Lie algebra extension




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


A short exact sequences 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 cocycles

We discuss how 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.



Discussion in the generality of super Lie algebras includes

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