nLab
Lie algebra extension

Idea

The notion of Lie algebra extension is a special case of the general notion of Extensions in ∞-Lie algebra cohomology.

Definition

A short exact sequences of Lie algebras is a diagram

$$0\to 𝔨\stackrel{i}{\to }𝔤\stackrel{p}{\to }𝔟\to 0$$0\to \mathfrak{k} \overset{i}\to \mathfrak{g}\overset{p}\to\mathfrak{b}\to 0

where $𝔨,𝔤,𝔟$ are Lie algebras, $i,p$ are homomorphisms of Lie algebras and the underlying diagram of vector spaces is exact, i.e. $\mathrm{Ker}(p)=\mathrm{Im}(i)$, $\mathrm{Ker}(i)=0$ and $\mathrm{Im}(p)=0$.

We also say that this diagram (and sometimes, loosely speaking, $𝔤$ itself) is a Lie algebra extension of $𝔟$ by the “kernel” $𝔨$.

Lie algebra extensions may be obtained from Lie group group extensions via the tangent Lie algebra functor.

Classification by nonabelian Lie algebra cocycles

We discuss how Lie algebra extensions are classified by cocycles in nonabelian Lie algebra cohomology.

Each element $g\in 𝔤$ defines a derivative $\varphi (g)$ on $𝔨$ by $\varphi (g)(k)=[g,k]$. The rule $g\mapsto \varphi (g)$ defines a homomorphism of Lie algebras $\varphi :𝔤\to \mathrm{Der}(𝔨)$. Indeed,

$$\varphi ([g_1,g_2])(k)=[[g_1,g_2],k]=[[g_1,k],g_2]+[g_1,[g_2,k]]=-\varphi (g_2)([g_1,k])+\varphi (g_1)([g_2,k])=[-\varphi (g_2)\circ \varphi (g_1)+\varphi (g_1)\circ \varphi (g_2)](k)=[\varphi (g_1),\varphi (g_2)](k),$$\phi([g_1,g_2])(k) = [[g_1,g_2],k] = [[g_1,k],g_2] + [g_1,[g_2,k]] = -\phi(g_2)([g_1,k]) + \phi(g_1)([g_2,k]) = [-\phi(g_2)\circ\phi(g_1) + \phi(g_1)\circ\phi(g_2)](k) = [\phi(g_1),\phi(g_2)](k),

for all $g_1,g_2\in 𝔤$, for all $k\in 𝔨$. The restriction $\varphi {\mid }_{𝔨}$ takes (by definition) values in the Lie subalgebra $\mathrm{Int}(𝔨)$ of inner derivatives of $𝔨$. If $g_1$ and $g_2$ are in the same coset, that is $g_1+𝔨=g_2+frakk$, then there is $k\in 𝔨$ with $g_1+k=g_2$ and such that for all $k\prime \in 𝔨$ we have $\varphi (g_1)+\varphi (k)=\varphi (g_1+k\prime )=\varphi (g_2+k+k\prime )=\varphi (g_2)+\varphi (k+k\prime )$ and therefore

$$\begin{array}{ccc}\varphi (g_1)+\mathrm{Int}(𝔨)& =& \varphi (g_1)+\varphi (𝔨)\\ & =& \varphi (g_1+𝔨)\\ & =& \varphi (g_2+𝔨)\\ & =& \varphi (g_2)+\varphi (𝔨)\\ & =& \varphi (g_2)+\mathrm{Int}(𝔨).\end{array}$$\array{\phi(g_1) + Int(\mathfrak{k}) &=& \phi(g_1) + \phi(\mathfrak{k})\\ &=& \phi(g_1 + \mathfrak{k}) \\ &=& \phi(g_2 + \mathfrak{k}) \\ &=& \phi(g_2) + \phi(\mathfrak{k})\\ &=& \phi(g_2) + Int(\mathfrak{k}).}

Thus we obtain a well-defined map ${\varphi }_{*}:𝔤/𝔨\to \mathrm{Der}(𝔨)/\mathrm{Int}(𝔨)$.

