Lie Algebraic Homotopy 3-Types

Abstract

In this work, we study the relationship among Lie algebraic homotopy 3-types. We investigate the categorical connection between crossed squares, internal crossed modules, 2-crossed modules, quadratic modules of Lie algebras, and simplicial and bisimplicial Lie algebras.

1. Introduction

The definition of crossed modules was first given by Whitehead for groups in 1949 [1]. Crossed modules of groups provide a useful tool for developing new techniques to study the homotopy groups of spaces. Later on, in the context of Lie algebras, Christian Kassel and Jean-Louis Loday extended the idea of crossed modules to this field in their paper published in 1984 [2].

Simplicial groups, first given by Kan in [3], provide a well-organized homotopy theory and describe all homotopy types of connected spaces. In [4], Conduché introduced an algebraic model for connected 3-types known as 2-crossed modules. These models have useful properties, as mentioned in [4], and form a category that is equivalent to simplicial groups with a Moore complex of length 2. Additionally, in [5], Conduché established a connection between these two concepts by constructing a 2-crossed module from a crossed square of groups, which was introduced by Loday and Guin-Waléry in [6].

This connection broadens our knowledge of the relationship between simplicial groups and 2-crossed modules.

There are numerous connections and applications for Lie algebraic structures in geometry. Tangent spaces in differential geometry can be thought of as the algebraic equivalents of Lie algebras. Ellis in [7] defined 2-crossed modules of Lie algebras as a way to capture the algebraic structure of a Moore complex of length 2. This provides a Lie algebraic counterpart to the group theoretic concept introduced by Conduché. Shuffles of Lie products provide the Samelson and Whitehead product equivalents in the homotopy theory of simplicial Lie algebras. In [8] for Lie algebras, Akça and Arvasi examined the connection between these shuffles and the crossed and 2-crossed modules.

Ellis and Steiner defined crossed n-cubes of groups as an algebraic representation of connected (n + 1)-types [9]. They showed that these crossed n-cubes are equivalent to the $ca{t}^n$-groups in [10]. Porter established an equivalence between crossed n-cubes and n-types of simplicial groups [11]. Ellis extended this idea for crossed n-cubes of Lie algebras [12], demonstrating their equivalence for cat-algebras. Similar to the functor obtained by Porter in the group situation, Akça and Arvasi obtained a functor that maps simplicial Lie algebras to crossed n-cubes of Lie algebras [8]. In Section 3, we investigate the 2-dimensional case of this functor specifically for Lie algebras, which sheds light on the relationships between crossed n-cubes, simplicial Lie algebras, and $ca{t}^n$-algebras.

In Section 4, the “Artin–Mazur” codiagonal functor [13] will be used to describe the functor from internal crossed modules to quadratic modules of Lie algebras. We directly demonstrate the Lie algebraic quadratic module structure in terms of the bisimplicial nerve of an internal crossed module. The functor from quadratic modules to internal crossed modules will be briefly explained in Section 5 using the equivalence between 2-crossed modules and simplicial Lie algebras as defined by Ellis [7] and the functor from simplicial Lie algebras to crossed squares [8]. Similar equivalencies can be seen in [14,15,16].

The following diagram provides a summary of the paper’s major ideas and findings.

Here ${\mathbf{Crs}}_{\mathbf{Lie}}^{\mathbf{2}}$ denotes crossed squares of Lie algebras, $\mathbf{Int}\left({\mathbf{XMod}}_{\mathbf{Lie}}\right)$ internal crossed modules of Lie algebras, ${\mathbf{BiSimp}}_{\mathbf{LieAlg}}$ bisimplicial Lie algebras, ${\mathbf{Simp}}_{\mathbf{LieAlg}}$ simplicial Lie algebras whose Moore complex has length at most 2, ${\mathbf{X}}_{\mathbf{2}}{\mathbf{Mod}}_{\mathbf{Lie}}$ 2-crossed modules of Lie algebras, and ${\mathbf{Quad}}_{\mathbf{Lie}}$ quadratic modules of Lie algebras. The dotted arrows are the composites of the functors constructed in Section 3, Section 4 and Section 5.

The complex relationship between Lie algebraic homotopy 3-types is explored in this article, with a focus on the connections between different algebraic structures like crossed squares, internal crossed modules, 2-crossed modules, simplicial and bisimplicial modules, and quadratic modules for the Lie algebraic case. We can deduce the main results from this work by noting that crossed squares, internal crossed modules, 2-crossed modules, simplicial and bisimplicial modules, and quadratic modules can all be used to describe Lie algebraic homotopy 3-types, and there exist functors between these algebraic 3-types. Overall, this article offers an in-depth analysis of the relationship between Lie algebraic homotopy 3-types, shedding light on several representations of those concepts and their relationships.

2. Crossed Modules of Lie Algebras

In this section, first we will review the fundamental definitions for Lie algebras as given in [17].

Definition 1. 

Let k be a unitary and commutative ring and let L be an k - module. If there exists a bilinear transformation $[-,-]:L\times L\to L$ satisfying the properties

i. $[l,l]=0$

ii. $[l,[l^{\prime },l^{″}]]+[l^{\prime },[l^{″},l]]+[l^{\prime \prime },[l,l^{\prime }\left]\right]=0$

L is called a Lie algebra over A. The function $[-,-]$ is called a Lie bracket or product, and the property in (ii) is called the Jacobi identity.

Definition 2. 

Let $L_1$ and $L_2$ be two Lie algebras; then a Lie algebra action of $L_1$ on $L_2$ is a k-bilinear map

$$\begin{array}{ccc}\hfill L_1\times L_2& ⟶& L_2\\ \hfill (l_1,l_2)& \mapsto & l_1·l_2\end{array}$$

that satisfies the following two axioms:

$$[l_1,l_1^{\prime }]·l_2=l_1·(l_1^{\prime }·l_2)-l_1^{\prime }·(l_1·l_2)$$

$$l_1·[l_2,l_2^{\prime }]=[l_1·l_2,l_2^{\prime }]+[l_2,l_1·l_2^{\prime }]$$

for $l_1,l_1^{\prime }\in L_1$ and $l_2,l_2^{\prime }\in L_2$.

