Spatial realization of a Lie algebra and Bar construction of a group

TL;DR

This paper establishes an isomorphism between the spatial realization of a rational Lie algebra and the simplicial bar construction of a group, linking algebraic and topological structures through the Baker-Campbell-Hausdorff product.

Contribution

It demonstrates a novel isomorphism connecting the spatial realization of a Lie algebra with the bar construction of a group, advancing the understanding of their algebraic-topological relationship.

Findings

Spatial realization of a Lie algebra is isomorphic to the bar construction of a group.

The isomorphism is established via the Baker-Campbell-Hausdorff product.

The work bridges Lie algebra structures with simplicial group constructions.

Abstract

We prove that the spatial realization of a rational complete Lie algebra $L$, concentrated in degree 0, is isomorphic to the simplicial bar construction on the group, obtained from the Baker-Campbell-Hausdorff product on $L$.

Equations

$$⟨-⟩:\mathbf{cdgl}\to \mathbf{Sset},$$

$$\hat{\mathbb{L}}(V)=\underset{n}{\underleftarrow{\lim }}\mathbb{L}(V)/{\mathbb{L}}^{\ge n}(V),$$

$${\mathfrak{L}}_0=(\mathbb{L}(a_0),d),d{a}_0=-\frac{1}{2}[a_0,a_0].$$

$$d{a}_{01}=[a_{01},a_1]+\frac{{\mathrm{ad}}_{01}}{e^{{\mathrm{ad}}_{01}}-1}(a_1-a_0).$$

$${\mathfrak{L}}_2=(\hat{\mathbb{L}}(a_0,a_1,a_2,a_{01},a_{02},a_{12},a_{012}),d)\text{ with }d(a_{012})=a_{01}\ast a_{12}\ast a_{02}^{-1}-[a_0,a_{012}].$$

$$⟨L⟩:={\mathrm{Hom}}_{\mathbf{cdgl}}({\mathfrak{L}}_{\bullet },L).$$

$$\delta^{i}a_{i_{0}\dots i_{p}}=a_{j_{0}\dots j_{p}}\quad\text{with}\quad j_{k}=\left\{\begin{array}[]{ll}i_{k}&\text{if }i_{k}<i,\\ i_{k}+1&\text{if }i_{k}\geq i,\end{array}\right.$$

$${\sigma }^i{a}_{i_0\dots i_p}$$

$$d_0[g_1|\dots |g_n]$$

$$d_i[g_1|\dots |g_n]$$

$$d_n[g_1|\dots |g_n]$$

$$0=df(a_{0rs})=f(a_{0r})\ast f(a_{rs})\ast f(a_{0s}{)}^{-1}.$$

$$f(a_{rs})=f(a_{0r}{)}^{-1}\ast f(a_{0s}).$$

$$\Phi :{\mathrm{Hom}}_{\mathbf{cdgl}}({\mathfrak{L}}_n,L)\to L^n,\text{defined by}f\mapsto (f(a_{01}),f(a_{02}),\dots ,f(a_{0n})).$$

$$\delta^{i}(a_{0r})=\left\{\begin{array}[]{ll}a_{0,r}&\text{if}\;r<i\\ a_{0,r+1}&\text{if}\;r\geq i\end{array}\right.\quad\text{for}\quad i>0,\quad\text{and}\quad\delta^{0}(a_{0,r})=a_{1,r+1}.$$

$$d_0(f(a_{01}),\dots ,f(a_{0(n-1)}))$$

$$d_0(x_1,\dots ,x_n)=(x_1^{-1}\ast x_2,\dots ,x_1^{-1}\ast x_n).$$

$$d_i(x_1,\dots ,x_n)=(x_1,\dots ,{\hat{x}}_i,\dots ,x_n).$$

$$\sigma^{i}(a_{0r})=\left\{\begin{array}[]{ll}a_{0r}&\text{if}\;r\leq i\\ a_{0,r-1}&\text{if}\;r>i\end{array}\right.\quad\text{for}\quad i>0,$$