Choose a $k$-linear section of the projection $𝔤\to 𝔤/𝔨\cong 𝔟$ and denote by $\psi$ the composition $\varphi \circ \sigma$ where $\sigma :𝔤/𝔨=𝔟\to 𝔤$. One considers the problem of reconstructing the group $frakg$ from the knowledge of $\psi :𝔤/𝔨\to \mathrm{Der}(𝔨)$ and $𝔨$. In order to derive the necessary relations we will identify $𝔤$ with $𝔟\times 𝔨$ (as a set).

Indeed, write each element $g\in G$ as $\sigma (b)+k,b\in 𝔤/𝔨$, $k\in 𝔨$ by setting $b:=[g],k:=-\sigma ([g])+g$. Elements $b\in 𝔟$ and $k\in 𝔨$ in that decomposition are unique. Thus we obtain a bijection $𝔤\to 𝔟\times 𝔨$, $g\mapsto ([g],-\sigma ([g])+g)$. The commutation rule has to be figured out. If $(b_1,k_1)=g_1$, and $(b_2,k_2)=g_2$, then

(1)$$[g_1,g_2]=[\sigma (b_1)+k_1,\sigma (b_2)+k_2]=[\sigma (b_1),\sigma (b_2)]+[\sigma (b_1),k_2]-[\sigma (b_2),k_1]+[k_1,k_2].$$[g_1,g_2] = [\sigma(b_1) + k_1,\sigma(b_2) + k_2] = [\sigma(b_1),\sigma(b_2)] + [\sigma(b_1),k_2] - [\sigma(b_2),k_1] +[k_1,k_2].

Now $[\sigma (b_1),\sigma (b_2)]\in [b_1{b}_2]$ so it can be represented uniquely in the form $\sigma ([b_1,b_2])+k$ where $k\in 𝔨$ can be obtained by evaluating the antisymmetric $k$-bilinear form $\chi :𝔟\wedge 𝔟\to 𝔨$ defined by $\chi (b_1\wedge b_2)=-\sigma ([b_1,b_2])+[\sigma (b_1),\sigma (b_2)]$ on $(b_1,b_2)$. Then formula (1) becomes

$$\begin{array}{ccc}[g_1,g_2]& =& \sigma ([b_1,b_2])+\chi (b_1\wedge b_2)+\varphi (\sigma (b_1))(k_2)+\varphi (-\sigma (b_2))(k_1)+[k_1,k_2]\\ & =& \sigma ([b_1,b_2])+\chi (b_1\wedge b_2)+\psi (b_1)(k_2)-\psi (b_2)(k_1)+[k_1,k_2].\end{array}$$\array{ [g_1,g_2] & = & \sigma([b_1,b_2]) + \chi(b_1\wedge b_2) + \phi(\sigma(b_1))(k_2) + \phi(-\sigma(b_2))(k_1) + [k_1,k_2] \\ & = & \sigma([b_1,b_2]) + \chi(b_1\wedge b_2) + \psi(b_1)(k_2) -\psi(b_2)(k_1) + [k_1,k_2]. }

so that

(2)$$(b_1,k_1)(b_2,k_2)=([b_1,b_2],\chi (b_1\wedge b_2)+\psi (b_1)(k_2)-\psi (b_2)(k_1)+[k_1,k_2]).$$(b_1,k_1)(b_2,k_2) = ([b_1,b_2],\chi(b_1\wedge b_2) + \psi(b_1)(k_2) - \psi(b_2)(k_1) + [k_1,k_2]).

Thus all the information about the commutators is encoded in functions $\chi :𝔫\wedge 𝔟\to \mathrm{Der}(𝔨)$ and $\psi :𝔟\to \mathrm{Der}(𝔨)$, without knowledge of $\sigma$.

However, not every pair $(\chi ,\psi )$ will give some commutation rule on $𝔟\times k$ satisfying Jacobi identity, and also some different pairs may lead to the isomorphic extensions.

In order to satisfy the Jacobi identity, this pair needs to form a nonabelian 2-cocycle in the sense of nonabelian Lie algebra cohomology.

Examples

• Kostant Souriau extension