Definition 3. 

A precrossed module of Lie algebras consists of a homomorphism of Lie algebras $\partial :L_2\to L_1$ with the action of $L_1$ on $L_2$, denoted by $(l_1,l_2)\mapsto l_1·l_2$, for $l_1\in L_1$, $l_2\in L_2$. One requires that the following identity hold:

$$\partial (l_1·l_2)=[l_1,\partial \left(l_2\right)]$$

(1)

for all $l_2\in L_2$ and $l_1\in L_1$. If additionally we have

$$\partial \left(l_2\right)·l_2^{\prime }=[l_2,l_2^{\prime }]$$

(2)

for $l_2$,$l_2^{\prime }\in L_2$ then $\partial :L_2\to L_1$ is called a crossed module.

Internal crossed modules can be regarded as the category object in the category of crossed modules. Next, for the Lie algebraic case, we provide a precise definition of an internal crossed module. For a comprehensive understanding of internal objects and their applications, readers are encouraged to refer to the relevant studies, such as [18,19].

Definition 4. 

Let ${\alpha }_1:M_1⟶N_1$ and ${\alpha }_0:M_0⟶N_0$ be crossed modules of Lie algebras. An internal crossed module of Lie algebras is an internal category in ${\mathbf{XMod}}_{\mathrm{Lie}}$, represented by the commutative diagram

where $s=(s_1,s_0),t=(t_1,t_0),\epsilon =({\epsilon }_1,{\epsilon }_0)$ are morphisms of crossed modules satisfying

$${\alpha }_0{s}_1=s_0{\alpha }_1,{\alpha }_0{t}_1=t_0{\alpha }_1,{\alpha }_1{\epsilon }_1={\epsilon }_0{\alpha }_0$$

and

$$s_i{\epsilon }_i=t_i{\epsilon }_i=\mathrm{id},i=0,1.$$

There are composition morphisms

$${\circ }_1:M_{1s_1\times t_1}{M}_1⟶M_1,{\circ }_0:N_{1s_0\times t_0}{N}_1⟶N_1,$$

such that $({\circ }_1,{\circ }_0)$ is a morphism of crossed modules and

$$s_i\left(y{\circ }_{i}x\right)=s_i\left(x\right),t_i\left(y{\circ }_{i}x\right)=t_i\left(y\right),$$

$$z{\circ }_i\left(y{\circ }_{i}x\right)=\left(z{\circ }_{i}y\right){\circ }_{i}x,x{\circ }_i{\epsilon }_i\left(s_i\left(x\right)\right)=x={\epsilon }_i\left(t_i\left(x\right)\right){\circ }_{i}x$$

whenever compositions are defined for $i=0,1$.

Example 1. 

Let $\alpha :M\to N$ be a crossed module of Lie algebras. Similar to [20], consider the crossed modules

$$\alpha \times \alpha :M\times M⟶N\times N\mathit{and}\alpha :M⟶N,$$

where the actions on the direct products are componentwise. Define

$$\begin{array}{cccccc}\hfill s_1(m^{\prime },m)& =m,\hfill & \hfill t_1(m^{\prime },m)& =m^{\prime },\hfill & \hfill {\epsilon }_1\left(m\right)& =(m,m),\hfill \\ \hfill s_0(n^{\prime },n)& =n,\hfill & \hfill t_0(n^{\prime },n)& =n^{\prime },\hfill & \hfill {\epsilon }_0\left(n\right)& =(n,n).\hfill \end{array}$$

For composable pairs, define the compositions

$$(m^{″},m^{\prime }){\circ }_1(m^{\prime },m)=(m^{″},m),(n^{″},n^{\prime }){\circ }_0(n^{\prime },n)=(n^{″},n).$$

These maps are morphisms of crossed modules and satisfy the internal category identities.

Example 2. 

Let I be an ideal of a Lie algebra L. The inclusion $\partial :I\hookrightarrow L$, together with the action $l·a=[l,a]$, is a crossed module. Indeed,

$$\partial (l·a)=[l,\partial (a\left)\right],\partial \left(a\right)·b=[a,b]$$

for all $l\in L$ and $a,b\in I$.

3. Internal Crossed Modules from Simplicial Algebras

Based on concepts from Loday [10], Porter defined a functor from simplicial groups to crossed n-cubes in [11]. Ellis [12] defined crossed n-cubes in Lie algebras, Jordan algebras, and commutative algebras, among other algebraic contexts. Ellis’ definition of the crossed n-cube of Lie algebras is undoubtedly significant, but in this section, we only apply it to the case where n = 2, or crossed squares. As a result, we do not mention the general meaning of a crossed n-cube. According to [12,21], in low dimensions, a crossed 1-cube is equivalent to a crossed module, and a crossed 2-cube is equivalent to a crossed square. An analogous functor to Porter’s group case from [11] has been given in [8]. We can obtain crossed squares of Lie algebras using this functor in dimension 2.

For a simplicial algebra $\mathit{L}$, the diagram

is an underlying square of a crossed square of Lie algebras. Where $\overline{N{L}_1}=ker{d}_1^1$ and $N{L}_1=ker{d}_0^1$. Due to the fact that $L_1$ acts on $\overline{N{L}_1}$, $N{L}_2/{\partial }_{3}N{l}_3$ and $N{L}_1$ there are Lie actions on $N{L}_2/{\partial }_{3}N{l}_3$ and $N{L}_1$ of $\overline{N{L}_1}$ via ∂, as well as $N{L}_2/{\partial }_3$ and $\overline{N{L}_1}$ of $N{L}_1$ via ${\partial }^{\prime }$. All actions are obtained by the Lie bracket, and both ∂ and ${\partial }^{\prime }$ are inclusions. The h-map is defined by

$$\begin{array}{ccc}h:N{L}_1\times \overline{N{L}_1}\hfill & ⟶\hfill & N{L}_2/{\partial }_{3}N{l}_3\hfill \\ \left(a,\overline{b}\right)\hfill & \mapsto \hfill & h(a,b)=[s_{1}a,s_{1}b-s_{0}b]+{\partial }_{3}N{l}_3\hfill \end{array}$$

That is, we obtain a functor

$$M(-,2):{\mathbf{Simp}}_{\mathbf{LieAlg}}⟶{\mathbf{Crs}}_{\mathbf{Lie}}^{\mathbf{2}}.$$

Proposition 1. 

