Lie algebra extension




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

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

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

𝔞 𝔤^ 𝔤. \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

0𝔞𝔤^𝔤0. 0 \to \mathfrak{a} \longrightarrow \hat \mathfrak{g} \longrightarrow \mathfrak{g} \to 0 \,.

When 𝔞\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 𝔤^\hat\mathfrak{g} vanishes as soon as already one of its arguments is in 𝔞\mathfrak{a}, then one has a central extension (𝔞\mathfrak{a} is in the center of 𝔤^\hat \mathfrak{g}).

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

μ 2:𝔤𝔤 \mu_2 \colon \mathfrak{g} \wedge \mathfrak{g} \longrightarrow \mathbb{R}

satisfying some conditions. The corresponding extension of 𝔤\mathfrak{g} is then, at the level of underlying vector space, the direct sum 𝔤^=𝔤\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],μ 2(x 1,x 2)) [(x_1,t_1), (x_2,t_2)] = ([x_1,x_2], \mu_2(x_1,x_2))

for all x 1,x 2𝔤x_1,x_2 \in \mathfrak{g} and t 1,t 2t_1,t_2 \in \mathbb{R}. The condition on μ 2\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 μ 2\mu_2 may itself be thought of as a homomorphism, namely from 𝔤\mathfrak{g} to the line Lie 2-algebra bb\mathbb{R}. With this, then 𝔤^\hat \mathfrak{g} given by the above formula is simply the homotopy fiber of μ 2\mu_2, and the whole story comes down to saying that there is a homotopy fiber sequence of L-∞ algebras of the form

𝔤^ 𝔤 μ 2 b. \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 𝔤\mathfrak{g} on 𝔞\mathfrak{a}, hence a homomorphism ρ:𝔤𝔡𝔢𝔯(𝔞)\rho \colon \mathfrak{g}\longrightarrow \mathfrak{der}(\mathfrak{a}) to the derivations on 𝔞\mathfrak{a} and with this the bracket on 𝔤𝔞\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]+ρ(x 1)(t 2)ρ(x 2)(t 1))). [(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)) ) \,.


A short exact sequence of Lie algebras is a diagram

0𝔨i𝔤p𝔟0 0\to \mathfrak{k} \overset{i}\to \mathfrak{g}\overset{p}\to\mathfrak{b}\to 0

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

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

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


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

ϕ([g 1,g 2])(k)=[[g 1,g 2],k]=[[g 1,k],g 2]+[g 1,[g 2,k]]=ϕ(g 2)([g 1,k])+ϕ(g 1)([g 2,k])=[ϕ(g 2)ϕ(g 1)+ϕ(g 1)ϕ(g 2)](k)=[ϕ(g 1),ϕ(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 2𝔤g_1,g_2 \in \mathfrak{g}, for all k𝔨k \in \mathfrak{k}. The restriction ϕ| 𝔨\phi|_{\mathfrak{k}} takes (by definition) values in the Lie subalgebra Int(𝔨)Int(\mathfrak{k}) of inner derivatives of 𝔨\mathfrak{k}. If g 1g_1 and g 2g_2 are in the same coset, that is g 1+𝔨=g 2+frakkg_1 + \mathfrak{k} = g_2 + {\frak k}, then there is k𝔨k \in \mathfrak{k} with g 1+k=g 2g_1 + k = g_2 and such that for all k𝔨k' \in \mathfrak{k} we have ϕ(g 1)+ϕ(k)=ϕ(g 1+k)=ϕ(g 2+k+k)=ϕ(g 2)+ϕ(k+k)\phi(g_1) + \phi(k) = \phi(g_1 + k') = \phi(g_2 + k + k') = \phi(g_2)+\phi(k + k') and therefore

ϕ(g 1)+Int(𝔨) = ϕ(g 1)+ϕ(𝔨) = ϕ(g 1+𝔨) = ϕ(g 2+𝔨) = ϕ(g 2)+ϕ(𝔨) = ϕ(g 2)+Int(𝔨).\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 ϕ *:𝔤/𝔨Der(𝔨)/Int(𝔨)\phi_* : \mathfrak{g}/\mathfrak{k} \to Der(\mathfrak{k})/Int(\mathfrak{k}).

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

Indeed, write each element gGg \in G as σ(b)+k,b𝔤/𝔨\sigma(b) + k, b \in \mathfrak{g}/\mathfrak{k}, k𝔨k \in \mathfrak{k} by setting b:=[g],k:=σ([g])+gb := [g], k := -\sigma([g]) + g. Elements b𝔟b \in \mathfrak{b} and k𝔨k \in \mathfrak{k} in that decomposition are unique. Thus we obtain a bijection 𝔤𝔟×𝔨\mathfrak{g} \rightarrow \mathfrak{b} \times \mathfrak{k}, g([g],σ([g])+g)g \mapsto ([g], -\sigma([g]) + g ). The commutation rule has to be figured out. If (b 1,k 1)=g 1(b_1,k_1) = g_1, and (b 2,k 2)=g 2(b_2,k_2) = g_2, then

(1)[g 1,g 2]=[σ(b 1)+k 1,σ(b 2)+k 2]=[σ(b 1),σ(b 2)]+[σ(b 1),k 2][σ(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 [σ(b 1),σ(b 2)][b 1b 2][\sigma(b_1),\sigma(b_2)] \in [b_1b_2] so it can be represented uniquely in the form σ([b 1,b 2])+k\sigma([b_1,b_2]) + k where k𝔨k \in \mathfrak{k} can be obtained by evaluating the antisymmetric kk-bilinear form χ:𝔟𝔟𝔨\chi : \mathfrak{b} \wedge \mathfrak{b} \rightarrow \mathfrak{k} defined by χ(b 1b 2)=σ([b 1,b 2])+[σ(b 1),σ(b 2)]\chi(b_1 \wedge b_2) = - \sigma([b_1,b_2]) + [\sigma(b_1),\sigma(b_2)] on (b 1,b 2)(b_1,b_2). Then formula (1) becomes

[g 1,g 2] = σ([b 1,b 2])+χ(b 1b 2)+ϕ(σ(b 1))(k 2)+ϕ(σ(b 2))(k 1)+[k 1,k 2] = σ([b 1,b 2])+χ(b 1b 2)+ψ(b 1)(k 2)ψ(b 2)(k 1)+[k 1,k 2].\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],χ(b 1b 2)+ψ(b 1)(k 2)ψ(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 χ:𝔫𝔟Der(𝔨)\chi : \mathfrak{n} \wedge \mathfrak{b} \rightarrow Der(\mathfrak{k}) and ψ:𝔟Der(𝔨)\psi : \mathfrak{b} \to Der(\mathfrak{k}), without knowledge of σ\sigma.

However, not every pair (χ,ψ)(\chi,\psi) will give some commutation rule on 𝔟×k\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.


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

See also

Revised on August 27, 2015 04:25:20 by Urs Schreiber (