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nonabelian Lie algebra cohomology

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Contents

Idea

Abstractly, nonabelian Lie algebra cohomology is the restriction of the general notion of ∞-Lie algebra cohomology to cocycles of the form 𝔤der𝔥\mathfrak{g} \to der \mathfrak{h}, where 𝔤\mathfrak{g} and 𝔥\mathfrak{h} are ordinary Lie algebras and der()der(-) denotes the Lie algebra of derivations.

Traditionally abelian Lie algebra cohomology is conceived as the cohomology of the Chevalley-Eilenberg complex of a Lie algebra and some nonabelian generalizations of this model have been given in the literature. We show below how these definitions are the nonabelian cohomology special cases of the general abstract definition of ∞-Lie algebra cohomology.

The coefficients are not now in a Lie algebra module (which is viewed here as an abelian Lie algebra with action of another Lie algebra), but an arbitrary Lie algebra with something that is action of another Lie algebra up to an inner automorphism.

For example the problem of extensions of Lie algebras by nonabelian Lie algebras leads to 1,2,3 nonabelian cocycles; 2-cocycles are analogues of factor systems.

Below, in the section Abstract definition we discuss how a nonabelian Lie algebra cocylce is a morphism

(ψ,χ):𝔤Der(𝔨) (\psi,\chi) : \mathfrak{g} \to Der(\mathfrak{k})

of L-∞-algebras to the strict Lie 2-algebra of derivations of 𝔨\mathfrak{k}.

A generalization (indeed a horizontal categorification) is nonabelian Lie algebroid cohomology.

Nonabelian 2-cocycles

Let FF be a field. Lie algebra factor system (or a nonabelian 2-cocycle) on a FF-Lie algebra 𝔟\mathfrak{b} with coefficients in FF-Lie algebra 𝔨\mathfrak{k} is a pair (χ,ψ)(\chi,\psi) where χ:𝔟𝔟𝔨\chi: \mathfrak{b}\wedge \mathfrak{b}\to\mathfrak{k} and ψ:𝔟Der(𝔨)\psi:\mathfrak{b}\to Der(\mathfrak{k}) are FF-linear maps satisfying

χ([b 1,b 2]b 3)χ(b 1[b 2,b 3])+χ(b 2[b 1,b 3]) =ψ(b 3)(χ(b 1b 2))ψ(b 1)(χ(b 2b 3))+ψ(b 2)(χ(b 1b 3)) \begin{aligned} & \chi([b_1,b_2]\wedge b_3)-\chi(b_1\wedge [b_2,b_3])+\chi(b_2\wedge[b_1,b_3]) \\& = \psi(b_3)(\chi(b_1\wedge b_2))-\psi(b_1)(\chi(b_2\wedge b_3))+\psi(b_2)(\chi(b_1\wedge b_3)) \end{aligned}

for all b 1,b 2,b 3Bb_1,b_2,b_3\in B and

[ψ(a),ψ(b)]=ψ([a,b])+ad 𝔨(χ(ab)) [\psi(a),\psi(b)]=\psi([a,b])+ad_{\mathfrak{k}}(\chi(a\wedge b))

where a,bBa,b\in B and ad 𝔨:𝔨Int(𝔨)ad_{\mathfrak{k}}:\mathfrak{k}\to Int(\mathfrak{k}) is the canonical map into inner automorphisms k[k,]k\mapsto [k,].

Schreier’s theory for Lie algebras

Otto Schreier (1926) and Eilenberg-Mac Lane (late 1940-s) developed a theory of nonabelian extensions of abstract groups leading to the low dimensional nonabelian group cohomology. For Lie algebras, the theory can be developed in the same manner. One tries to classify extensions of Lie algebras

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

Theorem. To every Lie algebra extension as above, and a choice of FF-linear section σ:𝔟𝔤\sigma:\mathfrak{b}\to\mathfrak{g} of pp, one can assign a nonabelian 2-cocycle (factor system) on 𝔟\mathfrak{b} with values in 𝔨\mathfrak{k} as follows: set

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)]

and define ϕ:𝔤Der(𝔨)\phi:\mathfrak{g}\to Der(\mathfrak{k}) by ϕ(g)(k):=[g,k]\phi(g)(k):=[g,k]. Then set ψ:=ϕσ\psi:=\phi\circ\sigma. Then (χ,ψ)(\chi,\psi) is a nonabelian 2-cocycle on 𝔟\mathfrak{b} with values in 𝔨\mathfrak{k}.

Theorem. (cocycle crossed product of Lie algebras) Let (χ,ψ)(\chi,\psi) be a factor system as above. Then define a FF-linear bracket on the FF-vector space 𝔟𝔨\mathfrak{b}\oplus\mathfrak{k} by

[(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])

Then

(i) [,][,] is a antisymmetric and satisfies the Jacobi identity, i.e. 𝔤:=(𝔟𝔨,[,])\mathfrak{g}:=(\mathfrak{b}\oplus\mathfrak{k},[,]) is an FF-Lie algebra.

