, revised April 2, 2002; published on May 8, 2002. 2000 Mathematics Subject Classification: 18G50, 18G55. Key words and phrases: Simplicial Lie Algebra, Peiffer Elements, 2-Crossed Modules. \begin{matrix}c \\ ◯ \end{matrix} 2002, İ. Akça and Z. Arvasi. Permission to copy for private use granted.

where I ∪ J = {0, 1, 2} , I ∩ J = ∅. (iii) If L_{n} = D_{n} then

∂_{n}(NL_{n}) ⊇ ∑_{I,J}[K_{I}, K_{J}]

where I, J are nonempty subsets of {0, …, n − 1} with I ∪ J = {0, …, n − 1} . Part (i) of this theorem is an analogue of a group-theoretic result of Brown and Loday [2]

The paper is organised as follows. In Section 1 we recall some basics on simplicial Lie algebras and crossed modules. In Section 2 we prove Proposition 2.3. In Section 3 we prove Theorem 1. In Section 4 we use Theorem 1 to construct a functor from simplicial Lie algebras to 2-crossed modules; we also explain how Theorem 1 yields a functor from simplicial Lie algebras to the crossed n-cubes of Lie algebras introduced in [8].

1. Review of simplicial Lie algebras

All Lie algebras will be over a fixed commutative ring k.

A simplicial Lie algebra [6] L is a sequence of Lie algebras,

L = {L_{0}, L_{1}, … L_{n}, …},

together with face and degeneracy maps

\begin{matrix}d_{i} = d_{i}^{n} : & L_{n} → L_{n−1}, & 0 ⩽ i ⩽ n (n ≠ 0) \\ s_{i} = s_{i}^{n} : & L_{n} → L_{n+1}, & 0 ⩽ i ⩽ n. \end{matrix}

These maps are required to satisfied the simplicial identities

d_{i}d_{j} = d_{j−1}d_{i}  for i < j
d_{i}s_{j} = \{ \begin{matrix}s_{j−1}d_{i} & for & i < j \\ identity & for & i = j, j + 1 \\ s_{j}d_{i−1} & for & i > j + 1 \end{matrix}
s_{i}s_{j} = s_{j+1}s_{i}  for i ⩽ j.

It can be completely described as a functor L: Δ^{op} →LieAlg_{k} where Δ is the category of finite ordinals [n] = {0 < 1 < ⋯ < n} and increasing maps.

Elements x ∈ L_{n} are called n-dimensional simplices. A simplex x is called degenerate if x = s_{i}(y) for some y.

A simplicial map f : L → L′ is a family of homomorphisms f_{n} : L_{n} → L′_{n} commuting with the d_{i} and s_{i}. We let SLA denote the category of simplicial Lie algebras.

An essential reference from our point of view is Carrasco’s thesis, [3], where many of the basic techniques used here were developed systematically for the first time and the notion of hypercrossed complex was defined (although in a different context).

The following notation and terminology is derived from the analogous group theoretic case treated in [3], [4]. For the ordered set [n] = {0 < 1 < … < n}, let α_{i}^{n} : [n + 1] → [n] be the increasing surjective map given by

α_{i}^{n}(j) = \{ \begin{matrix}j & if & j ⩽ i \\ j − 1 & if & j > i. \end{matrix}

Let S(n, n − r) be the set of all monotone increasing surjective maps from [n] to [n−r]. This can be generated from the various α_{i}^{n} by composition. The composition of these generating maps is subject to the following rule α_{j}α_{i} = α_{i−1}α_{j}, j < i. This implies that every element α ∈ S(n, n − r) has a unique expression as α = α_{i_{1}} ∘ α_{i_{2}} ∘ … ∘ α_{i_{r}} with 0 ⩽ i_{1} < i_{2} < … < i_{r} ⩽ n − 1, where the indices i_{k} are the elements of [n] such that {i_{1}, …, i_{r}} = {i : α(i) = α(i + 1)}. We thus can identify S(n, n − r) with the set {(i_{r}, …, i_{1}) : 0 ⩽ i_{1} < i_{2} < … < i_{r} ⩽ n − 1}. In particular, the single element of S(n, n), defined by the identity map on [n], corresponds to the empty 0-tuple ( ) denoted by ∅_{n}. Similarly the only element of S(n, 0) is (n − 1, n − 2, …, 0). For all n ⩾ 0, let

S(n) = ⋃_{0⩽r⩽n} S(n, n − r).

We say that α = (i_{r}, …, i_{1}) < β = (j_{s}, …, j_{1}) in S(n)

if i_{1} = j_{1}, …, i_{k} = j_{k} but i_{k+1} > j_{k+1} (k ⩾ 0) or
if i_{1} = j_{1}, …, i_{r} = j_{r} and r < s.