$${\sigma }^0(a_{0r})=a_{0,r-1}\text{if}r>1,\text{and}{\sigma }^0(a_{0,1})=0,$$

$$s_0(x_1,\dots ,x_n)=(0,x_1,\dots ,x_n)$$

$$s_i(x_1,\dots ,x_n)=(x_1,\dots ,x_i,x_i,\dots ,x_n).$$

$$\Psi :{\mathrm{Hom}}_{\mathbf{cdgl}}({\mathfrak{L}}_{\bullet },L)\to B_{\bullet }(\exp L)$$

$$\Psi (f)=[f(a_{01})|f(a_{01}{)}^{-1}f(a_{02})|f(a_{02}{)}^{-1}f(a_{03})|\dots |f(a_{0(n-1)}{)}^{-1}f(a_{0n})]$$

Full text

The authors are partially supported by the MINECO-FEDER grant MTM2016-78647-P. The second author is partially supported by the ANR-11-LABX-0007-01 “CEMPI”

Introduction

In [4], we construct a cosimplicial differential graded complete Lie algebra (henceforth ${\mathrm{cdgl}}$) $({\mathfrak{L}}_{\bullet},d)$, in which $({\mathfrak{L}}_{1},d)$ is the Lawrence-Sullivan model of the interval introduced in [7]. As in the work of Sullivan ([9]) for differential commutative graded algebras, the existence of this cosimplicial object gives an adjoint pair of functors between the category ${\mathbf{cdgl}}$ of cdgl’s and the category ${\mathbf{Sset}}$ of simplicial sets, see [4] or [2]. In this work, we focus on one of them, the spatial realization functor,

[TABLE]

defined by $\langle L\rangle={\mathrm{Hom}}_{{\mathbf{cdgl}}}({\mathfrak{L}}_{\bullet},L)$ for $L\in{\mathbf{cdgl}}$. (Let us also notice that $\langle L\rangle$ is isomorphic to the nerve of $L$, a deformation retract of the Getzler-Hinich realization, see [1], [8].)

More precisely, we are interested in the realization $\langle L\rangle$ of a complete Lie algebra, $L$, concentrated in degree 0 and (thus) with the differential 0. In this case, a group structure can be defined on the set $L$ from the Baker-Campbell-Hausdorff formula. We denote by $\exp\;L$ this group. The realization $\langle L\rangle$ is an Eilenberg-MacLane space of type $K(\pi,1)$, see [5]. The purpose of this work is the determination of $\langle L\rangle$ up to isomorphism.

Main Theorem ​​**.**

Let $L$ be a complete differential graded Lie algebra, concentrated in degree 0. Then, its spatial realization $\langle L\rangle$ is isomorphic to the simplicial bar construction on $\exp\;L$.

Let us emphasize that for an Eilenberg-Maclane space $K(G,1)$, of associated Lie algebra $L$, then the realizations of the $A_{PL}(K(G,1))$ of Sullivan ([9]) and ${\mathrm{MC}}_{*}(L)$ of Getzler ([6]) are weakly equivalent to the simplicial bar resolution. In our setting, the simplicial set ${\mathrm{Hom}}({\mathcal{L}}_{\bullet},L)$ is isomorphic to the simplicial bar resolution. With [2], we know that spaces more general than a $K(G,1)$ admits a cdgl model $L$. For them, the simplicial set ${\mathrm{Hom}}({\mathcal{L}}_{\bullet},L)$ appears as a natural extension of the bar construction. We will come back on this point in a forthcoming work.

In Section 1, we recall basic background on ${\mathbf{cdgl}}$ and our construction ${\mathfrak{L}}_{\bullet}$. Section 2 consists of the proof of the Main theorem.

1. Some reminders

We first recall the construction of the cosimplicial ${\mathrm{cdgl}}$ ${\mathfrak{L}}_{\bullet}$. Let $V$ be a finite dimensional graded vector space. The completion of the free graded Lie algebra on $V$, $\mathbb{L}(V)$, is the inverse limit,

[TABLE]

where $\mathbb{L}^{\geq n}(V)$ is the ideal generated by the Lie brackets of length $\geq n$. We call ${\widehat{\mathbb{L}}}(V)$ the free complete graded Lie algebra on $V$.

