Néron–Severi Lie algebra, autoequivalences of the derived category, and monodromy

Let $X$ be a smooth complex projective variety. The group of autoequivalences of the derived category of $X$ acts naturally on its singular cohomology $H^\bullet (X,\mathbb{Q})$ and we denote by $G^{\mathrm{eq}}(X)\subset \mathrm{Gl}(H^\bullet (X,\mathbb{Q}))$ its image. Let $\overline{G^{\mathrm{eq}}(X)}\subset \mathrm{Gl}(H^\bullet (X,\mathbb{Q})$ be its Zariski closure. We study the relation of the Lie algebra $\mathrm{Lie}\, \overline{G^{\mathrm{eq}}(X)}$ and the Néron–Severi Lie algebra $\mathfrak{g}_{\mathrm{NS}}(X)\subset \mathrm{End}\, (H(X,\mathbb{Q}))$ in case $X$ has trivial canonical line bundle.
At the same time for mirror symmetric families of (weakly) Calabi–Yau varieties we consider a conjecture of Kontsevich on the relation between the monodromy of one family and the group $G^{\mathrm{eq}}(X)$ for a very general member $X$ of the other family.