(ii) k(0,k)k\mapsto (0,k) defines an embedding i:𝔨𝔤i:\mathfrak{k}\to\mathfrak{g} of Lie algebras and (b,k)b(b,k)\mapsto b is a surjective homomorphism of Lie algebra p:𝔤𝔟p:\mathfrak{g}\to\mathfrak{b} whose kernel is the Lie ideal i(𝔨)=0𝔨𝔤i(\mathfrak{k})=0\oplus\mathfrak{k}\subset\mathfrak{g}. This way 0𝔨i𝔤p𝔟00\to\mathfrak{k}\overset{i}\to\mathfrak{g}\overset{p}\to\mathfrak{b}\to 0 is an extension of the base Lie algebra 𝔟\mathfrak{b} by the kernel Lie algebra 𝔨\mathfrak{k}.

(iii) If the 2-cocycle is obtained from a Lie algebra extension 0𝔨i 0𝔤 0p 0𝔟00\to \mathfrak{k}\overset{i_0}\to \mathfrak{g}_0\overset{p_0}\to\mathfrak{b}\to 0 and an arbitrary FF-linear section σ 0\sigma_0 of p 0p_0, then the map can σ:𝔤 0𝔤can_\sigma:\mathfrak{g}_0\to\mathfrak{g} given by g(p(g),σ(p(g))+g)g\mapsto (p(g),-\sigma(p(g))+g) is well-defined and a Lie algebra isomorphism such that can σi 0=ican_\sigma\circ i_0=i, p 0=pcan σp_0=p\circ can_\sigma, hence the two extensions are isomorphic.

In addition to the problem of extensions, nonabelian 2-cocycles appear in a more general problem of liftings of Lie algebras.

Abstract definition

We claim that the above definition of nonabelian Lie algebra cocycles may be understood naturally in terms of the general notion of cohomology and in particular is the image of the story of nonabelian group cohomology under Lie differentiation:

The following observation is not in the literature.

Proposition

Let Lie\infty Lie be the (∞,1)-category of L-∞-algebras. Let 𝔤,𝔨\mathfrak{g}, \mathfrak{k} be Lie algebras. Then the degree 2 nonabelian Lie algebra cohomology of 𝔤\mathfrak{g} with coefficients in 𝔨\mathfrak{k} is

H nonab 2(𝔤,𝔨)π 0Lie(𝔤,Der(𝔨)), H^2_{nonab}(\mathfrak{g}, \mathfrak{k}) \simeq \pi_0 \infty Lie(\mathfrak{g}, Der(\mathfrak{k})) \,,

where Der(𝔨)Der(\mathfrak{k}) is the strict Lie 2-algebra of derivations on 𝔨\mathfrak{k}.

More in detail:

  • nonabelian degree 2 Lie algebra cocycles (ψ,ξ)(\psi,\xi) are in natural bijections with morphisms

    𝔤Der(𝔨) \mathfrak{g} \to Der(\mathfrak{k})
  • coboundaries η\eta between cocycles (ψ 1,ξ 1)(\psi_1,\xi_1) and (ψ 2,ξ 2)(\psi_2,\xi_2) correspond to homotopies between these

    (ψ 1,χ 1) 𝔤 η Der(𝔨) (ψ 2,χ 2) \array{ & \nearrow\searrow^{\mathrlap{(\psi_1,\chi_1)}} \\ \mathfrak{g} &\Downarrow^{\eta}& Der(\mathfrak{k}) \\ & \searrow\nearrow^{\mathrlap{(\psi_2,\chi_2)}} }

    and this correspondence is precise if we take the homotopy to be induced from the “standard cylinder object”, described below.

Proof

Checking this is a straightforward matter of unwinding the definitions of morphisms of L L_\infty-algebras.

Which is what we indicate.

We model Lie\infty Lie as usual a subcategory of dg-algebras of semifree dgas, by representing each L L_\infty-algebra 𝔤\mathfrak{g} by its Chevalley-Eilenberg algebra CE(𝔤)CE(\mathfrak{g}).

For the Lie algebra 𝔤\mathfrak{g} itself with Lie bracket [,]:𝔤𝔤𝔤[-,-] : \mathfrak{g} \wedge \mathfrak{g} \to \mathfrak{g} this is the semifree dga

CE(𝔤)=( 𝔤 *,d=[,] *), CE(\mathfrak{g}) = (\wedge^\bullet \mathfrak{g}^* , \; d = [-,-]^* ) \,,

where the differential is on generators the dual of the Lie bracket, [,] *:𝔤 *𝔤 *𝔤 *[-,-]^* : \mathfrak{g}^* \to \mathfrak{g}^* \wedge \mathfrak{g}^* extended as a graded derivation to all of 𝔤 *\wedge^\bullet \mathfrak{g}^*.

