An introduction to Lie algebra cohomology/lecture 12
Construction of an extension
Now one turns the question around. What is the situation if one has Lie algebras \({\mathfrak{g}}\) (projective) and \({\mathfrak{h}}\ .\)
definition
If \(\alpha^0\in End(\mathfrak{h})\) can be written as \[ \alpha^0 h=[\alpha,h],\quad \alpha,h\in\mathfrak{h}\] one says that \(\alpha^0\) is inner and we denote the space of inner endomorphisms by \(\mathfrak{ad}(\mathfrak{h})\ .\) If \[ \alpha^0[h_1,h_2]=[\alpha^0 h_1,h_2]+[h_1,\alpha^0 h_2]\] then \(\alpha^0\) is called a derivation and the space of derivations is denoted by \(\mathfrak{der}(\mathfrak{h})\ .\)
proposition
If \(\alpha^0, \beta^0\in\mathfrak{der}(\mathfrak{h})\) then \([\alpha^0,\beta^0]\in\mathfrak{der}(\mathfrak{h})\ .\)
proposition
\[[\mathfrak{der}(\mathfrak{h}),\mathfrak{ad}(\mathfrak{h})]\subset \mathfrak{ad}(\mathfrak{h})\]
proof
\[[\mathfrak{d},\mathrm{ad}(\alpha)]h=\mathfrak{d}[\alpha,h]-[\alpha,\delta h]=[\mathfrak{d}\alpha,h]=\mathrm{ad}(\mathfrak{d}\alpha)h\]
definition
Let \[\mathfrak{out}(\mathfrak{h})=\mathfrak{der}(\mathfrak{h})/\mathfrak{ad}(\mathfrak{h})\]
To see how one should define a Lie algebra structure on the sum of \({\mathfrak{g}}\) and \({\mathfrak{h}}\ ,\) it pays to have a look at \({\mathfrak{k}}\ .\) Every element in \(y\in{\mathfrak{k}}\) can be uniquely written as \(\sigma(x)+\iota(h)\ ,\) \(x\in{\mathfrak{g}}, h\in{\mathfrak{h}}\ :\) take \(x=p(y)\) and\(h=\iota^{-1} (1-\sigma p)(y) \ .\) Let \(\phi:{\mathfrak{k}}\rightarrow {\mathfrak{h}}\oplus_R{\mathfrak{g}}\) be defined by \(\phi(\sigma(x)+\iota(h))=(h,x)\ .\) Then \[\phi([\sigma(x)+\iota(h_1),\sigma(y)+\iota(h_2)])=\phi([\sigma(x),\sigma(y)]+[\iota(h_1),\sigma(y)]+[\sigma(x),\iota(h_2)]+\iota([h_1,h_2]))\] \[=\phi(\sigma([x,y])-\iota(h^2(x,y))+[\iota(h_1),\sigma(y)]+[\sigma(x),\iota(h_2)]+\iota([h_1,h_2]))\] \[=([h_1,h_2]+{\mathbf{d}^{(0)}}(x)h_2-{\mathbf{d}^{(0)}}(y)h_1-h^2(x,y),[x,y])\] Suppose \(d^{(0)}:\mathfrak{g}\rightarrow\mathfrak{out}(\mathfrak{h})\) is a Lie algebra homomorphism. Let \[d^{(0)}=[\mathbf{d}^{(0)}]\] with \[\tag{1} \mathbf{d}^{(0)}([x,y])=ad(h^2(x,y))+\mathbf{d}^{(0)}(x)\mathbf{d}^{(0)}(y) -\mathbf{d}^{(0)}(y)\mathbf{d}^{(0)}(x)\]
(Since \(d^{(0)}\in\mathfrak{out}(\mathfrak{h})\) is a representation, \( \mathbf{d}^{(0)}([x,y])-\mathbf{d}^{(0)}(x)\mathbf{d}^{(0)}(y) +\mathbf{d}^{(0)}(y)\mathbf{d}^{(0)}(x)\) must be an inner derivation, represented by an element in \(\mathfrak{h}\) depending linearly on \(x\) and \(y\ ,\) and denoted by \(h^2(x,y)\ .\))
definition
One defines \[[(h_1,x_1),(h_2,x_2)]=([h_1,h_2]+{\mathbf{d}^{(0)}}(x_1)h_2-{\mathbf{d}^{(0)}}(x_2)h_1-h^2(x_1,x_2),[x_1,x_2])\]
lemma
