I am reading the book Integrable Systems: the r-matrix Approach. I have a question about the Killing form.
Let $\mathfrak{g}=\mathfrak{gl}_n$ be the full matrix algebra. Consider the Killing form $\langle X, Y \rangle = tr(XY)$, where $X, Y \in \mathfrak{gl}_n$. In Example 1.5, it is said that the dual $\mathfrak{b}_+^*$ of the subalgebra $\mathfrak{b}_+$ consisting of all upper triangular matrices in $\mathfrak{g}$ can be identified with $\mathfrak{b}_-$ consisting of all lower triangular matrices.
I think that the isomorphism $\varphi: \mathfrak{b}_- \to \mathfrak{b}_+^*$ is given by $\varphi(\xi)(\eta) = \langle \xi, \eta \rangle = tr(\xi \eta)$, $\xi \in \mathfrak{b}_-$, $\eta \in \mathfrak{b}_+$.
My question is: is the following map $\psi: \mathfrak{b}_- \to \mathfrak{b}_-^*$ which is given by $\psi(\xi)(\eta) = \langle \xi, \eta \rangle = tr(\xi \eta)$, $\xi \in \mathfrak{b}_-$, $\eta \in \mathfrak{b}_-$, also an isomorphism?
Thank you very much.