cohomology

∞-Lie theory

# Contents

## 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

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

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

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

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 $g \in \mathfrak{g}$ defines a derivative $\phi(g)$ on $\mathfrak{k}$ by $\phi(g)(k) = [g,k]$. The rule $g \mapsto \phi(g)$ defines a homomorphism of Lie algebras $\phi : \mathfrak{g} \rightarrow Der(\mathfrak{k})$. Indeed,

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

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

$\array{\phi(g_1) + Int(\mathfrak{k}) &=& \phi(g_1) + \phi(\mathfrak{k})\\ &=& \phi(g_1 + \mathfrak{k}) \\ &=& \phi(g_2 + \mathfrak{k}) \\ &=& \phi(g_2) + \phi(\mathfrak{k})\\ &=& \phi(g_2) + Int(\mathfrak{k}).}$

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

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

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

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

Now $[\sigma(b_1),\sigma(b_2)] \in [b_1b_2]$ so it can be represented uniquely in the form $\sigma([b_1,b_2]) + k$ where $k \in \mathfrak{k}$ can be obtained by evaluating the antisymmetric $k$-bilinear form $\chi : \mathfrak{b} \wedge \mathfrak{b} \rightarrow \mathfrak{k}$ 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

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

so that

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

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

However, not every pair $(\chi,\psi)$ will give some commutation rule on $\mathfrak{b} \times k$ 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.

## References

Discussion in the generality of super Lie algebras includes

Revised on March 25, 2015 11:02:00 by Urs Schreiber (195.113.30.252)