For any strict Lie 2-algebra coming from a differential crossed module (𝔥 1δ𝔥 1)(\mathfrak{h}_1 \stackrel{\delta}{\to} \mathfrak{h}_1) with action ρ:𝔥 1der(𝔥 2)\rho : \mathfrak{h}_1 \to der(\mathfrak{h}_2) – that we think of in the following as equivalently a linear map ρ:𝔥 1𝔥 2𝔥 2\rho : \mathfrak{h}_1 \otimes \mathfrak{h}_2 \to \mathfrak{h}_2 – the Chevalley-Eilenberg algebra is

CE(𝔥 2δ𝔥 1)=( (𝔥 1 *𝔥 2 *),d δ) CE(\mathfrak{h}_2 \stackrel{\delta}{\to} \mathfrak{h}_1) = \left( \wedge^\bullet ( \mathfrak{h}_1^* \oplus \mathfrak{h}_2^* ) , \; d_{\delta} \right)

with 𝔥 1 *\mathfrak{h}_1^* in degree 1 and 𝔥 2 *\mathfrak{h}_2^* in degree 2, and with the differential given on degree 1 generators by

d δ| 𝔥 1 *=[,] 𝔥 1 *+δ *:𝔥 1 *𝔥 1 *𝔥 1 *𝔥 2 * d_\delta |_{\mathfrak{h}_1^*} = [-,-]_{\mathfrak{h}_1}^* + \delta^* : \mathfrak{h}_1^* \to \mathfrak{h}_1^* \wedge \mathfrak{h}_1^* \oplus \mathfrak{h}_2^*

and on degree 2 generators by

d δ| 𝔥 2 *=ρ *:𝔥 2 *𝔥 1 *𝔥 2 *. d_\delta |_{\mathfrak{h}_2^*} = \rho^* : \mathfrak{h}_2^* \to \mathfrak{h}_1^* \otimes \mathfrak{h}_2^* \,.

The case of the derivation strict Lie 2-algebra of a Lie algebra 𝔨\mathfrak{k} is the special case of this for

Der(𝔨)=(𝔨adder(𝔨)). Der(\mathfrak{k}) = (\mathfrak{k} \stackrel{ad}{\to} der(\mathfrak{k})) \,.

Now a morphism

(ψ,χ):𝔤Der(𝔨) (\psi, \chi) : \mathfrak{g} \to Der(\mathfrak{k})

of \infty-Lie algebras is given by a morphism

CE(𝔤)CE(Der(𝔨)):(ψ *,χ *) CE(\mathfrak{g}) \leftarrow CE(Der(\mathfrak{k})) : (\psi^*, \chi^*)

of dg-algebras.

Morphisms of dg-algebras are given by morphisms of the underlying graded algebras, subject to the respect for the differentials. Morphisms of the underlying graded Grassmann algebras are given by grading preserving linear maps on the spaces of generators.

So the underlying maps

𝔤 * (𝔥 1 *𝔥 2 *):(ψ *,χ *) \wedge^\bullet \mathfrak{g}^* \leftarrow \wedge^\bullet (\mathfrak{h}_1^* \oplus \mathfrak{h}_2^*) : (\psi^* , \chi^*)

come from linear maps

𝔤 *𝔥 1 *:ψ * \mathfrak{g}^* \leftarrow \mathfrak{h}_1^* : \psi^*

and

𝔤 *𝔤 *𝔥 2 *:χ * \mathfrak{g}^* \wedge \mathfrak{g}^* \leftarrow \mathfrak{h}_2^* : \chi^*

i.e. form linear maps

ψ:𝔤der(𝔨) \psi : \mathfrak{g} \to der(\mathfrak{k})

and

χ:𝔤𝔤𝔨. \chi : \mathfrak{g} \wedge \mathfrak{g} \to \mathfrak{k} \,.

This is the underlying data of the nonabelian 2-cocycle. Now the respect for the differentials on the Chevalley-Eilenberg algebras will give the cocycle condition:

let ω𝔥 2 *CE(𝔥 1𝔥 2)\omega \in \mathfrak{h}_2^* \subset CE(\mathfrak{h}_1 \to \mathfrak{h}_2) be any degree 2 element, then respect for the differential implies that

ω(χ([,],))=ω(ρ(ψ()(χ(,)))) (ψ *,χ *) ω(ρ()())) d 𝔤=[,] * d δ ω(ξ(,)) (ψ *,χ *) ω. \array{ \omega(\chi([-,-],-)) = \omega(\rho(\psi(-)(\chi(-,-)))) &\stackrel{(\psi^*, \chi^*)}{\leftarrow}& \omega(\rho(-)(-))) \\ \uparrow^{d_\mathfrak{g} = [-,-]^*} && \uparrow^{\mathrlap{d_{\delta}}} \\ \omega(\xi(-,-)) &\stackrel{(\psi^*, \chi^*)}{\leftarrow}& \omega } \,.