If the diagram

is a crossed square of Lie algebras, then

$${\lambda }^{\prime }\times {\delta }^{\prime }:L⋉M⟶N⋉P,({\lambda }^{\prime }\times {\delta }^{\prime })(l,m)=\left({\lambda }^{\prime }\left(l\right),{\delta }^{\prime }\left(m\right)\right),$$

is a crossed module of Lie algebras, with the action of $N⋉P$ on $L⋉M$ is defined by

$$(n,p)·(l,m)=\left(n·l+p·l-h(m,n),p·m\right)$$

for $(n,p)\in N⋉P$ and $(l,m)\in L⋉M$.

Proof. 

$CM1.$ First note that the Lie bracket in $N⋉P$ is

$$[(n_1,p_1),(n_2,p_2)]=\left([n_1,n_2]+p_1·n_2-p_2·n_1,[p_1,p_2]\right).$$

For $(n,p)\in N⋉P$ and $(l,m)\in L⋉M$, we obtain

$$\begin{array}{ccc}\hfill ({\lambda }^{\prime }\times {\delta }^{\prime })\left((n,p)·(l,m)\right)& \hfill =\hfill & ({\lambda }^{\prime }\times {\delta }^{\prime })\left(n·l+p·l-h(m,n),p·m\right)\hfill \\ & =& \left({\lambda }^{\prime }(n·l)+{\lambda }^{\prime }(p·l)-{\lambda }^{\prime }h(m,n),{\delta }^{\prime }(p·m)\right)\hfill \\ & =& \left([n,{\lambda }^{\prime }\left(l\right)]+p·{\lambda }^{\prime }\left(l\right)-m·n,[p,{\delta }^{\prime }\left(m\right)]\right)\hfill \\ & =& \left([n,{\lambda }^{\prime }\left(l\right)]+p·{\lambda }^{\prime }\left(l\right)-{\delta }^{\prime }\left(m\right)·n,[p,{\delta }^{\prime }\left(m\right)]\right)\hfill \\ & =& \left[(n,p),({\lambda }^{\prime }\left(l\right),{\delta }^{\prime }\left(m\right))\right]\hfill \\ & =& \left[(n,p),({\lambda }^{\prime }\times {\delta }^{\prime })(l,m)\right].\hfill \end{array}$$

$CM2.$ For $(l_1,m_1),(l_2,m_2)\in L⋉M$, we obtain

$$\begin{array}{cc}\hfill ({\lambda }^{\prime }\times {\delta }^{\prime })(l_1,m_1)·(l_2,m_2)& =({\lambda }^{\prime }\left(l_1\right),{\delta }^{\prime }\left(m_1\right))·(l_2,m_2)\hfill \\ & =\left({\lambda }^{\prime }\left(l_1\right)·l_2+{\delta }^{\prime }\left(m_1\right)·l_2-h(m_2,{\lambda }^{\prime }\left(l_1\right)),{\delta }^{\prime }\left(m_1\right)·m_2\right)\hfill \\ & =\left([l_1,l_2]+{\delta }^{\prime }\left(m_1\right)·l_2-m_2·l_1,{\delta }^{\prime }\left(m_1\right)·m_2\right)\hfill \\ & =\left([l_1,l_2]+m_1·l_2-m_2·l_1,[m_1,m_2]\right)\hfill \\ & =[(l_1,m_1),(l_2,m_2)]\hfill \end{array}$$

□

Proposition 2. 

If the diagram

is a crossed square of Lie algebras, then

is an internal crossed module of Lie algebras, where

$${\alpha }_1={\lambda }^{\prime }\times {\delta }^{\prime }:L⋉M⟶N⋉P,{\alpha }_0={\delta }^{\prime }:M⟶P.$$

and

$$\begin{array}{cccccc}\hfill s_1(l,m)& =m,\hfill & \hfill t_1(l,m)& =\lambda \left(l\right)+m,\hfill & \hfill {\epsilon }_1\left(m\right)& =(0,m),\hfill \\ \hfill s_0(n,p)& =p,\hfill & \hfill t_0(n,p)& =\delta \left(n\right)+p,\hfill & \hfill {\epsilon }_0\left(p\right)& =(0,p).\hfill \end{array}$$

The compositions are defined by

$$(l^{\prime },\lambda \left(l\right)+m){\circ }_1(l,m)=(l^{\prime }+l,m)$$

and

$$(n^{\prime },\delta \left(n\right)+p){\circ }_0(n,p)=(n^{\prime }+n,p).$$

Proof. 

We show that the interchange law holds. Suppose that $(l^{\prime },m^{\prime }){\circ }_1(l,m)$ and $(n^{\prime },p^{\prime }){\circ }_0(n,p)$ are defined. Thus

$$m^{\prime }=\lambda \left(l\right)+m,p^{\prime }=\delta \left(n\right)+p.$$

Since the composition $\left((n^{\prime },p^{\prime })·(l^{\prime },m^{\prime })\right){\circ }_1\left((n,p)·(l,m)\right)$ should exists we must have

$$\lambda \left(n·l+p·l-h(m,n)\right)+p·m=(\delta \left(n\right)+p)·(\lambda \left(l\right)+m)=p^{\prime }·m^{\prime }$$

