Contents

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.

Discussion in the generality of super Lie algebras includes