This makes S(n) an ordered set. For instance, the orders in S(2) and in S(3) are respectively:

\begin{aligned}S(2) & = & \{∅_{2} < (1) < (0) < (1, 0)\}; \\ S(3) & = & \{∅_{3} < (2) < (1) < (2, 1) < (0) < (2, 0) < (1, 0) < (2, 1, 0)\}. \end{aligned}

We also define α ∩ β as the set of indices which belong to both α and β.

The Moore complex The Moore complex NL of a simplicial Lie algebra L is the complex

NL : ⋯ → NL_{n} →^{∂_{n}} NL_{n−1} → ⋯ → NL_{1} →^{∂_{1}} NL_{0} →^{∂_{0}} 0

where

NL_{0} = L_{0},  NL_{n} = ⋂_{i=0}^{n−1} Kerd_{i},  ∂_{n} = d_{n} (restricted to NL_{n}).

Truncated Simplicial Lie Algebras

By an m-truncated Simplicial Lie Algebra, we mean a collection of Lie algebras {L_{0}, …, L_{m}} and homomorphisms d_{i} : L_{n} → L_{n−1} for 0 ⩽ i ⩽ n, 0 ⩽ n ⩽ m and s_{i} : L_{n} → L_{n+1} for 0 ⩽ i ⩽ n, 0 ⩽ n ⩽ m − 1 which satisfy the simplicial identities.

The Semidirect Decomposition of a Simplicial Lie Algebra

The fundamental idea behind this can be found in Conduché [5]. A detailed investigation of it for the case of a simplicial group is given in Carrasco and Cegarra [4]. The algebra case of that structure is also given in Carrasco’s thesis [3].

Proposition 1.1. If L is a Simplicial Lie Algebra, then for any n ⩾ 0

\begin{aligned}L_{n} & ≅ & (… (NL_{n} ⋊ s_{n−1}NL_{n−1}) ⋊ … ⋊ s_{n−2} … s_{0}NL_{1})⋊ \\ & & (… (s_{n−2}NL_{n−1} ⋊ s_{n−1}s_{n−2}NL_{n−2}) ⋊ … ⋊ s_{n−1}s_{n−2} … s_{0}NL_{0}). \end{aligned}

Proof: This is by repeated use of the following lemma. □

Lemma 1.2. Let L be a Simplicial Lie Algebra. Then L_{n} can be decomposed as a semidirect product:

L_{n} ≅ Kerd_{n}^{n} ⋊ s_{n−1}^{n−1}(L_{n−1}).

Proof: The isomorphism is defined as follows:

\begin{aligned}θ : & L_{n} & ⟶ & Kerd_{n}^{n} ⋊ s_{n−1}^{n−1}(L_{n−1}) \\ & l & ⟼ & (l − s_{n−1}d_{n}l, s_{n−1}d_{n}l). \end{aligned}

□

The bracketting and the order of terms in this multiple semidirect product are generated by the sequence:

L_{1} ≅ NL_{1} ⋊ s_{0}NL_{0}
L_{2} ≅ (NL_{2} ⋊ s_{1}NL_{1}) ⋊ (s_{0}NL_{1} ⋊ s_{1}s_{0}NL_{0})
L_{3} ≅ ((NL_{3} ⋊ s_{2}NL_{2}) ⋊ (s_{1}NL_{2} ⋊ s_{2}s_{1}NL_{1}))⋊
((s_{0}NL_{2} ⋊ s_{2}s_{0}NL_{1}) ⋊ (s_{1}s_{0}NL_{1} ⋊ s_{2}s_{1}s_{0}NL_{0})).

and

L_{4} ≅ (((NL_{4} ⋊ s_{3}NL_{3}) ⋊ (s_{2}NL_{3} ⋊ s_{3}s_{2}NL_{2}))⋊
((s_{1}NL_{3} ⋊ s_{3}s_{1}NL_{2}) ⋊ (s_{2}s_{1}NL_{2} ⋊ s_{3}s_{2}s_{1}NL_{1})))⋊
s_{0}(decomposition of L_{3}).

Note that the term corresponding to α = (i_{r}, …, i_{1}) ∈ S(n) is

s_{α}(NL_{n−#α}) = s_{i_{r}...i_{1}}(NL_{n−#α}) = s_{i_{r}}...s_{i_{1}}(NL_{n−#α}),

where #α = r. Hence any element x ∈ L_{n} can be written in the form

x = y + ∑_{α∈S(n)} s_{α}(x_{α}) with y ∈ NL_{n} and x_{α} ∈ NL_{n−#α}.

