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cohomology

# Contents

## Idea

An extension of a Lie algebra $\mathfrak{g}$ is another Lie algebra $\hat {\mathfrak{g}}$ that is equipped with a surjective Lie algebra homomorphism to $\mathfrak{g}$

$\array{ \hat{\mathfrak{g}} \\ \downarrow \\ \mathfrak{g} } \,.$

For non-tivial extensions, this homomorphism has a kernel $\mathfrak{a} \hookrightarrow \hat \mathfrak{g}$ , consisting of those elements of $\hat{\mathfrak{g}}$ that map to the zero element in $\mathfrak{g}$. That kernel is a sub-Lie algebra of $\hat{\mathfrak{g}}$ and hence one says that $\hat\mathfrak{g}$ is an extension of $\mathfrak{g}$ by $\mathfrak{a}$.

$\array{ \mathfrak{a} &\hookrightarrow& \hat{\mathfrak{g}} \\ &&\downarrow \\ && \mathfrak{g} } \,.$

This means equivalently that there is a short exact sequence of Lie algebras of the form

$0 \to \mathfrak{a} \longrightarrow \hat \mathfrak{g} \longrightarrow \mathfrak{g} \to 0 \,.$

When $\mathfrak{a}$ happens to be abelian, hence when its Lie bracket is trivial, then one speaks of an abelian extension, and when furthermore the Lie bracket of $\hat\mathfrak{g}$ vanishes as soon as already one of its arguments is in $\mathfrak{a}$, then one has a central extension ($\mathfrak{a}$ is in the center of $\hat \mathfrak{g}$).

Central extensions by the ground field (say $\mathbb{R}$) are equivalently induced by a 2-cocyle $\mu_2$ in the Lie algebra cohomology of $\mathfrak{g}$ with coefficients in the ground field, say $\mathbb{R}$, i.e. by linear maps

$\mu_2 \colon \mathfrak{g} \wedge \mathfrak{g} \longrightarrow \mathbb{R}$

satisfying some conditions. The corresponding extension of $\mathfrak{g}$ is then, at the level of underlying vector space, the direct sum $\hat \mathfrak{g} = \mathfrak{g} \oplus \mathbb{R}$, equipped with the Lie bracket given by the formula

$[(x_1,t_1), (x_2,t_2)] = ([x_1,x_2], \mu_2(x_1,x_2))$

for all $x_1,x_2 \in \mathfrak{g}$ and $t_1,t_2 \in \mathbb{R}$. The condition on $\mu_2$ to be a 2-cocycle is precisely the condition that this formula satisfies the Jacobi identity.

If one regards all Lie algebras here as being special cases of Lie 2-algebras, then the 2-cocycle $\mu_2$ may itself be thought of as a homomorphism, namely from $\mathfrak{g}$ to the line Lie 2-algebra $b\mathbb{R}$. With this, then $\hat \mathfrak{g}$ given by the above formula is simply the homotopy fiber of $\mu_2$, and the whole story comes down to saying that there is a homotopy fiber sequence of L-∞ algebras of the form

$\array{ \mathbb{R} &\hookrightarrow& \hat{\mathfrak{g}} \\ &&\downarrow \\ && \mathfrak{g} &\stackrel{\mu_2}{\longrightarrow}& b \mathbb{R} } \,.$

This perspective on Lie algebra extensions makes it evident how the concept generalizes to a concept of L-∞ algebra extensions.

Of course extensions need not be central or even abelian. An important class of non-abelian extensions are semidirect product Lie algebras. These are given by an Lie action of $\mathfrak{g}$ on $\mathfrak{a}$, hence a homomorphism $\rho \colon \mathfrak{g}\longrightarrow \mathfrak{der}(\mathfrak{a})$ to the derivations on $\mathfrak{a}$ and with this the bracket on $\mathfrak{g} \oplus \mathfrak{a}$ is given by the formula

$[(x_1,t_1), (x_2,t_2)] = ( [x_1,x_2], \;([t_1,t_2] + \rho(x_1)(t_2) - \rho(x_2)(t_1)) ) \,.$

## Definition

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

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.

## Properties

### Classification by nonabelian Lie algebra cohomology

We discuss how in general 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.

## Examples

• The Heisenberg Lie algebra is an extension of $\mathbb{R}^{2}$, regarded as an abelian Lie algebra, by $\mathbb{R}$ with the corresponding 2-cocycle $\mu_2$ being the canonical commutation relation $\mu_2(q,q)= 0$, $\mu_2(p,p)= 0$, $\mu_2(q,p) = 1$.

• More generally, the Kostant Souriau extension exhibits a Poisson bracket on a symplectic manifold as an extension of the Lie algebra of Hamiltonian vector fields.

For more discussion putting these two examples in perspective see also at quantization – Motivation from classical mechanics and Lie theory.

## References

Discussion in the generality of super Lie algebras includes