Since this has to hold for all ω\omega, we get the first part of the cocycle condition:

χ([,],)=ρ(ψ()χ(,)) \chi([-,-],-) = \rho(\psi(-)\chi(-,-))

(both sides here regarded as elements of a graded Grassmann algebra as indicated above, so with all antisymmetrization on the arguments implicit).

Similarly, for λ𝔥 1 *CE(𝔥 1𝔥 2)\lambda \in \mathfrak{h}_1^* \subset CE(\mathfrak{h}_1 \to \mathfrak{h}_2) be any degree 1 element, then respect for the differential implies that

λ(ψ([,]))=λ([ψ(),ψ()])+λ(ad(χ(,))) λ([,] 𝔥 1)+λ(ad()) λ(ψ()) (ψ *,χ *) λ. \array{ \lambda(\psi([-,-])) = \lambda([\psi(-), \psi(-)]) + \lambda(ad(\chi(-,-))) &\stackrel{}{\leftarrow}& \lambda([-,-]_{\mathfrak{h}_1}) + \lambda(ad(-)) \\ \uparrow && \uparrow \\ \lambda(\psi(-)) &\stackrel{(\psi^* , \chi^*)}{\leftarrow}& \lambda } \,.

Again, this has to hold for all λ\lambda, so we have the auxiliary condition on the cocycle

ψ([,])=[ψ(),ψ()]+ad(ξ(,)). \psi([-,-]) = [\psi(-),\psi(-)] + ad(\xi(-,-)) \,.

This shows that morphisms 𝔤Der(𝔨)\mathfrak{g} \to Der(\mathfrak{k}) are in bijection to the nonabelian cocycles.

It remains to show that the homotopies map to coboundaries. For that we may take in Lie\infty Lie the standard cylinder object of some CE(𝔤)CE(\mathfrak{g}) to be

CE(𝔤)CE(𝔤)C (Δ 1)CE(𝔤)CE(𝔤), CE(\mathfrak{g})\otimes CE(\mathfrak{g}) \leftarrow C^\bullet(\Delta^1)\otimes CE(\mathfrak{g}) \leftarrow CE(\mathfrak{g}) \,,

where C (Δ 1)C^\bullet(\Delta^1) is the semifree dga of cochains on the cellular 1-simplex, i.e.

C (Δ 1)=( (a,bc),da=db=dc), C^\bullet(\Delta^1) = (\wedge^\bullet (\langle a,b \rangle \oplus \langle c \rangle) , d a = - d b = d c ) \,,

with a,ba,b generators in degree 0 and cc in degree 1. Using this, write out the data implied by a morphism η\eta that is a left homotopy

CE(𝔤) (ψ 1 *,χ 1 *) C (Δ 1)CE(𝔤) η CE(Der(𝔨)) (ψ 2 *,χ 2 *) CE(𝔤) \array{ CE(\mathfrak{g}) \\ \downarrow & \nwarrow^{\mathrlap{(\psi_1^*, \chi_1^*)}} \\ C^\bullet(\Delta^1)\otimes CE(\mathfrak{g}) &\stackrel{\eta}{\leftarrow}& CE(Der(\mathfrak{k})) \\ \uparrow & \swarrow_{\mathrlap{(\psi_2^*, \chi_2^*)}} \\ CE(\mathfrak{g}) }

along the above lines.

Notice that in dgAlg opdgAlg^{op} every object is cofibrant, so that this is indeed a left homotopy. See ∞-Lie algebra cohomology for more on this.

References

On original source is

  • G. Hochschild, Lie algebra kernels and cohomology, Amer. J. Math. 76, n.3 (1954) 698–716.

The notation above is from personal notes of Z. Škoda (1997). A systematic theory has been many times partly rediscovered from soon after the Eilenberg–Mac Lane work on group extension, among first by Hochschild and then by many others till nowdays. Here is a recent online account emphasising parallels with differential geometry:

  • Dmitri Alekseevsky, Peter W. Michor, Wolfgang Ruppert, Extensions of Lie algebras math.DG/0005042

A more conceptual picture is in a work of Danny Stevenson which extends also to its categorification, extensions of Lie 2-algebras. See

There is also

  • N. Inassaridze, E. Khmaladze, and M. Ladra, Non-abelian Cohomology and Extensions of Lie Algebras Journal of Lie Theory, Volume 18 (2008) 413–432 (pdf)
Revised on February 2, 2011 22:05:07 by Urs Schreiber (89.204.153.113)