Crossed Modules of Lie Algebras The notion of crossed module of Lie algebras was defined by Kassel and Loday

Let M and P be two Lie algebras. By an action of P on M we mean a k-bilinear map P × M → M, (p, m) ↦ p · m satisfying

\begin{aligned}[p, p′] · m & = & p · (p′ · m) − p′(p · m) \\ p · [m, m′] & = & [p · m, m′] + [m, p · m′] \end{aligned}

for all m, m′ ∈ M, p, p′ ∈ P. For instance, if P is a subalgebra of some Lie algebra Q (including possibly the case P = Q), and if M is an ideal in Q, then Lie multiplication in Q yields an action of P on M.

Suppose that M and N are Lie algebras with an action of M on N and action of N on M. For any Lie algebra Q we call a bilinear function h : M × N → Q a Lie pairing [7] if

\begin{aligned}h([m, m′], n) & = & h(m, m′ · n) − h(m′, m · n), \\ h(m, [n, n′]) & = & h(n′ · m, n) − h(n · m, n′), \\ h(n · m, m′ · n′) & = & −[h(m, n), h(m′, n′)], \end{aligned}

for all m, m′ ∈ M, n, n′ ∈ N. For example if M and N are both ideals of some Lie algebra then the function M × N → M ∩ N, (m, n) ↦ [m, n] is a Lie pairing.

Recall from [10] the notion of a crossed module of Lie algebras. A crossed module of Lie algebras is a Lie homomorphism ∂ : M → P together with an action of P on M such that ,

CM1) ∂(p · m) = [p, ∂m] CM2) ∂m · m′ = [m, m′]

for all m, m′ ∈ M, p ∈ P.

The second condition (CM2) is called the Peiffer identity. A standard example of a crossed module is any ideal I in P giving an inclusion map I → P, which is a crossed module. Conversely, given any crossed module ∂ : M → P, the image I = ∂M of M is an ideal in P.

2. Hypercrossed Complex Pairings and Boundaries in the Moore Complex

Lemma 2.1. Let L be a simplicial Lie algebra and let \overline{NL}_{n}^{(r)} = ⋂n ; i≠r^{i=0} Kerd_{i} for 0 ⩽ r ⩽ n. Then the mapping

φ : NL_{n} ⟶ \overline{NL}_{n}^{(r)}

in L_{n}, given by

φ(l) = l − ∑_{k=0}^{n−r−1}(−1)^{k+1}s_{r+k}d_{n}l,

is a k-linear isomorphism. □

This easily implies:

Lemma 2.2.

d_{n}(NL_{n}) = d_{r}(\overline{NL}_{n}^{(r)}).

□

Proof of Theorem 1 (iii): For any J ⊂ [n−1], J ≠ ∅, let r be the smallest element of J. If r = 0, then replace J by I and restart and if 0 ∈ I ∩ J, then re-define r to be the smallest nonzero element of J. Otherwise continue.

Letting l_{0} ∈ ⋂_{j∈J} Kerd_{j} and l_{1} ∈ ⋂_{i∈I} Kerd_{i}, one obtains

d_{i}[s_{r−1}l_{0}, s_{r}l_{1}] = 0 for i ≠ r

and hence [s_{r−1}l_{0}, s_{r}l_{1}] ∈ \overline{NL}_{n}^{(r)}. It follows that

[l_{0}, l_{1}] = d_{r}[s_{r−1}l_{0}, s_{r}l_{1}] ∈ d_{r}(\overline{NL}_{n}^{(r)}) = d_{n}NL_{n} by the previous lemma,

and this implies

[ ⋂_{i∈I} Kerd_{i}, ⋂_{j∈J} Kerd_{j}] ⊆ ∂_{n}NL_{n}.

Hypercrossed complex pairings

We recall from Carrasco [3] the construction of a family of k-linear morphisms. This was done there for associative algebras but adapts well to the Lie context. We define a set P (n) consisting of pairs of elements (α, β) from S(n) with α∩β = ∅, (for the definition of α ∩ β, see section 1), where α = (i_{r}, …, i_{1}), β = (j_{s}, ..., j_{1}) ∈ S(n). The k-linear morphisms that we will need,

