An introduction to Lie algebra cohomology/lecture 11
Noncentral extensions
A module \(\mathfrak{g}\) is projective if for every surjective morphism of modules \(\alpha:\mathfrak{k}\rightarrow\mathfrak{g}\rightarrow 0\) and every morphism \(\beta:\mathfrak{h}\rightarrow\mathfrak{g}\ ,\) there exists a morphism \(\gamma:\mathfrak{h}\rightarrow\mathfrak{k}\) such that \(\beta=\alpha\gamma\ .\) In particular, if \(\mathfrak{g}\) is projective and \(\alpha\) surjective, there exists a \(\gamma:\mathfrak{g}\rightarrow\mathfrak{k}\) such that \(id_{\mathfrak{g}}=\alpha\gamma\ ,\) that is, \(\gamma\) is a section of \(\alpha\ .\)
Consider the exact sequence of Lie algebras with Lie algebra morphisms
- \[ \iota \qquad \quad \ p\]
\[\tag{1} 0 \quad \rightarrow \quad \mathfrak{h} \quad \rightarrow \quad \mathfrak{k} \quad \rightarrow \quad \mathfrak{g} \quad \rightarrow 0 \ .\]
question
If \(\mathfrak{g}\) is projective as an \(R\)-module, is there a section of \(p\) which is also a Lie algebra morphism, that is, is it also projective as a Lie algebra?
Let \(\sigma\) be a section, but not necessarily a Lie algebra morphism (Exists, since \(\mathfrak{g}\) is projective). Define \(k^2\in C_\wedge^2(\mathfrak{g},\mathfrak{k})\) by \[\tag{2} k^2(x,y)=\sigma([x,y]_{\mathfrak{g}})-[\sigma(x),\sigma(y)]_{\mathfrak{k}}.\]
One has \[\tag{3} p k^2(x,y)=p\sigma([x,y]_{\mathfrak{g}})-p[\sigma(x),\sigma(y)]_{\mathfrak{k}}=[x,y]_{\mathfrak{g}}-[p\sigma(x),p\sigma(y)]_{\mathfrak{g}}=0. \]
Since \(k^2(x,y)\in\ker p=\mathrm{im\ }\iota\ ,\) one can define \(h^2\in C_{\wedge}^2(\mathfrak{g},\mathfrak{h})\) by \[\tag{4} \iota(h^2(x,y))=k^2(x,y)\]
Let \(\mathrm{der}(\mathfrak{h})\) denote the space of derivations on \(\mathfrak{h}\ ,\) that is, \(\alpha\in \mathrm{der}(\mathfrak{h})\) if \(\alpha[x,y]_\mathfrak{h}=[\alpha x, y ]_\mathfrak{h}+ [x,\alpha y]_\mathfrak{h}\ .\) Now define \(\mathbf{d}^{(0)}:\mathfrak{g}\rightarrow \mathrm{End}(\mathfrak{h})\) by \[\tag{5} \iota(\mathbf{d}^{(0)}(x)h)=[\sigma(x),\iota(h)]_{\mathfrak{k}}\]
Bold is used to indicate that this is not a representation, even though it is close to one. It is well-defined since, for instance, \(p[\sigma(x),\iota(h)]_{\mathfrak{k}}=[p\sigma(x),p\iota(h)]_{\mathfrak{g}}=0\ .\) It defines a derivation, since \[\iota(\mathbf{d}^{(0)}(x)[h_1,h_2]_\mathfrak{h})=[\sigma(x),\iota([h_1,h_2]_\mathfrak{h})]_{\mathfrak{k}}\ :\] \[=[\sigma(x),[\iota(h_1),\iota(h_2)]_\mathfrak{k}]_{\mathfrak{k}}\ :\] \[=[[\sigma(x),\iota(h_1),\iota(h_2)]_\mathfrak{k}]_{\mathfrak{k}}+[\iota(h_1),[\sigma(x),\iota(h_2)]_\mathfrak{k}]_{\mathfrak{k}}\ :\] \[=[\iota(\mathbf{d}(x)^{(0)}h_1),\iota(h_2)]_{\mathfrak{k}}+[\iota(h_1),\iota(\mathbf{d}^{(0)}(x)h_2)]_{\mathfrak{k}}\ :\] \[=\iota([\mathbf{d}(x)^{(0)}h_1,h_2]_{\mathfrak{h}})+\iota([h_1,\mathbf{d}^{(0)}(x)h_2]_{\mathfrak{h}})\] It follows that \[\mathbf{d}^{(0)}(x)[h_1,h_2]_\mathfrak{h}=[\mathbf{d}(x)^{(0)}h_1,h_2]_{\mathfrak{h}}+[h_1,\mathbf{d}^{(0)}(x)h_2]_{\mathfrak{h}}\] and the conlusion follows. Moreover, \(\mathbf{d}^{(0)}\) satisfies \[\tag{6} \mathbf{d}^{(0)}([x,y])=ad_+(h^2(x,y))+\mathbf{d}^{(0)}(x)\mathbf{d}^{(0)}(y) -\mathbf{d}^{(0)}(y)\mathbf{d}^{(0)}(x)\]