then

$$\begin{array}{cc}\hfill \left((n^{\prime },p^{\prime }){\circ }_0(n,p)\right)·\left((l^{\prime },m^{\prime }){\circ }_1(l,m)\right)& =(n^{\prime }+n,p)·(l^{\prime }+l,m)\hfill \\ \hfill & =\left(\begin{array}{c}(n^{\prime }+n)·(l^{\prime }+l)+p·(l^{\prime }+l)\hfill \\ -h(m,n^{\prime }+n),p·m\hfill \end{array}\right)\hfill \\ \hfill & =\left(\begin{array}{c}{n}^{\prime }·l^{\prime }+n^{\prime }·l+n·l^{\prime }+n·l+p·l^{\prime }\hfill \\ +p·l-h(m,n^{\prime })-h(m,n),p·m\hfill \end{array}\right)\hfill \\ \hfill & =\left(\begin{array}{c}{n}^{\prime }·l^{\prime }+(\delta \left(n\right)+p)·l^{\prime }-h(\lambda \left(l\right),n^{\prime })-h(m,n^{\prime })\hfill \\ +n·l+p·l-h(m,n),p·m\hfill \end{array}\right)\hfill \\ \hfill & =\left(\begin{array}{c}{n}^{\prime }·l^{\prime }+(\delta \left(n\right)+p)·l^{\prime }-h(\lambda \left(l\right)+m,n^{\prime })\hfill \\ +n·l+p·l-h(m,n),p·m\hfill \end{array}\right)\hfill \\ \hfill & =\left(n^{\prime }·l^{\prime }+p^{\prime }·l^{\prime }-h(m^{\prime },n^{\prime }),p^{\prime }·m^{\prime }\right)\hfill \\ \hfill & {\circ }_1\left(n·l+p·l-h(m,n),p·m\right)\hfill \\ \hfill & =\left((n^{\prime },p^{\prime })·(l^{\prime },m^{\prime })\right){\circ }_1\left((n,p)·(l,m)\right).\hfill \end{array}$$

Moreover, the composition preserves the crossed-module homomorphism

$$\begin{array}{cc}\hfill {\alpha }_1\left((l^{\prime },m^{\prime }){\circ }_1(l,m)\right)& =\left({\lambda }^{\prime }\left(l^{\prime }\right)+{\lambda }^{\prime }\left(l\right),{\delta }^{\prime }\left(m\right)\right)\hfill \\ \hfill & ={\alpha }_1(l^{\prime },m^{\prime }){\circ }_0{\alpha }_1(l,m),\hfill \end{array}$$

where commutativity of the crossed square gives ${\delta }^{\prime }\lambda =\delta {\lambda }^{\prime }$. We omit the remaining axioms as they are straightforward. □

That is, we obtain a functor

$$F:{\mathbf{Crs}}_{\mathbf{Lie}}^{\mathbf{2}}\to \mathbf{Int}\left({\mathbf{XMod}}_{\mathbf{Lie}}\right)$$

from crossed squares to internal crossed modules of Lie algebras.

The construction of this section is summarized by the diagram

(3)

where the dotted arrow is the composite of the functor $M(-,2)$ with the functor F of Proposition 2.

4. Quadratic Modules from Internal Crossed Modules of Lie Algebras

We will construct a quadratic module from internal crossed modules of Lie algebras as follows:

• To obtain a bisimplicial Lie algebra, we apply the nerves to an internal crossed module of Lie algebras in both directions.
• To obtain a simplicial Lie algebra from this bisimplicial Lie algebra and obtain the Moore complex, we apply the Artin–Mazur [13] codiagonal functor for simplicial Lie algebras.
• We show that the Moore complex of this simplicial Lie algebra formed in the preceding step is isomorphic to the mapping cone complex, analogous to Loday’s suggestions, of the internal crossed module, and this complex has the quadratic module structure for Lie algebras.

Step 1.

In this step, we demonstrate how to obtain a bisimplicial Lie algebra from an internal crossed module of Lie algebras. We achieve this through the use of the connection between crossed squares and cat-groups, which is expressed in terms of a bisimplicial nerve of a crossed square. The construction of the bisimplicial group from a crossed square was presented by Mutlu and Porter [22]. To obtain a simplicial Lie algebra, we begin by applying the Nerve functor to an internal crossed module of Lie algebras.

Let $\partial :M\to N$ be a crossed module of Lie algebras; the simplicial Lie algebra $\mathcal{N}$ can be given as

$$\mathrm{Ner}{\left(\mathcal{N}\right)}_0=N,\mathrm{Ner}{\left(\mathcal{N}\right)}_r=M⋉\mathrm{Ner}{\left(\mathcal{N}\right)}_{r-1}(r\ge 1).$$

We write an element of $\mathrm{Ner}{\left(\mathcal{N}\right)}_r$ as $(m_r,\dots ,m_1,n)$. The first semidirect product uses the given action of N on M, while for $r\ge 2$, the action is

$$(m_{r-1},\dots ,m_1,n)·m_r=n·m_r+{\displaystyle \sum _{j=1}^{r-1}}[m_j,m_r].$$

In particular, we have

$$\begin{array}{cc}\hfill \mathrm{Ner}{\left(\mathcal{N}\right)}_1& =M⋉N,\hfill \\ \hfill \mathrm{Ner}{\left(\mathcal{N}\right)}_2& =M⋉(M⋉N),\hfill \\ \hfill \mathrm{Ner}{\left(\mathcal{N}\right)}_3& =M⋉(M⋉(M⋉N\left)\right).\hfill \end{array}$$

For $n=1$, $\{M⋉N,N\}$ is a 1-truncated simplicial Lie algebra where the homomorphisms are defined as

$$d_0^1(m_1,n)=\partial \left(m_1\right)+n,d_1^1(m_1,n)=n,s_0^0\left(n\right)=(0,n).$$

For $n=2$, we have $Ner{\mathcal{N}}_2=M⋉(M⋉N)$ where the appropriate operators are

$$\begin{array}{cc}\hfill d_0^2(m_2,m_1,n)& =(m_2,\partial \left(m_1\right)+n),\hfill \\ \hfill d_1^2(m_2,m_1,n)& =(m_2+m_1,n),\hfill \\ \hfill d_2^2(m_2,m_1,n)& =(m_1,n),\hfill \\ \hfill s_0^1(m_1,n)& =(m_1,0,n),\hfill \\ \hfill s_1^1(m_1,n)& =(0,m_1,n).\hfill \end{array}$$