The new bracket \([\cdot,\cdot] \) obeys the Jacobi identity as long as \(h^2\in Z_\wedge^2(\mathfrak{g},\mathfrak{h})\ .\)
proof
- \[[[(h_1,x_1),(h_2,x_2)],(h_3.x_3)]-[(h_1,x_1),[(h_2,x_2),(h_3,x_3)]]+[(h_2,x_2),[(h_1,x_1),(h_3,x_3)]]\]
\[=[([h_1,h_2]+{\mathbf{d}^{(0)}}(x_1)h_2-{\mathbf{d}^{(0)}}(x_2)h_1-h^2(x_1,x_2),[x_1,x_2]),(h_3,x_3)]\ :\] \[-[(h_1,x_1),([h_2,h_3]+{\mathbf{d}^{(0)}}(x_2)h_3-{\mathbf{d}^{(0)}}(x_3)h_2-h^2(x_2,x_3),[x_2,x_3])]\ :\] \[+[(h_2,x_2),([h_1,h_3]+{\mathbf{d}^{(0)}}(x_1)h_3-{\mathbf{d}^{(0)}}(x_3)h_1-h^2(x_1,x_3),[x_1,x_3])]\] \[=([[h_1,h_2]+{\mathbf{d}^{(0)}}(x_1)h_2-{\mathbf{d}^{(0)}}(x_2)h_1-h^2(x_1,x_2),h_3]+\mathbf{d}^{(0)}([x_1,x_2])h_3 -\mathbf{d}^{(0)}(x_3)([h_1,h_2]+{\mathbf{d}^{(0)}}(x_1)h_2-{\mathbf{d}^{(0)}}(x_2)h_1-h^2(x_1,x_2))-h^2([x_1,x_2],x_3),[[x_1,x_2],x_3])\ :\] \[-([h_1,[h_2,h_3]+{\mathbf{d}^{(0)}}(x_2)h_3-{\mathbf{d}^{(0)}}(x_3)h_2-h^2(x_2,x_3)] +\mathbf{d}^{(0)}(x_1)([h_2,h_3]+{\mathbf{d}^{(0)}}(x_2)h_3-{\mathbf{d}^{(0)}}(x_3)h_2-h^2(x_2,x_3)) -\mathbf{d}^{(0)}([x_2,x_3])h_1-h^2(x_1,[x_2,x_3]),[x_1,[x_2,x_3]])\ :\] \[+([h_2,[h_1,h_3]+{\mathbf{d}^{(0)}}(x_1)h_3-{\mathbf{d}^{(0)}}(x_3)h_1-h^2(x_1,x_3)] +\mathbf{d}^{(0)}(x_2)([h_1,h_3]+{\mathbf{d}^{(0)}}(x_1)h_3-{\mathbf{d}^{(0)}}(x_3)h_1-h^2(x_1,x_3)) -\mathbf{d}^{(0)}([x_1,x_3])h_2-h^2(x_2,[x_1,x_3]), [x_2,[x_1,x_3]])\] \[=([{\mathbf{d}^{(0)}}(x_1)h_2,h_3]-[{\mathbf{d}^{(0)}}(x_2)h_1,h_3]-[h^2(x_1,x_2),h_3]+\mathbf{d}^{(0)}([x_1,x_2])h_3 -\mathbf{d}^{(0)}(x_3)[h_1,h_2]-\mathbf{d}^{(0)}(x_3){\mathbf{d}^{(0)}}(x_1)h_2+\mathbf{d}^{(0)}(x_3){\mathbf{d}^{(0)}}(x_2)h_1 +\mathbf{d}^{(0)}(x_3)h^2(x_1,x_2)-h^2([x_1,x_2],x_3)\ :\] \[-[h_1,{\mathbf{d}^{(0)}}(x_2)h_3]+[h_1,{\mathbf{d}^{(0)}}(x_3)h_2]+[h_1,h^2(x_2,x_3)] -\mathbf{d}^{(0)}(x_1)[h_2,h_3]-\mathbf{d}^{(0)}(x_1){\mathbf{d}^{(0)}}(x_2)h_3+\mathbf{d}^{(0)}(x_1){\mathbf{d}^{(0)}}(x_3)h_2 +\mathbf{d}^{(0)}(x_1)h^2(x_2,x_3) +\mathbf{d}^{(0)}([x_2,x_3])h_1+h^2(x_1,[x_2,x_3])\ :\] \[+[h_2,{\mathbf{d}^{(0)}}(x_1)h_3]-[h_2,{\mathbf{d}^{(0)}}(x_3)h_1]-[h_2,h^2(x_1,x_3)] +\mathbf{d}^{(0)}(x_2)[h_1,h_3]+\mathbf{d}^{(0)}(x_2){\mathbf{d}^{(0)}}(x_1)h_3-\mathbf{d}^{(0)}(x_2){\mathbf{d}^{(0)}}(x_3)h_1 -\mathbf{d}^{(0)}(x_2)h^2(x_1,x_3)-\mathbf{d}^{(0)}([x_1,x_3])h_2-h^2(x_2,[x_1,x_3]), 0)\] \[=([-\mathbf{d}^{(0)}(x_1)[h_2,h_3]+[h_2,{\mathbf{d}^{(0)}}(x_1)h_3]+[{\mathbf{d}^{(0)}}(x_1)h_2,h_3]\ :\] \[+\mathbf{d}^{(0)}(x_2)[h_1,h_3]-[h_1,{\mathbf{d}^{(0)}}(x_2)h_3] -[{\mathbf{d}^{(0)}}(x_2)h_1,h_3]\ :\] \[-[h_2,{\mathbf{d}^{(0)}}(x_3)h_1]+[h_1,{\mathbf{d}^{(0)}}(x_3)h_2]-\mathbf{d}^{(0)}(x_3)[h_1,h_2]\ :\] \[-[h^2(x_1,x_2),h_3]+\mathbf{d}^{(0)}([x_1,x_2])h_3 -\mathbf{d}^{(0)}(x_1){\mathbf{d}^{(0)}}(x_2)h_3+\mathbf{d}^{(0)}(x_2){\mathbf{d}^{(0)}}(x_1)h_3\ :\] \[ -\mathbf{d}^{(0)}(x_3){\mathbf{d}^{(0)}}(x_1)h_2 +\mathbf{d}^{(0)}(x_1){\mathbf{d}^{(0)}}(x_3)h_2-\mathbf{d}^{(0)}([x_1,x_3])h_2-[h_2,h^2(x_1,x_3)]\ :\] \[ +[h_1,h^2(x_2,x_3)]+\mathbf{d}^{(0)}(x_3){\mathbf{d}^{(0)}}(x_2)h_1 +\mathbf{d}([x_2,x_3])h_1-\mathbf{d}^{(0)}(x_2){\mathbf{d}^{(0)}}(x_3)h_1\ :\] \[ +\mathbf{d}^{(0)}(x_3)h^2(x_1,x_2)-h^2([x_1,x_2],x_3)\ :\] \[ +\mathbf{d}^{(0)}(x_1)h^2(x_2,x_3) +h^2(x_1,[x_2,x_3])\ :\] \[ -\mathbf{d}^{(0)}(x_2)h^2(x_1,x_3)-h^2(x_2,[x_1,x_3]), 0)\] \[=(\mathbf{d}^2h^2(x_1,x_2,x_3),0)\] \[=(0,0)\]
proposition
\[ \mathbf{d}^2h^2(x_1,x_2,x_3)\in\mathcal{C}(\mathfrak{h})\]
proof
\[\mathrm{ad}(\mathbf{d}^2h^2(x_1,x_2,x_3))\] \[=\mathrm{ad}(\mathbf{d}^{(0)}(x_1)h^2(x_2,x_3))-\mathbf{d}^{(0)}(x_2)h^2(x_1,x_3)+\mathbf{d}^{(0)}(x_3)h^2(x_1,x_2) -h^2([x_1,x_2],x_3)-h^2(x_2,[x_1,x_3])+h^2(x_1,[x_2,x_3]))\] \[=[\mathbf{d}^{(0)}(x_1),\mathrm{ad}(h^2(x_2,x_3))]-[\mathbf{d}^{(0)}(x_2),\mathrm{ad}(h^2(x_1,x_3))] +[\mathbf{d}^{(0)}(x_3),\mathrm{ad}(h^2(x_1,x_2))] -\mathrm{ad}(h^2([x_1,x_2],x_3))-\mathrm{ad}(h^2(x_2,[x_1,x_3]))+\mathrm{ad}(h^2(x_1,[x_2,x_3]))\] \[=[\mathbf{d}^{(0)}(x_1),\mathbf{d}^{(0)}([x_2,x_3])]-[\mathbf{d}^{(0)}(x_2),\mathbf{d}^{(0)}([x_1,x_3])] +[\mathbf{d}^{(0)}(x_3),\mathbf{d}^{(0)}([x_1,x_2])] -\mathrm{ad}(h^2([x_1,x_2],x_3))-\mathrm{ad}(h^2(x_2,[x_1,x_3]))+\mathrm{ad}(h^2(x_1,[x_2,x_3]))\] \[=\mathbf{d}^{(0)}([x_1,[x_2,x_3]]) -\mathbf{d}^{(0)}([[x_1,x_2],x_3])-\mathbf{d}^{(0)}([x_2,[x_1,x_3]])\] \[=0\]
corollary
If \(\mathfrak{h}\) has no center, one has automatically \( \mathbf{d}^2h^2=0\ ,\) which implies that the Jacobi identity is always satisfied. Another way of saying this, is that the choice \(d^{(0)}\in\mathfrak{out}(\mathfrak{h})\) determines an extension
- \[ \iota_0 \qquad \quad \ p_0\]
\[\tag{2} 0 \quad \rightarrow \quad \mathfrak{ad}(\mathfrak{h}) \quad \rightarrow \quad \mathfrak{k}_0 \quad \rightarrow \quad \mathfrak{g} \quad \rightarrow 0 \ .\]
proposition
\(\bar{\mathbf{d}}^2 \bar{h}^2-\mathbf{d}^2h^2\in B_\wedge^3(\mathfrak{g},\mathcal{Z}(\mathfrak{h}))\ .\)
proof
Let \(\bar{\mathbf{d}}^{(0)}(x)=\mathbf{d}^{(0)}+\mathrm{ad}(h^1(x))\ .\) Then \[ \mathrm{ad}(\bar{h}^2(x_1,x_2))=\bar{\mathbf{d}}^{(0)}([x_1,x_2])-[\bar{\mathbf{d}}^{(0)}(x_1),\bar{\mathbf{d}}^{(0)}(x_2)]\ :\] \[=\mathbf{d}^{(0)}([x_1,x_2])+\mathrm{ad}(h^1([x_1,x_2]))-[\mathbf{d}^{(0)}(x_1),\mathbf{d}^{(0)}(x_2)] -[\mathbf{d}^{(0)}(x_1),\mathrm{ad}(h^1(x_2))]-[\mathrm{ad}(h^1(x_1)),\mathbf{d}^{(0)}(x_2)]- [\mathrm{ad}(h^1(x_1)),\mathrm{ad}(h^1(x_2))]\ :\] \[=\mathrm{ad}(h^2(x_1,x_2))+\mathrm{ad}(h^1([x_1,x_2]))-\mathrm{ad}(\mathbf{d}^{(0)}(x_1)h^1(x_2)]+ \mathrm{ad}(\mathbf{d}^{(0)}(x_2)h^1(x_1))-\mathrm{ad}([h^1(x_1),h^1(x_2)])\ :\] \[ =\mathrm{ad}(h^2(x_1,x_2)-\mathbf{d}^1h^1(x_1,x_2)-[h^1(x_1),h^1(x_2)])\] This implies \[ \bar{h}^2=h^2-\mathbf{d}^1h^1-[h^1,h^1]+\eta^2.