Some conjectures about LLV Lie algebra, group of autoequivalences of the derived category and monodromy

I will discuss some recent work on the conjectural relation of the group of autoequivalences of the derived category and the LLV Lie algebra for (weak) CY varieties. Another conjecture (essentially due to Kontsevich) describes the relation of the above group to the monodromy of the mirror symmetric family.