For $n=3$,

$$\begin{array}{cc}\hfill d_0^3(m_3,m_2,m_1,n)& =(m_3,m_2,\partial \left(m_1\right)+n),\hfill \\ \hfill d_1^3(m_3,m_2,m_1,n)& =(m_3,m_2+m_1,n),\hfill \\ \hfill d_2^3(m_3,m_2,m_1,n)& =(m_3+m_2,m_1,n),\hfill \\ \hfill d_3^3(m_3,m_2,m_1,n)& =(m_2,m_1,n),\hfill \\ \hfill s_0^2(m_2,m_1,n)& =(m_2,m_1,0,n),\hfill \\ \hfill s_1^2(m_2,m_1,n)& =(m_2,0,m_1,n),\hfill \\ \hfill s_2^2(m_2,m_1,n)& =(0,m_2,m_1,n).\hfill \end{array}$$

More generally, we have the following pattern

$$\begin{array}{cccc}\hfill d_0^r(m_r,\dots ,m_1,n)& =(m_r,\dots ,m_2,\partial \left(m_1\right)+n),\hfill & & \\ \hfill d_i^r(m_r,\dots ,m_1,n)& =(m_r,\dots ,m_{i+2},m_{i+1}+m_i,m_{i-1},\dots ,m_1,n),\hfill & & 1\le i\le r-1,\hfill \\ \hfill d_r^r(m_r,\dots ,m_1,n)& =(m_{r-1},\dots ,m_1,n),\hfill & & \\ \hfill s_i^r(m_r,\dots ,m_1,n)& =(m_r,\dots ,m_{i+1},0,m_i,\dots ,m_1,n),\hfill & & 0\le i\le r.\hfill \end{array}$$

Taking this pattern and extending it to higher dimensions, we obtain a simplicial Lie algebra

Next, we construct a bisimplicial Lie algebra by applying the Nerve functor to an internal crossed module of Lie algebras. Let

be an internal crossed module. From this internal crossed module, we will obtain a bisimplicial Lie algebra, denoting it by $(\mathbf{L}.,.)$. We obtain the following diagram by applying the Nerve functor to the crossed modules of Lie algebras ${\alpha }_1:M_1\to N_1$ and ${\alpha }_0:M_0\to N_0$

where the left side stands for the simplicial Lie algebra $Ner\left(M_1\right)$, which was obtained from the crossed module ${\alpha }_1$, and the right side stands for the simplicial Lie algebra $Ner\left(M_0\right)$, which was obtained from the crossed module ${\alpha }_0$. Consequently, the elements of the bisimplicial Lie algebra $(\mathbf{L}.,.)$ can be given as

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{0,q}& =& M_0⋉\dots M_0⋉N_0=M_0^{\left(q\right)}⋉N_0=Ner{\left(M_0\right)}_q\hfill \\ & & \\ \hfill {\mathcal{L}}_{p,0}& =& M_1⋉\dots M_1⋉N_1=M_1^{\left(p\right)}⋉N_1=Ner{\left(M_1\right)}_p\hfill \end{array}$$

and in general for any two non-zero p and q, ${\mathcal{L}}_{p,q}$ can be written

$${\mathcal{L}}_{p,q}={(M_1^{\left(q\right)}⋉N_1)}^{\left(p\right)}⋉(M_0^{\left(q\right)}⋉N_0).$$

Applying the nerve in both directions gives a bisimplicial Lie algebra $\mathbf{L}=\left({\mathcal{L}}_{p,q}\right)$ with

$$\begin{array}{cc}\hfill {\mathcal{L}}_{0,q}& =M_0^{\left(q\right)}⋉N_0,\hfill \\ \hfill {\mathcal{L}}_{p,0}& =N_1^{\left(p\right)}⋉N_0,\hfill \\ \hfill {\mathcal{L}}_{p,q}& ={\left(M_1^{\left(q\right)}⋉N_1\right)}^{\left(p\right)}⋉\left(M_0^{\left(q\right)}⋉N_0\right),p,q>0\hfill \end{array}$$

where $X^{\left(r\right)}⋉Y$ denotes the semidirect product with r copies of X, and $X^{\left(0\right)}⋉Y=Y$. In low degrees,

$${\mathcal{L}}_{0,0}=N_0,{\mathcal{L}}_{1,0}=N_1⋉N_0,{\mathcal{L}}_{0,1}=M_0⋉N_0,$$

and

$${\mathcal{L}}_{1,1}=(M_1⋉N_1)⋉(M_0⋉N_0).$$

In low dimensions this bisimplicial Lie algebra can be given as

Step 2.

First, we adapt the Artin–Mazur codiagonal functor [13] to bisimplicial Lie algebras. Let $\mathbf{L}=\left({\mathcal{L}}_{p,q}\right)$ be a bisimplicial Lie algebra and define

$${\mathcal{L}}_{\left(n\right)}=\underset{p+q=n}{⨁}{\mathcal{L}}_{p,q}$$

$$\nabla {\left(\mathbf{L}\right)}_n=\left\{(x_0,\dots ,x_n)\in {\mathcal{L}}_{\left(n\right)}:d_0^v{x}_p=d_{p+1}^h{x}_{p+1},0\le p<n\right\}$$

where element of ${\mathcal{L}}_{\left(n\right)}$ is written as $x=(x_0,\dots ,x_n),$ in which $x_p\in {\mathcal{L}}_{p,n-p}.$ The face and degeneracy maps are given by

$$\begin{array}{cc}\hfill D_j\left(x\right)=& \left(d_j^v{x}_0,d_{j-1}^v{x}_1,\dots ,d_1^v{x}_{j-1},d_j^h{x}_{j+1},\dots ,d_j^h{x}_n\right),\hfill \\ \hfill S_j\left(x\right)=& \left(s_j^v{x}_0,s_{j-1}^v{x}_1,\dots ,s_0^v{x}_j,s_j^h{x}_j,s_j^h{x}_{j+1},\dots ,s_j^h{x}_n\right).\hfill \end{array}$$

Remark 1. 

