How to define inverse of a non-degenerate triangular structure?

Question

Let $\mathfrak {g}$ be a non-degenerate triangular Lie bialgebra with the non-degenerate triangular structure $r \in \bigwedge^2 \mathfrak {g}.$ Then how does it induce $r^{-1} \in \bigwedge^2 \mathfrak {g}^{\ast}\ ?$

Since $r$ is a non-degenerate triangular structure on $\mathfrak {g}$ it induces an isomorphism $r^{\sharp} : \mathfrak {g}^{\ast} \longrightarrow \mathfrak {g}.$ So ${r^{\sharp}}^{-1} : \mathfrak {g} \longrightarrow \mathfrak {g}^{\ast}$ exists as an isomorphism of vector spaces. But how does it give rise to an element $r^{-1} \in \bigwedge^2 \mathfrak {g}^{\ast}\ $? Is it $\left ({r^{\sharp}}^{-1} \right )^{\otimes 2} (r)\ $?

Any help in this regard would be warmly appreciated. Thanks for your time.

Answer 1

The answer to your last question is yes.

Note that the property of being triangular for $r$ reads as $[r,r]=0$, where $[-,-]$ is the Lie bracket of $\mathfrak{g}$ that one extends to $\wedge^\bullet\mathfrak{g}$ by the graded Leibniz rule.

This condition translates on $\omega:=r^{-1}$ in the following way: $d_{CE}\omega=0$. Here $d_{CE}$ is the differential for the Chevalley-Eilenberg cochain complex.

The whole story is parallel to how one identifies non-degenerate Poisson bivectors with symplectic forms (non-degenerate closed $2$-forms).