{M_{α,β} : NL_{n−#α} × NL_{n−#β} ⟶ NL_{n} : (α, β) ∈ P (n), n ⩾ 0}

are given as composites by the diagrams

\xymatrix{NL_{n−#α} × NL_{n−#β} \ar[d]_{s_{α}×s_{β}} \ar[r]^{M_{α,β}} & NL_{n} \\ L_{n} × L_{n} \ar[r]_{[ , ]} & L_{n} \ar[u]_{p}}

where

s_{α} = s_{i_{r}} … s_{i_{1}} : NL_{n−#α} ⟶ L_{n} , s_{β} = s_{j_{s}} … s_{j_{1}} : NL_{n−#β} ⟶ L_{n},

p : L_{n} → NL_{n} is defined by composite projections p = p_{n−1} … p_{0}, where

p_{j} = 1 − s_{j}d_{j} with j = 0, 1, … n − 1

and we denote the Lie bracket by [ , ] : L_{n} × L_{n} → L_{n}. Thus

M_{α,β}(x_{α}, y_{β}) = p[ , ](s_{α} × s_{β})(x_{α}, y_{β})
= p([s_{α}(x_{α}), s_{β}(y_{β})])
= (1 − s_{n−1}d_{n−1}) … (1 − s_{0}d_{0})([s_{α}(x_{α}), s_{β}(y_{β})]).

We now define the ideal I_{n} to be that generated by all elements of the form

M_{α,β}(x_{α}, y_{β})

where x_{α} ∈ NL_{n−#α} and y_{β} ∈ NL_{n−#β} and for all (α, β) ∈ P (n).

Consider M_{α,β}(x_{α}, y_{β}) and M_{β,α}(y_{β}, x_{α}), here one uses [s_{α}(x_{α}), s_{β}(y_{β})], the other

[s_{β}(y_{β}), s_{α}(x_{α})] = −[s_{α}(x_{α}), s_{β}(y_{β})],

so the changing α and β gives the only minus sign.

Example For n = 2, suppose α = (1), β = (0) and x, y ∈ NL_{1} = Kerd_{0}. It follows that

M_{(1)(0)}(x, y) = p_{1}p_{0}[s_{1}x, s_{0}y]
= [s_{1}x, s_{0}y] − [s_{1}x, s_{1}y]
= [s_{1}x, s_{0}y − s_{1}y]

and these give the generator elements of the ideal I_{2}.

For n = 3, the linear morphisms are the following

\begin{matrix}M_{(1,0)(2)}, & M_{(2,0)(1)}, & M_{(2,1)(0)}, \\ M_{(2)(0)}, & M_{(2)(1)}, & M_{(1)(0)}. \end{matrix}

For all x ∈ NL_{1}, y ∈ NL_{2}, the corresponding generators of I_{3} are:

\begin{aligned}M_{(1,0)(2)}(x, y) & = & [s_{1}s_{0}x − s_{2}s_{0}x, s_{2}y], \\ M_{(2,0)(1)}(x, y) & = & [s_{2}s_{0}x − s_{2}s_{1}x, s_{1}y − s_{2}y], \\ M_{(2,1)(0)}(x, y) & = & [s_{2}s_{1}x, s_{0}y − s_{1}y + s_{2}y]; \end{aligned}

whilst for all x, y ∈ NL_{2},

\begin{aligned}M_{(1)(0)}(x, y) & = & [s_{1}x, s_{0}y − s_{1}y] + [s_{2}x, s_{2}y], \\ M_{(2)(0)}(x, y) & = & [s_{2}x, s_{0}y], \\ M_{(2)(1)}(x, y) & = & [s_{2}x, s_{1}y − s_{2}y]. \end{aligned}

Proposition 2.3. Let L be a simplicial Lie algebra and n > 0, and D_{n} the ideal in L_{n} generated by degenerate elements. We suppose L_{n} = D_{n}, and let I_{n} be ideal generated by elements of the form

M_{α,β}(x_{α}, y_{β}) with (α, β) ∈ P (n)

where x_{α} ∈ NL_{n−#α}, y_{β} ∈ NL_{n−#β}. Then

NL_{n} = I_{n}.

As corollary we, of course, have that the image of N_{n} is equal to the image of NL_{n}, i.e., ∂_{n}(NL_{n}) = ∂_{n}(I_{n}).

We omit its proof which can be obtained by changing slightly the corresponding results in [1]

3. Proof of first two parts of Theorem 1

Proof of Theorem 1 (i): We know that any element l_{2} of L_{2} can be expressed in the form

l_{2} = b + s_{1}y + s_{0}x + s_{0}u

with b ∈ NL_{2}, x, y ∈ NL_{1} and u ∈ s_{0}L_{0}. We suppose D_{2} = L_{2}. For n = 1, we take α = (1), β = (0) and x, y ∈ NL_{1} =Kerd_{0}. The ideal I_{2} is generated by elements of the form

M_{(1)(0)}(x, y) = [s_{1}x, s_{0}y − s_{1}y].