As a graded Lie algebra, ${\mathfrak{L}}_{n}$ is the free complete graded Lie algebra on the rational vector space generated by the elements $a_{i_{0}\dots i_{k}}$ of degree $|a_{i_{0}\dots i_{k}}|=k-1$, with $0\leq i_{0}<\dots<i_{k}\leq n$. We denote by ${\rm ad}_{i_{0}\dots i_{k}}$ the Lie derivation $[a_{i_{0}\dots i_{k}},-]$. The ${\mathrm{cdgl}}$ ${\mathfrak{L}}_{n}$ satisfies the following properties.

– ${\mathfrak{L}}_{0}$ is the free ${\mathrm{cdgl}}$ on a Maurer-Cartan element $a_{0}$, that is:

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– ${\mathfrak{L}}_{1}=({\widehat{\mathbb{L}}}(a_{0},a_{1},a_{01}),d)$ is the Lawrence-Sullivan interval (see [7]), where $a_{0}$ and $a_{1}$ are Maurer-Cartan elements and

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– A model ${\mathfrak{L}}_{2}$ for the triangle has been described in [4] (see also [3]):

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Here $*$ denotes the Baker-Campbell-Hausdorff product defined for any pair of elements $a$, $b$ of degree 0 by $a\ast b=\log(\exp^{a}\,\exp^{b})$.

– Moreover these structures appear in each ${\mathfrak{L}}_{n}$: each vertex $a_{r}$ is a Maurer-Cartan element, each triple $(a_{r},a_{s},a_{rs})$ is a Lawrence-Sullivan interval and each family $(a_{r},a_{s},a_{t},a_{rs},a_{rt},a_{st},a_{rst})$ is a triangle as above.

The family $({\mathfrak{L}}_{n})_{n\geq 0}$ forms a cosimplicial ${\mathrm{cdgl}}$ which allows the definition of the spatial realization of $L\in{\mathbf{cdgl}}$ by,

[TABLE]

The cofaces $\delta^{i}$ and the codegeneracies $\sigma^{i}$ of the cosimplicial ${\mathrm{cdgl}}$ ${\mathfrak{L}}_{\bullet}$ are defined by

[TABLE]

[TABLE]

2. Proof of the main theorem

The simplicial bar construction on a group $G$, is the simplicial set $B_{\bullet}G$ with set of $n$-simplices $B_{n}G=G^{n}$. Its elements are denoted $[g_{1}|\dots|g_{n}]$, with $g_{i}\in G$. The faces $d_{i}$ and degeneracies $s_{i}$ of $B_{\bullet}G$ are defined as follows:

[TABLE]

The degeneracy $s_{i}$ inserts the identity $e$ of $G$ in position $i$.

Let $L\in{\mathbf{cdgl}}$ be generated in degree 0 and $f\colon{\mathfrak{L}}_{n}\to L$ a morphism in ${\mathbf{cdgl}}$. For degree reasons, we have $f(a_{i_{0}\dots i_{k}})=0$ if $k\neq 1$. Moreover, since $f$ commutes the differential, from the definition of the differential in ${\mathfrak{L}}_{2}$, we get

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Therefore, for any $r,s>0$, we have

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The map $f$ being entirely defined by its values on the $a_{0i}$, we have a bijection

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We now determine the image of the faces and degeneracies on ${\mathrm{Hom}}_{{\mathbf{cdgl}}}({\mathfrak{L}}_{\bullet},L)$, induced from (1.1) and (1.6). For the face operators, as only the elements $(a_{0r})$ play a role, it suffices to consider,

[TABLE]

Let $f\colon{\mathfrak{L}}_{n}\to L$ be specified by $(f(a_{01}),\dots,f(a_{0n}))$. Then $d_{0}f=f\circ\delta^{0}\colon{\mathfrak{L}}_{n-1}\to L$ is described by

[TABLE]

Therefore, the face operator $d_{0}$ on $L^{\bullet}$, induced from $\Phi$, is

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Similar arguments give, for $i>0$,

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As for the degeneracies, starting from

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[TABLE]

we get

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and for $i>0$,

[TABLE]

Now a straightforward and easy computation shows that the morphism

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defined by

[TABLE]

is an isomorphism of simplicial sets.