An introduction to Lie algebra cohomology/lecture 9
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the symplectic form
definition
If \(a^2\in Z^2(\mathfrak{g},\mathbb{C})\) is nondegenerate, then one says that \( a^2\) is a proto-symplectic form. If, moreover, \(a^2\in Z_\wedge^2(\mathfrak{g},\mathbb{C})\ ,\) one says that \( a^2\) is a symplectic form
definition
Let the symplectic operator \( \mathcal{S}:\mathfrak{g}\rightarrow \mathfrak{g}^\star=C^1(\mathfrak{g},\mathbb{C})\) be defined by \[ \mathcal{S}(x)=\iota^2(x)a^2\] If \(c^1\in C^1(\mathfrak{g},\mathbb{C})\) can be written as \[ c^1=S(x)\] one defines the cosymplectic operator \(\mathcal{C}\) by \[ C(c^1)=x\] One has, by definition, that \( \mathcal{C}(\mathcal{S}(x))=x\ ,\) or \( \mathcal{C}\mathcal{S}\) equals the identity on \(\mathfrak{g}\ .\)
definition
One defines a Leibniz algebra structure on the domain of \(\mathcal{C}\) in \(\mathfrak{g}^\star\) as follows. Let \[ \{a_1^1,a_2^1\}=\mathcal{S}([\mathcal{C}(a_1^1),\mathcal{C}(a_2^1)])\]
proof
\[ \{\{a_1^1,a_2^1\},a_3^1\}=\mathcal{S}([\mathcal{C}(\{a_1^1,a_2^1\}),\mathcal{C}(a_3^1)])\ :\] \[=\mathcal{S}([\mathcal{C}(\mathcal{S}([\mathcal{C}(a_1^1),\mathcal{C}(a_2^1)])),\mathcal{C}(a_3^1)])\ :\] \[=\mathcal{S}([[\mathcal{C}(a_1^1),\mathcal{C}(a_2^1)],\mathcal{C}(a_3^1)])\ :\] \[=\mathcal{S}([\mathcal{C}(a_1^1),[\mathcal{C}(a_2^1)],\mathcal{C}(a_3^1)])-\mathcal{S}([\mathcal{C}(a_2^1),[\mathcal{C}(a_1^1)],\mathcal{C}(a_3^1)])\ :\] \[=\mathcal{S}([\mathcal{C}(a_1^1),\mathcal{C}(\mathcal{S}([\mathcal{C}(a_2^1)],\mathcal{C}(a_3^1)])))-\mathcal{S}([\mathcal{C}(a_2^1),\mathcal{C}(\mathcal{S}([\mathcal{C}(a_1^1)],\mathcal{C}(a_3^1)])))\ :\] \[=\mathcal{S}([\mathcal{C}(a_1^1),\mathcal{C}(\mathcal{S}([\mathcal{C}(a_2^1)],\mathcal{C}(a_3^1)])))-\mathcal{S}([\mathcal{C}(a_2^1),\mathcal{C}(\mathcal{S}([\mathcal{C}(a_1^1)],\mathcal{C}(a_3^1)])))\]
definition
If \( a^2(x,y)=K(x,\omega^1 y)\ ,\) one says that \(\sharp \mathcal{S}=\omega^1\in C^1(\mathfrak{g},\mathfrak{g}) \) is a (proto-)symplectic map.
definition
When \( K\) is nondegenerate, and \( \alpha\in End(\mathfrak{g})\ ,\) then \(\alpha^\star\) (the adjoint of \( \alpha \)) is defined by \[ K(\alpha^\star x,y )=K(x,\alpha y)\]
proposition
When \(a^2\) is a symplectic form, \(\omega^1\) is an antisymmetric map, that is to say, \[ \omega^{1\star}=-\omega^1\]
proof
\[ K(x,\omega^1 y)= a^2(x,y)= - a^2(y,x)= -K(y,\omega^1 x)=-K(\omega^{1\star} y ,x )= -K(x,\omega^{1\star}y) \] It follows that \( \omega^1 y = - \omega^{1\star}y\ .\)
lemma
When \( K\) is nondegenerate, \[\omega^1([y,z]) = [y,\omega^1 (z)]-[\omega^{1\star}( y),z]\]
proof
\[ 0 = d^2 a^2 (x,y,z)=-a^2([x,y],z)-a^2(y,[x,z])+a^2(x,[y,z])\ :\] \[ =-K([x,y],\omega^1 z)-K(y,\mathcal{O}[x,z])+K(x,\omega^1 [y,z])\ :\] \[ =-K(x,[y,\omega^1z])-K(\omega^{1\star} y,[x,z])+K(x,\omega^1[y,z])\ :\] \[ =-K(x,[y,\omega^1 z])-K([x,z],\omega^{1\star} y)+K(x,\omega^1[y,z])\ :\] \[ =-K(x,[y,\omega^1z])-K(x,[z,\omega^{1\star} y])+K(x,\omega^1[y,z])\ :\] \[ =-K(x,[y,\omega^1z])+K(x,[\omega^{1\star} y,z])+K(x,\omega^1[y,z])\]
lemma
\[ d^1\omega^1(y,z)=-d_-^{(0)}(z)(\omega^1(y)+\omega^{1\star}( y))\]
proof
\[ d^1\omega^1(y,z) = [y,\omega^1 (z)]+[\omega^{1}( z),y]-\omega^1([y,z])\ :\] \[= [\omega^1(y)+\omega^{1\star}( y),z]\ :\] \[ = -d_-^{(0)}(z)(\omega^1(y)+\omega^{1\star}( y))\]
corollary
When \(a^2\) is symplectic, \( \omega^1\in Z_{\wedge}^1(\mathfrak{g},\mathfrak{g})\ .\)