The image of I_{2} by ∂_{2} is known to be [[Kerd_{0}, Kerd_{1}]] by direct calculation. Indeed,

\begin{aligned}d_{2}[M_{(1)(0)}(x, y)] & = & d_{2}[s_{1}x, s_{0}y − s_{1}y] \\ & = & [x, s_{0}d_{1}y − y] \end{aligned}

where x ∈ Kerd_{0} and [x, s_{0}d_{1}y − y] ∈ Kerd_{1} and all elements of Kerd_{1} have this form due to Lemma 2.1.

As ∂ = ∂_{1} restricted to NL_{1}, this is precisely d_{2}(M_{(1)(0)}(x, y)). In other words the ideal ∂I_{2} is the ‘Peiffer ideal’ of the precrossed module ∂ : NL_{1} → NL_{0}, whose vanishing is equivalent to this being a crossed module. The description of ∂I_{2} as [Kerd_{0},Kerd_{1}] gives that its vanishing in this situation is module-like behaviour since a module, M, is a Lie algebra with [M, M] = 0. Thus if (NL, ∂) yields a crossed module this fact will be reflected in the internal structure of L by the vanishing of [Kerd_{0},Kerd_{1}]. Because the image of this M_{(1)(0)}(x, y) is the Peiffer element determined by x and y, we will call the M_{α,β}(x, y) in higher dimensions higher order Peiffer elements and will seek similar internal conditions for their vanishing.

We have seen that in all dimensions

∑_{I,J}[K_{I}, K_{J}] ⊆ ∂_{n}(NL_{n}) = ∂I_{n}

and we will show shortly that this inclusion is an equality, not only in dimension 2 (as above), but also in dimension 3 and 4. The arguments are calculatory and do not generalise in an obvious way to higher dimensions although similar arguments can be used to get partial results there.

Proof of Theorem 1 (ii): By Proposition 2.3, we know the generator elements of the ideal I_{3} and ∂_{3}(I_{3}) = ∂_{3}(NL_{3}). For each pair α, β ∈ S(3) with ∅_{3} < α < β and α ∩ β = ∅, we take x ∈ NL_{3−#α}, y ∈ NL_{3−#β} and set M_{α,β}(x, y) = p_{3}p_{2}p_{1} [s_{α}(x), s_{β}(y)] where p_{i}(l) = l − s_{i}d_{i}(l). This element is thus in NL_{3}. The valid pairs together with their corresponding pairing functions is given in the following table:

α  β M_{α,β}(x, y)
1 (1,0) (2) [s_{1}s_{0}x − s_{2}s_{0}x, s_{2}y]
2 (2,0) (1) [s_{2}s_{0}x − s_{2}s_{1}x, s_{1}y − s_{2}y]
3 (2,1) (0) [s_{2}s_{1}x, s_{0}y − s_{1}y + s_{2}y]
4 (2) (1) [s_{1}x, s_{0}y − s_{1}y] + [s_{2}x, s_{2}y]
5 (2) (0) [s_{2}x, s_{0}y]
6 (1) (0) [s_{2}x, s_{1}y − s_{2}y]

The explanation of this table is the following:

∂_{3}M_{α,β}(x, y) is in [K_{I}, K_{J}] in the simple cases corresponding to the first 4 rows. In row 5, ∂_{3}M_{(2)(0)}(x, y) ∈ [K_{{0,1}}, K_{{1,2}}] + [K_{{0,1}}, K_{{0,2}}] and similarly in row 6, the higher Peiffer element is in the sum of the indicated [K_{I}, K_{J}]. To illustrate the sort of argument used we look at the case of α = (1, 0) and β = (2), i.e. row 1. For x ∈ NL_{1} and y ∈ NL_{2},

\begin{aligned}d_{3}[M_{(1,0)(2)}(x, y)] & = & d_{3}[s_{1}s_{0}x − s_{2}s_{0}x, s_{2}y] \\ & = & [s_{1}s_{0}d_{1}x − s_{0}x, y] \end{aligned}

and so

d_{3}[M_{(1,0)(2)}(x, y)] = [s_{1}s_{0}d_{1}x − s_{0}x, y] ∈ [Kerd_{2}, Kerd_{0} ∩ Kerd_{1}].