The direct sum ${\mathcal{L}}_{\left(n\right)}$ is a Lie algebra with the bracket

$$[(x_0,\dots ,x_n),(y_0,\dots ,y_n)]=([x_0,y_0],\dots ,[x_n,y_n]).$$

If $x,y\in \nabla {\left(\mathbf{L}\right)}_n$, then

$$\begin{array}{cc}\hfill d_0^v[x_p,y_p]& =[d_0^v{x}_p,d_0^v{y}_p]\hfill \\ & =[d_{p+1}^h{x}_{p+1},d_{p+1}^h{y}_{p+1}]\hfill \\ & =d_{p+1}^h[x_{p+1},y_{p+1}]\hfill \end{array}$$

which shows that $\nabla {\left(\mathbf{L}\right)}_n$ is a Lie subalgebra of ${\mathcal{L}}_{\left(n\right)}$.

Since all horizontal and vertical face and degeneracy maps are Lie algebra homomorphisms, so are $D_j$ and $S_j$. Therefore $\nabla \left(\mathbf{L}\right)$ is a simplicial Lie algebra, and the construction defines a functor

$$\nabla :{\mathbf{BiSimp}}_{\mathbf{LieAlg}}⟶{\mathbf{Simp}}_{\mathbf{LieAlg}}.$$

We now apply the Lie algebraic Artin–Mazur codiagonal functor ∇ to the bisimplicial Lie algebra obtained from the internal crossed module as described above to define the simplicial Lie algebra ${\mathbf{A}}^{\left(2\right)}$. Next, we define Lie algebras $A_n$ of n-simplices. For $n=0$, $A_0\cong N$. For $n=1$

$$A_1=((m_0,n_0),(n_1,n_0^{\prime }):d_0^v(m_0,n_0)={\alpha }_0\left(m_0\right)+n_0=n_0^{\prime }=d_1^h(n_1,n_0^{\prime }))\subset L_{0,1}\oplus L_{1,0}$$

Since $M_0$ acts on $N_1$ through $N_0$, the Lie algebra $A_1$ is isomorphic to $N_1⋉(M_0⋉N_0)$. Indeed $f:A_1\to N_1⋉(M_0⋉N_0)$ defined by $(m_0,n_0),(n_1,{\alpha }_0\left(m_0\right)+n_0)\mapsto (n_1,m_0,n_0)$ is an isomorphism. By associating $A_1$ with $N_1⋉(M_0⋉N_0)$ and using the Artin–Mazur codiagonal formula, we define

$$d_0^1(n_1,m_0,n_0)=t_0\left(n_1\right)+{\alpha }_0\left(m_0\right)+n_0$$

$$d_1^1(n_1,m_0,n_0)=n_0$$

$$s_0^0\left(n_0\right)=(0,0,n_0).$$

For $n=2$, $A_2$ is the subset of ${\mathcal{L}}_{0,2}\oplus {\mathcal{L}}_{1,1}\oplus {\mathcal{L}}_{2,0}$ in which its elements are

$$((m_0,m_0^{\prime },n_0),(m_1,n_1,m_0^{″},n_0^{\prime }),(n_1^{\prime },n_1^{″},n_0^{″}))$$

satisfying

$$d_0^v(m_0,m_0^{\prime },n_0)=d_1^h(m_1,n_1,m_0^{″},n_0^{\prime })$$

$$d_0^v(m_1,n_1,m_0^{″},n_0^{\prime })=d_2^h(n_1^{\prime },n_1^{\prime \prime },n_0^{″})$$

which implies

$$m_0^{\prime }=m_0^{″},n_0^{\prime }={\alpha }_0\left(m_0\right)+n_0,n_1^{\prime }={\alpha }_1\left(m_1\right)+n_1,n_0^{″}={\alpha }_0\left(m_0^{″}\right)+n_0^{\prime }.$$

Then ${\mathbf{a}}_2\in A_2$ is of the form

$${\mathbf{a}}_2=\left((m_0,m_0^{\prime },n_0),(m_1,n_1,m_0^{\prime },{\alpha }_0(m_0)+n_0),({\alpha }_1(m_1)+n_1,n_1^{″},{\alpha }_0(m_0^{\prime }+m_0)+n_0)\right)$$

and have

$$\begin{array}{ccc}{d}_0^2\left({\mathbf{a}}_2\right)& =& ((t_1(m_1)+m_0^{\prime },t_0(n_1)+{\alpha }_0(m_0)+n_0),\hfill \\ & & (n_1^{″},t_0{\alpha }_1(m_1)+t_0(n_1)+{\alpha }_0(m_0^{\prime }+m_0)+n_0)),\hfill \\ d_1^2\left({\mathbf{a}}_2\right)& =& \left((m_0^{\prime }+m_0,n_0),({\alpha }_1(m_1)+n_1+n_1^{″},{\alpha }_0(m_0^{\prime }+m_0)+n_0)\right),\hfill \\ d_2^2\left({\mathbf{a}}_2\right)& =& \left((m_0,n_0),(n_1,{\alpha }_0(m_0)+n_0)\right).\hfill \end{array}$$

