Let $(\mathfrak {g}, [\cdot, \cdot], \delta)$ be a triangular Lie bialgebra with classical $r$-matrix $r \in \bigwedge^2 \mathfrak {g}.$ In such a case we say that $r$ is a triangular structure on $\mathfrak {g}.$ A triangular structure $r$ on $\mathfrak {g}$ is said to be non-degenerate if it induces non-degenerate map $\underline {r} : \mathfrak {g}^{\ast} \longrightarrow \mathfrak {g}.$
This definition is given by Pavel Etingof in his lecture notes on compact quantum groups. But I don't know what's the map $\underline {r} : \mathfrak {g}^{\ast} \longrightarrow \mathfrak {g}$ induced by $r$ which the author is talking about. Could anyone please clear it up a bit?
Thanks for your time.