We have denoted [Kerd_{2},Kerd_{0} ∩ Kerd_{1}] by [K_{{2}}, K_{{0,1}}] where I = {2} and J = {0, 1}. Rows 2, 3 and 4 are similar. For Row 5, α = (2), β = (0) with x, y ∈ NL_{2} = Kerd_{0} ∩ Kerd_{1},

\begin{aligned}d_{3}[M_{(2)(0)}(x, y)] & = & d_{3}[s_{2}x, s_{0}y] \\ & = & [x, s_{0}d_{2}y]. \end{aligned}

We can assume, for x, y ∈ NL_{2},

x ∈ Kerd_{0} ∩ Kerd_{1} and y + s_{0}d_{2}y − s_{1}d_{2}y ∈ Kerd_{1} ∩ Kerd_{2}

and, multiplying them together,

\begin{aligned}[x, y + s_{0}d_{2}y − s_{1}d_{2}y] & = & [x, y − s_{1}d_{2}y] + [x, s_{0}d_{2}y] \\ & = & d_{3}[M_{(2)(1)}(x, y)] + d_{3}[M_{(2)(0)}(x, y)] \end{aligned}

and so

\begin{aligned}d_{3}[M_{(2)(0)}(x, y)] & ∈ & [K_{\{0,1\}}, K_{\{1,2\}}] + d_{3}[M_{(2)(1)}(x, y)] \\ & ⊆ & [K_{\{0,1\}}, K_{\{1,2\}}] + [K_{\{0,1\}}, K_{\{0,2\}}]. \end{aligned}

For Row 6, for α = (1), β = (0) and x, y ∈ NL_{2} = Kerd_{0} ∩ Kerd_{1},

\begin{aligned}d_{3}[M_{(1)(0)}(x, y)] & = & d_{3}([s_{1}x, s_{0}y − s_{1}y] + [s_{2}x, s_{2}y]) \\ & = & [s_{1}d_{2}x, s_{0}d_{2}y − s_{1}d_{2}x] + [x, y]. \end{aligned}

We can take the following elements

(s_{0}d_{2}y − s_{1}d_{2}y + y) ∈ Kerd_{1} ∩ Kerd_{2} and (s_{1}d_{2}x − x) ∈ Kerd_{0} ∩ Kerd_{2}.

When we multiply them together, we get

[s_{0}d_{2}y − s_{1}d_{2}y + y, s_{1}d_{2}x − x] = [s_{0}d_{2}y, s_{1}d_{2}x] − [s_{1}d_{2}y, s_{1}d_{2}x] + [y, x]
−[x, s_{0}d_{2}y] + [x, (s_{1}d_{2}y − y)]
+[y, (s_{1}d_{2}x − x)]
= d_{3}[M_{(1)(0)}(x, y)] − d_{3}[M_{(2)(0)}(x, y)]+
d_{3}[M_{(2)(1)}(x, y) + M_{(2)(1)}(y, x)]

and hence

d_{3}[M_{(1)(0)}(x, y)] ∈ [K_{{0,2}}, K_{{1,2}}] + [K_{{0,1}}, K_{{1,2}}] + [K_{{0,1}}, K_{{0,2}}].

So we have shown

∂_{3}I_{3} ⊆ ∑_{I,J}[K_{I}, K_{J}] + [K_{{0,1}}, K_{{0,2}}] + [K_{{0,2}}, K_{{1,2}}] + [K_{{0,1}}, K_{{1,2}}].

The opposite inclusion can be verified by using proposition 2.3. Therefore

∂_{3}(NL_{3}) = [Kerd_{2}, (Kerd_{0} ∩ Kerd_{1})] + [Kerd_{1}, (Kerd_{0} ∩ Kerd_{2})]+
[Kerd_{0}, (Kerd_{1} ∩ Kerd_{2})] + [(Kerd_{0} ∩ Kerd_{1}), (Kerd_{0} ∩ Kerd_{2})]+

[(Kerd_{1} ∩ Kerd_{2}), (Kerd_{0} ∩ Kerd_{2})] + [(Kerd_{1} ∩ Kerd_{2}), (Kerd_{0} ∩ Kerd_{1})]

This completes the proof of Theorem 1 (ii).

4. Application to 2-Crossed Modules and Crossed Squares of Lie Algebras

The following definition is due to Ellis [9].

Definition 4.1. A 2-crossed module of Lie algebras consists of a complex of Lie algebras

M_{2} ⟶^{∂_{2}} M_{1} ⟶^{∂_{1}} M_{0}

with ∂_{2}, ∂_{1} morphisms of Lie algebras, where the algebra M_{0} acts on itself by Lie bracket M_{0} acts on M_{1} and M_{2} such that

M_{2} ⟶^{∂_{2}} M_{1}

is a crossed module in which M_{1} acts on M_{2} via M_{0}, further, there is a M_{0}-bilinear function giving

{ , } : M_{1} × M_{1} ⟶ M_{2},

called a Peiffer lifting, which satisfies the following axioms:

P L1 :  ∂_{2}{y_{0}, y_{1}} = [y_{0}, y_{1}] − y_{0} · ∂_{1}(y_{1}),
P L2 : {∂_{2}(x_{1}), ∂_{2}(x_{2})} = [x_{1}, x_{2}],
P L3 :  {∂_{2}(x), y} = y · x − ∂_{1}(y) · x,
P L4 :  {y, ∂_{2}(x)} = y · x,
P L5 :  {y_{0}, y_{1}} · z = {y_{0} · z, y_{1}} + {y_{0}, y_{1} · z},
P L6 :  {[y_{0}, y_{1}], y_{2}} = {y_{0}, y_{2}} · ∂_{1}y_{1} + {y_{1}, [y_{0}, y_{2}]}
−{y_{1}, y_{2}} · ∂_{1}y_{0} − {y_{0}, [y_{1}, y_{2}]},
P L7 :  {y_{0}, [y_{1}, y_{2}]} = {y_{0}, y_{1}} · ∂_{1}y_{2} − {y_{0}, y_{2}} · ∂_{1}y_{1}

+{y_{1}, [y_{0}, y_{2}] − y_{0} · ∂_{1}y_{2}} − {y_{2}, [y_{0}, y_{1}] − y_{0} · ∂_{1}y_{1}},

for all x, x_{1}, x_{2} ∈ M_{2}, y, y_{0}, y_{1}, y_{2} ∈ M_{1} and z ∈ M_{0}.

We denote such a 2-crossed module of algebras by {M_{2}, M_{1}, M_{0}, ∂_{2}, ∂_{1}}.

The following result is an analogous result of the commutative algebra version, cf. [1].

Proposition 4.2. Let L be a simplicial Lie algebra with the Moore complex NL. Then the complex of Lie algebras

NL_{2}/∂_{3}(NL_{3} ∩ D_{3}) ⟶^{\overline{∂}_{2}} NL_{0}

is a 2-crossed module of Lie algebras, where the Peiffer map is defined as follows:

\begin{aligned}\{ , \} : & NL_{1} × NL_{1} & ⟶ & NL_{2}/∂_{3}(NL_{3} ∩ D_{3}) \\ & (y_{0}, y_{1}) & ⟼ & \overline{[s_{1}y_{0}, s_{1}y_{1} − s_{0}y_{1}]}. \end{aligned}

Here the right hand side denotes a coset in NL_{2}/∂_{3}(NL_{3} ∩ D_{3}) represented by the corresponding element in NL_{2}.

Proof: We will show that all axioms of a 2-crossed module are verified. It is readily checked that the morphism \overline{∂}_{2} : NL_{2}/∂_{3}(NL_{3} ∩D_{3}) → NL_{1} is a crossed module. (In the following calculations we display the elements omitting the overlines.)

PL1:

\begin{aligned}\overline{∂}_{2}\{y_{0}, y_{1}\} & = & ∂_{2}[s_{1}y_{0}, s_{1}y_{1} − s_{0}y_{1}] \\ & = & [y_{0}, y_{1}] − y_{0} · ∂_{1}y_{1}. \end{aligned}

PL2: From ∂_{3}(M_{(1)(0)}(x_{1}, x_{2})) = [s_{1}d_{2}(x_{1}), s_{0}d_{2}(x_{2}) − s_{1}d_{2}(x_{2})] + [x_{1}, x_{2}], one obtains

\begin{aligned}\{\overline{∂}_{2}(x_{1}), \overline{∂}_{2}(x_{2})\} & = & [s_{1}d_{2}x_{1}, s_{1}d_{2}x_{2} − s_{0}d_{2}x_{2}] \\ & ≡ & [x_{1}, x_{2}] mod ∂_{3}(NL_{3} ∩ D_{3}). \end{aligned}

PL3:

\{\overline{∂}_{2}(x), y\} = [s_{1}∂_{2}x, s_{1}y − s_{0}y],

but

∂_{3}(M_{(2,0)(1)}(y, x)) = [s_{0}y − s_{1}y, s_{1}d_{2}x] − [s_{0}y − s_{1}y, x] ∈ ∂_{3}(NL_{3} ∩ D_{3})

and

∂_{3}(M_{(1,0)(2)}(y, x)) = [s_{1}s_{0}d_{1}y − s_{0}y, x] ∈ ∂_{3}(NE_{3} ∩ D_{3}),

so then

\{\overline{∂}_{2}(x), y\} ≡ [s_{1}(y), x] − [s_{0}(y), x]  mod ∂_{3}(NL_{3} ∩ D_{3})
= y · x − ∂_{1}(y) · x  by the definition of the action,

PL4: since ∂_{3}(M_{(2,1)(0)}(y, x)) = [s_{1}y, s_{0}d_{2}x − s_{1}d_{2}x] + [s_{1}(y), x],

\{y, \overline{∂}_{2}(x)\} = [s_{1}y, s_{1}∂_{2}x − s_{0}∂_{2}x]
≡ [s_{1}(y), x] mod ∂_{3}(NL_{3} ∩ D_{3})
= y · x  by the definition of the action.