Furthermore,

$$f:A_2\to (M_1⋉(N_1⋉M_0)⋉(N_1⋉(M_0⋉N_0)))$$

$${\mathbf{a}}_2\mapsto ((m_1,(n_1,m_0^{\prime })),(n_1^{″},(m_0,n_0)))$$

is an isomorphism. By associating $A_2$ with $(M_1⋉(N_1⋉M_0)⋉(N_1⋉(M_0⋉N_0)))$ and $A_1$ with $N_1⋉(M_0⋉N_0)$ we obtain a 2-truncated simplicial Lie algebra

with the face and degeneracy maps;

$$d_0^1(n_1,m_0,n_0)=t_0\left(n_1\right)+{\alpha }_0\left(m_0\right)+n_0,d_1^1(n_1,m_0,n_0)=n_0,s_0^0\left(n_0\right)=(0,0,n_0).$$

and

$$\begin{array}{cc}\hfill d_0^2\left((m_1,(n_1,m_0^{\prime })),(n_1^{″},(m_0,n_0))\right)& =\left(n_1^{″},t_1\left(m_1\right)+m_0^{\prime },t_0\left(n_1\right)+{\alpha }_0\left(m_0\right)+n_0\right),\hfill \\ \hfill d_1^2\left((m_1,(n_1,m_0^{\prime })),(n_1^{″},(m_0,n_0))\right)& =\left({\alpha }_1\left(m_1\right)+n_1+n_1^{″},m_0^{\prime }+m_0,n_0\right),\hfill \\ \hfill d_2^2\left((m_1,(n_1,m_0^{\prime })),(n_1^{″},(m_0,n_0))\right)& =(n_1,m_0,n_0),\hfill \\ \hfill s_0^1(n_1,m_0,n_0)& =\left((0,(0,m_0)),(n_1,(0,n_0))\right),\hfill \\ \hfill s_1^1(n_1,m_0,n_0)& =\left((0,(n_1,0)),(0,(m_0,n_0))\right).\hfill \end{array}$$

Step 3.

The Moore complex of ${\mathbf{A}}^{\left(2\right)}$ that was previously constructed is examined in this step. We demonstrate that this Moore complex is isomorphic to the mapping cone of the internal crossed module and that it has the quadratic module structure for Lie algebras. Then we have $N{A}_0\cong N_0$.

The Moore complex of ${\mathbf{A}}^{\left(2\right)}$ has $N{A}_0\cong N_0.$ Moreover, $N{A}_1=\ker d_0^1.$ Since

$$d_0^1(n_1,m_0,n_0)=t_0\left(n_1\right)+{\alpha }_0\left(m_0\right)+n_0,$$

every element of $N{A}_1$ is of the form

$$\left(-n_1,-m_0,{\alpha }_0\left(n_1\right)+t_0\left(m_0\right)\right).$$

Consequently,

$$\begin{array}{ccc}\hfill f_1:N{A}_1& ⟶& M_0⋉N_1,\hfill \\ \hfill \left(-n_1,-m_0,t_0\left(n_1\right)+{\alpha }_0\left(m_0\right)\right)& ⟼& (m_0,n_1)\hfill \end{array}$$

is an isomorphism. Then, $d_1^1$ induces

$$\begin{array}{ccc}\hfill {\partial }_1:M_0⋉N_1& ⟶& N_0,\hfill \\ \hfill (m_0,n_1)& ⟼& {\alpha }_0\left(m_0\right)+t_0\left(n_1\right).\hfill \end{array}$$

Next, write an element of $N{A}_2$ as $\mathbf{a}=\left((m_1,(n_1,m_0^{\prime })),(n_1^{″},(m_0,n_0\left)\right)\right)$ the conditions $d_0^2\left(\mathbf{a}\right)=0$ and $d_1^2\left(\mathbf{a}\right)=0$ give

$$n_1^{″}=0,m_0^{\prime }=-t_1\left(m_1\right),n_1=-{\alpha }_1\left(m_1\right),m_0=t_1\left(m_1\right),n_0=0.$$

Thus every element of $N{A}_2$ is determined by an element $m_1\in M_1$ and has the form

$$\mathbf{a}\left(m_1\right)=\left((m_1,(-t_1\left(m_1\right),-{\alpha }_1\left(m_1\right))),(0,({\alpha }_1\left(m_1\right),0))\right).$$

Hence $N{A}_2\cong M_1$. Furthermore,

$$d_2^2\left(\mathbf{a}\left(m_1\right)\right)=\left(-{\alpha }_1\left(m_1\right),t_1\left(m_1\right),0\right)$$

and therefore

$$f_1{d}_2^2\left(\mathbf{a}\left(m_1\right)\right)=\left(-t_1\left(m_1\right),{\alpha }_1\left(m_1\right)\right).$$

It follows that

$$\begin{array}{ccc}\hfill {\partial }_2:M_1& ⟶& M_0⋉N_1,\hfill \\ \hfill m_1& ⟼& \left(-t_1\left(m_1\right),{\alpha }_1\left(m_1\right)\right).\hfill \end{array}$$

From ${\alpha }_0{t}_1=t_0{\alpha }_1,$ we have

$$\begin{array}{cc}\hfill {\partial }_1{\partial }_2\left(m_1\right)& ={\partial }_1\left(-t_1\left(m_1\right),{\alpha }_1\left(m_1\right)\right)\hfill \\ \hfill & =-{\alpha }_0{t}_1\left(m_1\right)+t_0{\alpha }_1\left(m_1\right)\hfill \\ \hfill & =0.\hfill \end{array}$$

Thus

$$M_1\stackrel{{\partial }_2}{\to }{M}_0⋉N_1\stackrel{{\partial }_1}{\to }{N}_0$$

is a complex of Lie algebras, where

$${\partial }_2\left(m_1\right)=\left(-t_1\left(m_1\right),{\alpha }_1\left(m_1\right)\right),{\partial }_1(m_0,n_1)={\alpha }_0\left(m_0\right)+t_0\left(n_1\right).$$

For $x=(m_0,n_1),y=(m_0^{\prime },n_1^{\prime })\in M_0⋉N_1,$ the Peiffer element is

$$\langle x,y\rangle =\left({\alpha }_0\left(m_0\right)+t_0\left(n_1\right)\right)·y-[x,y].$$

In particular, the Peiffer lifting satisfies

$$\begin{array}{cc}\hfill \{{\partial }_2\left(m_1\right),{\partial }_2\left(m_1^{\prime }\right)\}& =\left\{(-t_1\left(m_1\right),{\alpha }_1\left(m_1\right)),(-t_1\left(m_1^{\prime }\right),{\alpha }_1\left(m_1^{\prime }\right))\right\}\hfill \\ & =[m_1,m_1^{\prime }].\hfill \end{array}$$

Remark 2. 