\quad \eta^2\in C_\wedge^2(\mathfrak{g},\mathcal{Z}(\mathfrak{h}))\] Then \[\bar{\mathbf{d}}^2 \bar{h}^2(x_1,x_2,x_3)=\bar{\mathbf{d}}^{(0)}(x_1)\bar{h}^2(x_2,x_3)-\bar{\mathbf{d}}^{(0)}(x_2)\bar{h}^2(x_1,x_3) +\bar{\mathbf{d}}^{(0)}(x_3)\bar{h}^2(x_1,x_2)- \bar{h}^2([x_1,x_2],x_3)- \bar{h}^2(x_2,[x_1,x_3])+ \bar{h}^2(x_1,[x_2,x_3])\] \[=(\mathbf{d}^{(0)}(x_1)+\mathrm{ad}(h^1(x_1)))(h^2(x_2,x_3)-\mathbf{d}^1h^1(x_2,x_3)-[h^1,h^1](x_2,x_3)+ \eta^2(x_2,x_3))\ :\] \[-(\mathbf{d}^{(0)}(x_2)+\mathrm{ad}(h^1(x_2)))(h^2(x_1,x_3)-\mathbf{d}^1h^1(x_1,x_3)-[h^1,h^1](x_1,x_3)+ \eta^2(x_1,x_3))\ :\] \[+(\mathbf{d}^{(0)}(x_3)+\mathrm{ad}(h^1(x_3)))(h^2(x_1,x_2)-\mathbf{d}^1h^1(x_1,x_2)-[h^1,h^1](x_1,x_2)+ \eta^2(x_1,x_2))\ :\] \[+h^2([x_1,x_2],x_3)-\mathbf{d}^1h^1([x_1,x_2],x_3)-[h^1([x_1,x_2]),h^1(x_3)]+\eta^2([x_1,x_2],x_3)\ :\] \[+h^2(x_2,[x_1,x_3])-\mathbf{d}^1h^1(x_2,[x_1,x_3])-[h^1(x_2),h^1([x_1,x_3])]+\eta^2(x_2,[x_1,x_3])\ :\] \[+h^2(x_1,[x_2,x_3])-\mathbf{d}^1h^1(x_1,[x_2,x_3])-[h^1(x_1),h^1([x_2,x_3])]+\eta^2(x_1,[x_2,x_3])\] \[=\mathbf{d}^{(0)}(x_1)(h^2(x_2,x_3)-\mathbf{d}^1h^1(x_2,x_3)-[h^1()x_2),h^1(x_3)]+ \eta^2(x_2,x_3))\ :\] \[-\mathbf{d}^{(0)}(x_2)(h^2(x_1,x_3)-\mathbf{d}^1h^1(x_1,x_3)-[h^1(x_1),h^1(x_3)]+ \eta^2(x_1,x_3))\ :\] \[+\mathbf{d}^{(0)}(x_3)(h^2(x_1,x_2)-\mathbf{d}^1h^1(x_1,x_2)-[h^1(x_1),h^1(x_2)]+ \eta^2(x_1,x_2))\ :\] \[+\mathrm{ad}(h^1(x_1))(h^2(x_2,x_3)-\mathbf{d}^1h^1(x_2,x_3)-[h^1(x_2),h^1(x_3)]+ \eta^2(x_2,x_3))\ :\] \[-\mathrm{ad}(h^1(x_2))(h^2(x_1,x_3)-\mathbf{d}^1h^1(x_1,x_3)-[h^1(x_1),h^1(x_3)]+ \eta^2(x_1,x_3))\ :\] \[+\mathrm{ad}(h^1(x_3))(h^2(x_1,x_2)-\mathbf{d}^1h^1(x_1,x_2)-[h^1(x_1),h^1(x_2)]+ \eta^2(x_1,x_2))\ :\] \[-h^2([x_1,x_2],x_3)+\mathbf{d}^1h^1([x_1,x_2],x_3)+[h^1([x_1,x_2]),h^1(x_3)]+\eta^2([x_1,x_2],x_3)\ :\] \[-h^2(x_2,[x_1,x_3])+\mathbf{d}^1h^1(x_2,[x_1,x_3])+[h^1(x_2),h^1([x_1,x_3])]+\eta^2(x_2,[x_1,x_3])\ :\] \[+h^2(x_1,[x_2,x_3])-\mathbf{d}^1h^1(x_1,[x_2,x_3])-[h^1(x_1),h^1([x_2,x_3])]+\eta^2(x_1,[x_2,x_3])\] \[=\mathbf{d}^2 h^2(x_1,x_2,x_3)-\mathbf{d}^{(0)}(x_1)(\mathbf{d}^{(0)}(x_2)h^1(x_3) -\mathbf{d}^{(0)}(x_3)h^1(x_2)-h^1([x_2,x_3]))-[\mathbf{d}^{(0)}(x_1)h^1(x_2),h^1(x_3)] -[h^1(x_2),\mathbf{d}^{(0)}(x_1)h^1(x_3)]+ {d}^{(0)}(x_1)\eta^2(x_2,x_3))\ :\] \[+\mathbf{d}^{(0)}(x_2)(\mathbf{d}^{(0)}(x_1)h^1(x_3)-\mathbf{d}^{(0)}(x_3)h^1(x_1)-h^1([x_1,x_3]))+[\mathbf{d}^{(0)}(x_2)h^1(x_1),h^1(x_3)] +[h^1(x_1),\mathbf{d}^{(0)}(x_2)h^1(x_3)] -{d}^{(0)}(x_2) \eta^2(x_1,x_3))\ :\] \[-\mathbf{d}^{(0)}(x_3)(\mathbf{d}^{(0)}(x_1)h^1(x_2)-\mathbf{d}^{(0)}(x_2)h^1(x_1)-h^1([x_1,x_2]))-[\mathbf{d}^{(0)}(x_3)h^1(x_1),h^1(x_2)] -[h^1(x_1),\mathbf{d}^{(0)}(x_3)h^1(x_2)]+ {d}^{(0)}(x_3)\eta^2(x_1,x_2))\ :\] \[+\mathrm{ad}(h^1(x_1))(h^2(x_2,x_3)-(\mathbf{d}^{(0)}(x_2)h^1(x_3)-\mathbf{d}^{(0)}(x_3)h^1(x_2)-h^1([x_2,x_3])))\ :\] \[-\mathrm{ad}(h^1(x_2))(h^2(x_1,x_3)-(\mathbf{d}^{(0)}(x_1)h^1(x_3)-\mathbf{d}^{(0)}(x_3)h^1(x_1)-h^1([x_1,x_3])))\ :\] \[+\mathrm{ad}(h^1(x_3))(h^2(x_1,x_2)-(\mathbf{d}^{(0)}(x_1)h^1(x_2)-\mathbf{d}^{(0)}(x_2)h^1(x_1)-h^1([x_1,x_2])))\ :\] \[+\mathbf{d}^1h^1([x_1,x_2],x_3)+[h^1([x_1,x_2]),h^1(x_3)]+\eta^2([x_1,x_2],x_3)\ :\] \[+\mathbf{d}^1h^1(x_2,[x_1,x_3])+[h^1(x_2),h^1([x_1,x_3])]+\eta^2(x_2,[x_1,x_3])\ :\] \[-\mathbf{d}^1h^1(x_1,[x_2,x_3])-[h^1(x_1),h^1([x_2,x_3])]+\eta^2(x_1,[x_2,x_3])\] \[=\mathbf{d}^2 h^2(x_1,x_2,x_3)\ :\] \[-\mathbf{d}^{(0)}(x_1)\mathbf{d}^{(0)}(x_2)h^1(x_3) +\mathbf{d}^{(0)}(x_2)\mathbf{d}^{(0)}(x_1)h^1(x_3)+[h^1(x_3),h^2(x_1,x_2)] + {d}^{(0)}(x_1)\eta^2(x_2,x_3))\ :\] \[+\mathbf{d}^{(0)}(x_1)\mathbf{d}^{(0)}(x_3)h^1(x_2)-\mathbf{d}^{(0)}(x_3)\mathbf{d}^{(0)}(x_1)h^1(x_2)-[h^1(x_2),h^2(x_1,x_3)] -{d}^{(0)}(x_2) \eta^2(x_1,x_3))\ :\] \[-\mathbf{d}^{(0)}(x_2)\mathbf{d}^{(0)}(x_3)h^1(x_1)+\mathbf{d}^{(0)}(x_3)\mathbf{d}^{(0)}(x_2)h^1(x_1) +[h^1(x_1),h^2(x_2,x_3)] +{d}^{(0)}(x_3)\eta^2(x_1,x_2))\ :\] \[+\mathbf{d}^1h^1([x_1,x_2],x_3)+\mathbf{d}^{(0)}(x_3)h^1([x_1,x_2])+\eta^2([x_1,x_2],x_3)\ :\] \[+\mathbf{d}^1h^1(x_2,[x_1,x_3])-\mathbf{d}^{(0)}(x_2)h^1([x_1,x_3])+\eta^2(x_2,[x_1,x_3])\ :\] \[-\mathbf{d}^1h^1(x_1,[x_2,x_3])+\mathbf{d}^{(0)}(x_1)h^1([x_2,x_3])+\eta^2(x_1,[x_2,x_3])\] \[=\mathbf{d}^2 h^2(x_1,x_2,x_3) + {d}^{(0)}(x_1)\eta^2(x_2,x_3))-{d}^{(0)}(x_2) \eta^2(x_1,x_3))+{d}^{(0)}(x_3)\eta^2(x_1,x_2))\ :\] \[+\mathbf{d}^1h^1([x_1,x_2],x_3)+\mathbf{d}^{(0)}(x_3)h^1([x_1,x_2])-\mathbf{d}^{(0)}([x_1,x_2])h^1(x_3)\ :\] \[+\mathbf{d}^1h^1(x_2,[x_1,x_3])-\mathbf{d}^{(0)}(x_2)h^1([x_1,x_3])+\mathbf{d}^{(0)}([x_1,x_3])h^1(x_2)\ :\] \[-\mathbf{d}^1h^1(x_1,[x_2,x_3])+\mathbf{d}^{(0)}(x_1)h^1([x_2,x_3])-\mathbf{d}^{(0)}([x_2,x_3])h^1(x_1)\] \[=\mathbf{d}^2 h^2(x_1,x_2,x_3) + {d}^2\eta^2(x_1,x_2,x_3)\ :\] \[-h^1([[x_1,x_2],x_3])-h^1([x_2,[x_1,x_3]])+h^1([x_1,[x_2,x_3]])\] \[=\mathbf{d}^2 h^2(x_1,x_2,x_3)+ {d}^2\eta^2(x_1,x_2,x_3)\]
proposition
\[ \mathbf{d}^2 h^2\in Z^3(\mathfrak{g},\mathcal{Z}(\mathfrak{h}))\]
proof