PL5: By the definition of the action, we get

{y_{0}, y_{1}} · z = {y_{0} · z, y_{1}} + {y_{0}, y_{1} · z}

with x, x_{1}, x_{2} ∈ NL_{2}/∂_{3}(NL_{3} ∩ D_{3}), y, y_{0}, y_{1}, y_{2} ∈ NL_{1} and z ∈ NL_{0}. Verification of axioms PL6 and PL7 are omitted as they are routine. This completes the proof of the proposition. □

This only used the higher dimension Peiffer elements. A result in terms of [K_{I}, K_{J}] vanishing can also be given:

Proposition 4.3. If in a simplicial Lie algebra L, one has [K_{I}, K_{J}] = 0 in dimension 2 for the following cases: I ∪ J = [2], I ∩ J = ∅; I = {0, 1}, J = {0, 2} or I = {1, 2}; and I = {0, 2}, J = {1, 2} then

NL_{2} ⟶ NL_{1} ⟶ NL_{0}

can be given the structure of a 2-crossed module.

Another application of higher order Peiffer elements is a Lie crossed square. First we recall from [8] the notion of crossed n-cubes of Lie algebras.

A crossed n-cube of Lie algebras is a family of Lie algebras, M_{A} for A ⊆< n >= {1, ..., n} together with homomorphisms μ_{i} : M_{A} → M_{A−{i}} for i ∈< n > and for A, B ⊆< n >, functions

h : M_{A} × M_{B} ⟶ M_{A∪B}

such that for all λ ∈ Λ, a, a′ ∈ M_{A}, b, b′ ∈ M_{B}, c ∈ M_{C}, i, j ∈< n > and A ⊆ B

\begin{aligned}1) & μ_{i}a & = & a if i ∉A \\ 2) & μ_{i}μ_{j}a & = & μ_{j}μ_{i}a \\ 3) & μ_{i}h(a, b) & = & h(μ_{i}a, μ_{i}b) \\ 4) & h(a, b) & = & h(μ_{i}a, b) = h(a, μ_{i}b) if i ∈ A ∩ B \\ 5) & h(a, a′) & = & [a, a′] \\ 6) & h(a, a) & = & 0 \\ 7) & h(a + a′, b) & = & h(a, b) + h(a′, b) \\ 8) & h(a, b + b′) & = & h(a, b) + h(a, b′) \\ 9) & λh(a, b) & = & h(λa, b) = h(a, λb) \\ 10) & h(h(a, b), c) & + & h(h(b, c), a) + h(h(c, a), b) = 0. \end{aligned}

A morphism of crossed n-cubes is defined in the obvious way. We thus denote a category of crossed n-cubes by Crs^{n}.

Theorem 1 (iii) can be used to verify that the following construction in a functor from simplicial Lie algebras to crossed n-cubes.

For a simplicial Lie algebra L and a given n, we write M(L, n) for the crossed n-cube, arising from the functor

M(−, n) : SLA ⟶ Crs^{n}.

Then the crossed n-cube M(L,n) is determined by:

(i) for A ⊆< n >,

M(L, n)_{A} = \frac{⋂_{j∈A} Kerd_{j−1}^{n}}{d_{n+1}^{n+1}(Kerd_{0}^{n+1} ∩ \{⋂_{j∈A} Kerd_{j}^{n+1}\});}

(ii) the inclusion

⋂_{j∈A} Kerd_{j−1}^{n} ⟶ ⋂_{j∈A−{i}} Kerd_{j−1}^{n}

induces the morphism

μ_{i} : M(L, n)_{A} ⟶ M(L, n)_{A−{i}};

(iii) the functions, for A, B ⊆< n >,

h : M(L, n)_{A} × M(L, n)_{B} ⟶ M(L, n)_{A∪B}

given by

h(x̄, ȳ) = \overline{[x, y]},

where an element of M(L, n)_{A} is denoted by x̄ with x ∈ ⋂_{j∈A}Kerd_{j−1}^{n}.

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This article may be accessed via WWW at http://www.rmi.acnet.ge/hha/ or by anonymous ftp at

ftp://ftp.rmi.acnet.ge/pub/hha/volumes/2002/n1a4/v4n1a4.(dvi,ps,pdf)

İ. Akça iakca@ogu.edu.tr

Department of Mathematics, Faculty of Science, Osmangazi University, 26480, Eskişehir, Turkey

Z. Arvasi zarvasi@ogu.edu.tr

Department of Mathematics, Faculty of Science, Osmangazi University, 26480, Eskişehir, Turkey