In [20] the functor from internal crossed modules to crossed squares is given. Analogous calculations will also be valid for Lie algebras. That is, from the internal crossed module given in Definition 4, we obtain a crossed square

This is an alternative method for mapping an internal crossed module to a 2-crossed module of Lie algebra, distinct from the previously established method outlined in reference [7]. Furthermore, the functor mapping a 2-crossed module to a quadratic module is given in [23] for the Lie algebra case.

The construction from an internal crossed module to a quadratic module is the diagram

(4)

where the functors $\mathcal{E}$ and $\mathcal{U}$ are the standard constructions recalled from [7,23].

5. Internal Crossed Modules from Quadratic Modules of Lie Algebras

In this section, we construct an internal crossed module from a quadratic module in three steps. We first associate a 2-truncated simplicial Lie algebra to the quadratic module. Applying the functor $M(-,2)$ gives a crossed square of Lie algebras, and Proposition 2 then yields the required internal crossed module. Let

be a quadratic module of Lie algebras. Let us consider its associated 2-truncated simplicial algebra

with face and degeneracy maps defined as:

$$d_0^1(c_1,c_0)=c_0,d_1^1(c_1,c_0)=\partial \left(c_1\right)+c_0,s_0^0\left(c_0\right)=(0,c_0)$$

and

$$\begin{array}{cccccccccccc}{s}_0^1(c_1,c_0)=(0,0,c_1,c_0)& & & & & d_0^2(c_2,c_1^{\prime },c_1,c_0)& =(c_1,c_0)\hfill & & & & & \\ s_1^1(c_1,c_0)=(0,c_1,0,c_0)& & & & & d_1^2(c_2,c_1^{\prime },c_1,c_0)& =(c_1^{\prime }+c_1,c_0)\hfill & & & & & \\ & & & & & d_2^2(c_2,c_1^{\prime },c_1,c_0)& =(c_1^{\prime },\partial \left(c_1\right)+c_0)\hfill & & & & & \end{array}$$

Additionally, Ellis’s work in [15] demonstrated that the Moore complex of the simplicial Lie algebra ${\mathbf{G}}^{\left(3\right)}$ is trivial in dimensions $n\ge 3$, which means that the length of the Moore complex of ${\mathbf{G}}^{\left(3\right)}$ is 2. Now consider the functor $M(-,2)$ that is provided in [8] to convert simplicial Lie algebras to crossed squares. Consequently, the crossed square that corresponds to this 2-truncated simplicial Lie algebra is

where

$$N{\mathbf{G}}_1^{\left(3\right)}=\ker d_0^1=\{(c_1,0)\mid c_1\in C_1\}\cong C_1$$

since from the definition of $d_0^1$, $(c_1,c_0)\in ker{d}_0^1$ implies $c_0=0$,

$$N{\mathbf{G}}_2^{\left(3\right)}=\ker d_0^2\cap \ker d_1^2=\{(c_2,0,0,0)\mid c_2\in C_2\}\cong C_2$$

since $d_0^2(c_2,c_1^{\prime },c_1,c_0)=(c_1,c_0)$ implies $c_1=c_0=0$ and $d_1^2(c_2,c_1^{\prime },c_1,c_0)=(c_1^{\prime }+c_1,c_0)$ implies $c_1^{\prime }+c_1=0$ for $(c_2,c_1^{\prime },c_1,c_0)\in ker{d}_0^2\cap ker{d}_1^2$, and $d_1^1(c_1,c_0)=\partial \left(c_1\right)+c_0$ we obtain

$${\overline{N\mathbf{G}}}_1^{\left(3\right)}:=\ker d_1^1=\left\{(c_1,c_0)\in C_1⋉C_0|\partial \left(c_1\right)+c_0=0\right\}$$

for $(c_1,c_0)\in ker{d}_1^1$.

Thus, the crossed square associated with the quadratic module can be given as follows:

$C_1$ acts on $C_2$ and $\overline{{\mathbf{NG}}_1^{\left(3\right)}}$ and $C_1⋉C_0$ acts on $C_2$, $\overline{{\mathbf{NG}}_1^{\left(3\right)}}$, and $C_1$.

Given the functor chain

$$\mathbf{Int}\left({\mathbf{XMod}}_{\mathbf{Lie}}\right)⟶{\mathbf{BiSimp}}_{\mathbf{LieAlg}}⟶{\mathbf{Simp}}_{\mathbf{LieAlg}}⟶{\mathrm{X}}_{\mathbf{2}}{\mathbf{Mod}}_{\mathbf{Lie}}$$

established in Section 4, the reverse direction is obtained by departing from a quadratic module and constructing the simplicial Lie algebra $G\left(3\right)$. Since Ellis demonstrated in [7] that the Moore complex of $G\left(3\right)$ has length at most 2, an application of the functor $M(-,2)$ yields the corresponding crossed square. The associated internal crossed module is then produced from this crossed square by Proposition 2.

The construction beginning with a quadratic module is summarized by

(5)

where the dotted arrow is the composite $F\circ M(-,2)\circ \mathcal{G}$ constructed in this section.

6. Conclusions

We have constructed relations among crossed squares of Lie algebras, internal crossed modules of Lie algebras, simplicial and bisimplicial Lie algebras, and quadratic modules of Lie algebras. These functors enable definitions of Lie algebraic homotopy 3-types defined by Moore complexes to be analyzed.

Crossed module techniques also appear in strict 2-term structures such as Nijenhuis conformal and Rota–Baxter Lie H-pseudoalgebras [24,25,26], as well as in the cohomology and homotopy of embedding tensors and tensor hierarchies [27,28].

This suggests extending the present constructions to these settings and studying their relation with cohomology, deformations, and non-abelian extensions. A further study is to determine conditions under which the comparison functors give equivalences.