\[ d^3 \mathbf{d}^2 h^2(x_1,x_2,x_3,x_4)=d^{(0)}(x_1)\mathbf{d}^2 h^2(x_2,x_3,x_4) -d^{(0)}(x_2)\mathbf{d}^2 h^2(x_1,x_3,x_4)+d^{(0)}(x_3)\mathbf{d}^2 h^2(x_1,x_2,x_4)-d^{(0)}(x_4)\mathbf{d}^2 h^2(x_1,x_2,x_3) \ :\] \[-\mathbf{d}^2 h^2([x_1,x_2],x_3,x_4)-\mathbf{d}^2 h^2(x_2,[x_1,x_3],x_4)-\mathbf{d}^2 h^2(x_2,x_3,[x_1,x_4])+\mathbf{d}^2 h^2(x_1,[x_2,x_3],x_4)+\mathbf{d}^2 h^2(x_1,x_3,[x_2,x_4])-\mathbf{d}^2 h^2(x_1,x_2,[x_3,x_4])\] \[=d^{(0)}(x_1)(\mathbf{d}^{(0)}(x_2)h^2(x_3,x_4)-\mathbf{d}^{(0)}(x_3)h^2(x_2,x_4)+\mathbf{d}^{(0)}(x_4)h^2(x_2,x_3) -h^2([x_2,x_3],x_4)-h^2(x_3,[x_2,x_4])+h^2(x_2,[x_3,x_4]))\ :\] \[-d^{(0)}(x_2)(\mathbf{d}^{(0)}(x_1)h^2(x_3,x_4)-\mathbf{d}^{(0)}(x_3)h^2(x_1,x_4)+\mathbf{d}^{(0)}(x_4)h^2(x_1,x_3) -h^2([x_1,x_3],x_4)-h^2(x_3,[x_1,x_4])+h^2(x_1,[x_3,x_4]))\ :\] \[+d^{(0)}(x_3)(\mathbf{d}^{(0)}(x_1)h^2(x_2,x_4)-\mathbf{d}^{(0)}(x_2)h^2(x_1,x_4)+\mathbf{d}^{(0)}(x_4)h^2(x_1,x_2) -h^2([x_1,x_2],x_4)-h^2(x_2,[x_1,x_4])+h^2(x_1,[x_2,x_4]))\ :\] \[-d^{(0)}(x_4)(\mathbf{d}^{(0)}(x_1)h^2(x_2,x_3)-\mathbf{d}^{(0)}(x_2)h^2(x_1,x_3)+\mathbf{d}^{(0)}(x_3)h^2(x_1,x_2) -h^2([x_1,x_2],x_3)-h^2(x_2,[x_1,x_3])+h^2(x_1,[x_2,x_3]))\ :\] \[-\mathbf{d}^{(0)}([x_1,x_2])h^2(x_3,x_4)+\mathbf{d}^{(0)}(x_3)h^2([x_1,x_2],x_4) -\mathbf{d}^{(0)}(x_4)h^2([x_1,x_2],x_3)\ :\] \[+h^2([[x_1,x_2],x_3],x_4)+h^2(x_3,[[x_1,x_2],x_4])-h^2([x_1,x_2],[x_3,x_4])\ :\] \[-\mathbf{d}^{(0)}(x_2)h^2([x_1,x_3],x_4)+\mathbf{d}^{(0)}([x_1,x_3])h^2(x_2,x_4) -\mathbf{d}^{(0)}(x_4)h^2(x_2,[x_1,x_3])\ :\] \[+h^2([x_2,[x_1,x_3]],x_4)+h^2([x_1,x_3],[x_2,x_4])-h^2(x_2,[[x_1,x_3],x_4])\ :\] \[ -\mathbf{d}^{(0)}(x_2)h^2(x_3,[x_1,x_4])+\mathbf{d}^{(0)}(x_3)h^2(x_2,[x_1,x_4]) -\mathbf{d}^{(0)}([x_1,x_4])h^2(x_2,x_3)\ :\] \[+h^2([x_2,x_3],[x_1,x_4])+h^2(x_3,[x_2,[x_1,x_4]])-h^2(x_2,[x_3,[x_1,x_4]])\ :\] \[+\mathbf{d}^{(0)}(x_1)h^2([x_2,x_3],x_4)-\mathbf{d}^{(0)}([x_2,x_3])h^2(x_1,x_4) +\mathbf{d}^{(0)}(x_4)h^2(x_1,[x_2,x_3])\ :\] \[-h^2([x_1,[x_2,x_3]],x_4)-h^2([x_2,x_3],[x_1,x_4])+h^2(x_1,[[x_2,x_3],x_4])\ :\] \[+\mathbf{d}^{(0)}(x_1)h^2(x_3,[x_2,x_4])-\mathbf{d}^{(0)}(x_3) h^2(x_1,[x_2,x_4]) +\mathbf{d}^{(0)}([x_2,x_4])h^2(x_1,x_3)\ :\] \[-h^2([x_1,x_3],[x_2,x_4])-h^2(x_3,[x_1,[x_2,x_4]])+h^2(x_1,[x_3,[x_2,x_4]])\ :\] \[-\mathbf{d}^{(0)}(x_1)h^2(x_2,[x_3,x_4])+\mathbf{d}^{(0)}(x_2)h^2(x_1,[x_3,x_4]) -\mathbf{d}^{(0)}([x_3,x_4])h^2(x_1,x_2)\ :\] \[+h^2([x_1,x_2],[x_3,x_4])+h^2(x_2,[x_1,[x_3,x_4]])-h^2(x_1,[x_2,[x_3,x_4]])\] \[=-\mathbf{d}^{(0)}(x_4)\mathbf{d}^{(0)}(x_3)h^2(x_1,x_2)+\mathbf{d}^{(0)}(x_3) \mathbf{d}^{(0)}(x_4)h^2(x_1,x_2)-\mathbf{d}^{(0)}([x_3,x_4])h^2(x_1,x_2)\ :\] \[-\mathbf{d}^{(0)}(x_2)\mathbf{d}^{(0)}(x_4)h^2(x_1,x_3)+\mathbf{d}^{(0)}(x_4)\mathbf{d}^{(0)}(x_2)h^2(x_1,x_3) +\mathbf{d}^{(0)}([x_2,x_4])h^2(x_1,x_3)\ :\] \[-\mathbf{d}^{(0)}(x_3) \mathbf{d}^{(0)}(x_2)h^2(x_1,x_4)+\mathbf{d}^{(0)}(x_2)\mathbf{d}^{(0)}(x_3)h^2(x_1,x_4) -\mathbf{d}^{(0)}([x_2,x_3])h^2(x_1,x_4) \ :\] \[+\mathbf{d}^{(0)}(x_1)\mathbf{d}^{(0)}(x_4)h^2(x_2,x_3)-\mathbf{d}^{(0)}(x_4)\mathbf{d}^{(0)}(x_1)h^2(x_2,x_3) -\mathbf{d}^{(0)}([x_1,x_4])h^2(x_2,x_3)\ :\] \[-\mathbf{d}^{(0)}(x_1)\mathbf{d}^{(0)}(x_3)h^2(x_2,x_4)\mathbf{d}^{(0)}(x_3) \mathbf{d}^{(0)}(x_1)h^2(x_2,x_4)+\mathbf{d}^{(0)}([x_1,x_3])h^2(x_2,x_4) \ :\] \[ +\mathbf{d}^{(0)}(x_1)\mathbf{d}^{(0)}(x_2)h^2(x_3,x_4) -\mathbf{d}^{(0)}(x_2)\mathbf{d}^{(0)}(x_1)h^2(x_3,x_4) -\mathbf{d}^{(0)}([x_1,x_2])h^2(x_3,x_4) \] \[=-\mathrm{ad}(h^2(x_3,x_4))h^2(x_1,x_2)+\mathrm{ad}(h^2(x_2,x_4))h^2(x_1,x_3)-\mathrm{ad}(h^2(x_2,x_3))h^2(x_1,x_4)\ :\] \[-\mathrm{ad}(h^2(x_1,x_4))h^2(x_2,x_3)+\mathrm{ad}(h^2(x_1,x_3))h^2(x_2,x_4)-\mathrm{ad}(h^2(x_1,x_2))h^2(x_3,x_4)\] \[=0\]
corollary
The choice of \(d^{(0)}\in\mathfrak{out}(\mathfrak{h})\) determines a cohomology class \( [\mathbf{d}^2 h^2]\in H^3(\mathfrak{g},\mathcal{Z}(\mathfrak{h}))\ .\)
proposition
One can see \(\mathfrak{h}\) as a semidirect sum \(\mathcal{Z}(\mathfrak{h})\oplus_{[\nu]}\mathfrak{ad}(\mathfrak{h})\ ,\) where \([\nu]\in H^2(\mathfrak{ad}(\mathfrak{h}),\mathcal{Z}(\mathfrak{h}))^\mathfrak{g}\ .\)
question
Why \(\mathfrak{g}\) invariant?
definition
We denote the new Lie algebra by \({\mathfrak{g}}{\oplus}_{h^2} {\mathfrak{h}}\ ,\) the semidirect product of \({\mathfrak{g}}\) and \({\mathfrak{h}}\) induced by \(h^2 \in Z_\wedge^2({\mathfrak{g}},{\mathfrak{h}})\ .\)
theorem
Let \(a^2\) be as defined in lecture two. Then
- \[ \mathfrak{k}\simeq \mathfrak{a}\oplus_{a^2} \mathfrak{g}\ .\]
proof
Consider \(\mathfrak{a}\oplus_{a^2}\mathfrak{g}\ ,\) where the representation \(d^{(0)}\) and the form \(a^2\in Z^2(\mathfrak{g},\mathfrak{a})\) are constructed as in the second lecture. Let \( \phi: \mathfrak{a}\oplus_{a^2}\mathfrak{g}\rightarrow \mathfrak{k}\) be defined by \(\phi((a,x))=\iota(a)+\sigma(x)\ .\) Then \[ \phi([(a_1,x_1),(a_2,x_2)])=\phi((d_+^{(0)}(x_1)a_2-d_-^{(0)}(x_2)a_1-a^2(x_1,x_2),[x_1,x_2]))\ :\] \[=\iota(d_+^{(0)}(x_1)a_2-d_-^{(0)}(x_2)a_1-a^2(x_1,x_2))+\sigma([x_1,x_2])\ :\] \[=[\sigma(x_1),\iota(a_2)]+[\iota(a_1),\sigma(x_2)]+[\sigma(x_1),\sigma(x_2)]\ :\] \[=[\iota(a_1)+\sigma(x_1),\iota(a_2)+\sigma(x_2)]\ :\] \[=[\phi((a_1,x_1)),\phi((a_2,x_2))].\]
Let now \(\psi:\mathfrak{k}\rightarrow \mathfrak{a}\oplus_{a^2}\mathfrak{g}\) be defined by \[\psi(x)=(\iota^{-1}(x-\sigma(p(x))),p(x))\ .\] Then \[\psi([x,y])=(\iota^{-1}([x,y]-\sigma(p([x,y]))),p([x,y]))\ :\] \[=(\iota^{-1}([x,y]-\sigma([p(x),p(y)])),[p(x),p(y)])\ :\] \[=(\iota^{-1}([x,y]-[\sigma(p(x)),\sigma(p(y))]-\iota a^2(p(x),p(y))),[p(x),p(y))])\ :\] \[=(\iota^{-1}([x-\sigma(p(x)),\sigma(p(y))]+[\sigma(p(x)),y-\sigma(p(y))]-\iota a^2(p(x),p(y))),[p(x),p(y))])\ :\] \[=(\iota^{-1}([\sigma(p(x)),y-\sigma(p(y))])+\iota^{-1}([x-\sigma(p(x),\sigma(p(y))])-a^2(p(x),p(y)),[p(x),p(y)])\ :\] \[=(d_+^{(0)}(p(x))\iota^{-1}(y-\sigma(p(y)))-d_-^{(0)}(p(y))\iota^{-1}(x-\sigma(p(x))-a^2(p(x),p(y)),[p(x),p(y)])\ :\] \[=[(\iota^{-1}(x-\sigma(p(x))),p(x)),(\iota^{-1}(y-\sigma(p(y))),p(y))]\ :\] \[=[\psi(x),\psi(y)]\] This implies that \(\phi\) and \(\psi\) are Leibniz algebra homomorphisms. Furthermore, \[ \phi(\psi(x))=\phi((\iota^{-1}(x-\sigma(p(x))),p(x)))\ :\] \[= x-\sigma(p(x))+\sigma(p(x)\ :\] \[ =x\] and \[\psi(\phi(a,x))=\psi(\iota(a)+\sigma(x))\ :\] \[=(\iota^{-1}(\iota(a)+\sigma(x)-\sigma(p(\iota(a)+\sigma(x))),\sigma(p(\iota(a)+\sigma(x)))\ :\] \[=(\iota^{-1}(\iota(a)+\sigma(x)-\sigma(x)),\sigma(x))\ :\] \[=(a,\sigma(x))\] This \(\phi\) and \(\psi\) are Leibniz algebra isomorphisms. One has
- \[ \mathfrak{k}\simeq \mathfrak{a}\oplus_{a^2} \mathfrak{g}\quad\square\]
What if one now applies the construction in the second lecture to \(\mathfrak{a}\oplus_{a^2} \mathfrak{g}\ ?\) The maps \(\iota\) and \(p\) are given by \[\iota(a)=(a,0)\] \[p((a,x))=x\] From the definition of the bracket it follows that \(p\) is a Lie algebra homomorphism, for \(\iota\) this is trivial since \(\mathfrak{a}\) is abelian. One chooses a section \(\tilde{\sigma}\) as follows. \[\tilde{\sigma}(x)=(a^1(x),x),\quad a^1\in C^1(\mathfrak{g},\mathfrak{a})\] Then \[\iota(\tilde{d}_+^{(0)}(x)a=[\tilde{\sigma}(x),\iota(a)]\ :\] \[=[(a^1(x),x),(a,0)]\ :\] \[=(d_+^{(0)}(x)a,0)\] and \[\iota(\tilde{d}_-^{(0)}(x)a=-[\iota(a),\tilde{\sigma}(x)]\ :\] \[=-[((a,0),a^1(x),x)]\ :\] \[=(d_-^{(0)}(x)a,0)\] It follows that \(\tilde{d}_\pm^{(0)}=d_\pm^{(0)}\ ,\) which implies that the induced coboundary operators will be the same. Now \[\iota(\tilde{a}^2(x_1,x_2))=\tilde{\sigma}([x_1,x_2])-[\tilde{\sigma}(x_1),\tilde{\sigma}(x_2)]\ :\] \[=(a^1([x_1,x_2]),[x_1,x_2])-[(a^1(x_1),x_1),(a^1(x_2),x_2)]\ :\] \[=(a^1([x_1,x_2]),[x_1,x_2])-(d_+^{(0)}(x_1)a^1(x_2)-d_-^{(0)}(x_2)a^1(x_1)-a^2(x_1,x_2),[x_1,x_2])\ :\] \[=(a^2(x_1,x_2)-d^1 a^1(x_1,x_2),0)\] and this implies \[\tilde{a}^2=a^2-d^1 a^1\ .\] Furthermore
- \[\mathfrak{k}\simeq \mathfrak{a}\oplus_{a^2} \mathfrak{g}\simeq \mathfrak{a}\oplus_{\tilde{a}^2} \mathfrak{g}\]
and if \(a^1\in Z^1(\mathfrak{g},\mathfrak{a})\) one has
- \[\mathfrak{a}\oplus_{a^2} \mathfrak{g}= \mathfrak{a}\oplus_{\tilde{a}^2} \mathfrak{g}\]
theorem
To the short exact sequence of Leibniz algebras \[0\rightarrow\mathfrak{a}\rightarrow\mathfrak{k}\rightarrow\mathfrak{g}\rightarrow 0\] is associated an \[[a^2]\in H^2(\mathfrak{g},\mathfrak{a})\] such that for any \(a^2\in[a^2]\) one has
- \[\mathfrak{k}\simeq \mathfrak{a}\oplus_{a^2} \mathfrak{g}\]
references
- Alekseevsky, Dmitri ; Michor, Peter W. ; Ruppert, W. A. F. Extensions of super Lie algebras. J. Lie Theory 15 (2005), no. 1, 125--134.[1]