since \[\iota(\mathbf{d}^{(0)}([x,y])h) = [\sigma([x,y]),\iota(h)]_{\mathfrak{k}}\ :\]
- \[ = [k^2(x,y),\iota(h)]_{\mathfrak{k}} +[[\sigma(x),\sigma(y)],\iota(h)]_{\mathfrak{k}}\ :\]
- \[ = [\iota h^2(x,y),\iota(h)]_{\mathfrak{k}} +[\sigma(x),[\sigma(y),\iota(h)]_{\mathfrak{k}}]_{\mathfrak{k}} -[\sigma(y),[\sigma(x),\iota(h)]_{\mathfrak{k}}]_{\mathfrak{k}}\ :\]
- \[ = \iota[h^2(x,y),h]_{\mathfrak{h}}+[\sigma(x),\iota(\mathbf{d}^{(0)}(y)h)]_{\mathfrak{k}} -[\sigma(y),\iota(\mathbf{d}^{(0)}(x)h)]_{\mathfrak{k}}\ :\]
- \[ = \iota(ad(h^2(x,y))h)+\iota(\mathbf{d}^{(0)}(x)\mathbf{d}^{(0)}(y)h) -\iota(\mathbf{d}^{(0)}(y)\mathbf{d}^{(0)}(x)h)\]
or \(\mathbf{d}^{(0)}([x,y])=ad_+(h^2(x,y))+\mathbf{d}^{(0)}(x)\mathbf{d}^{(0)}(y) -\mathbf{d}^{(0)}(y)\mathbf{d}^{(0)}(x)\ ,\) The fact that \(\mathbf{d}^{(0)}\) is not a representation has as a consequence that \( \mathbf{d}^\cdot\) (following the constructing for representations) is not a coboundary operator. One can think of \(\mathbf{d}^\cdot\) as a kind of connection with curvature \(\mathbf{d}^{\cdot+1} \mathbf{d}^{\cdot}\ .\) \[\mathbf{d}^1 \mathbf{d}^0 h(x,y)=\mathbf{d}^{(0)}(x)\mathbf{d}^0h(y)-\mathbf{d}_-^{(0)}(y)\mathbf{d}^0 h(x)-\mathbf{d}^0 h([x,y])\ :\] \[=\mathbf{d}^{(0)}(x)\mathbf{d}^{(0)}(y)h-\mathbf{d}^{(0)}(y)\mathbf{d}^{(0)}(x)h-\mathbf{d}^{(0)}([x,y])h\ :\] \[=- ad(h^2(x,y))h\]
Suppose now there exists a \(k^1\in C^1(\mathfrak{g},\mathfrak{k})\) such that \(\sigma+k^1\) is a Leibniz algebra homomorphism and a section of \(p\ .\) Then \[ x=p(\sigma+k^1)(x)=p\sigma(x)+p k^1(x)=x+p k^1(x), \] implying that \(k^1 (x)\in\ker(p)=\mathrm{im\ }\iota\ .\) Define \(h^1\in C^1(\mathfrak{g},\mathfrak{h})\) by \(\iota h^1(x)=k^1 (x)\ .\) Then (by assumption!) \[ 0 =(\sigma+k^1)([x,y])-[(\sigma+k^1)(x),(\sigma+k^1)(y)]\ :\] \[ =\sigma([x,y])+k^1([x,y])-[\sigma(x),\sigma(y)]-[k^1(x),\sigma(y)]-[\sigma(x),k^1(y)]-[k^1(x),k^1(y)]\ :\]
\[ = k^2(x,y) +k^1([x,y])-[\iota h^1(x),\sigma(y)]-[\sigma(x),\iota h^1(y)]-[\iota h^1(x),\iota h^1(y)]\ :\] \[ = \iota(h^2(x,y)-(\mathbf{d}_+^{(0)}(x)h^1(y)-\mathbf{d}_-^{(0)}(y)h^1(x)-h^1([x,y])-[h^1(x),h^1(y)])\ :\] \[= \iota((h^2-\mathbf{d}^1 h^1-[h^1,h^1])(x,y)).\] This implies that the existence of such a \(k^1\) is equivalent to \(h^2=\mathbf{d}^1 h^1+[h^1,h^1]\ .\) On the other hand one has \(\mathbf{d}^2h^2=0\ :\) \[ \iota \mathbf{d}^2 h^2(x,y,z)= \iota \mathbf{d}^{(0)}(x)h^2(y,z) -\iota \mathbf{d}^{(0)}(y)h^2(x,z) +\iota \mathbf{d}^{(0)}(z)h^2(x,y)-\iota h^2([x,y],z)-\iota h^2(y,[x,z])+\iota h^2(x,[y,z])\ :\]
- \[=[\sigma(x),k^2(y,z)]-[\sigma(y),k^2(x,z)]-[k^2(x,y),\sigma(z)]-k^2([x,y],z)-k^2(y,[x,z])+k^2(x,[y,z])\ :\]
- \[=[\sigma(x),\sigma([y,z])-[\sigma(y),\sigma(z)]]-[\sigma(y),\sigma([x,z])-[\sigma(x),\sigma(z)]]-[\sigma([x,y])-[\sigma(x),\sigma(y)],\sigma(z)]\ :\]
- \[ -\sigma([[x,y],z])+[\sigma([x,y]),\sigma(z)]-\sigma([y,[x,z]])+[\sigma(y),\sigma([x,z]]+\sigma([x,[y,z]])-[\sigma(x),\sigma([y,z])]\ :\]
- \[= -[\sigma(x),[\sigma(y),\sigma(z)]]+[\sigma(y),[\sigma(x),\sigma(z)]] +[[\sigma(x),\sigma(y)],\sigma(z)]\ :\]
- \[ -\sigma([[x,y],z]+[y,[x,z]]-[x,[y,z]]) \ :\]
- \[=0.\]
