For a positively graded algebra A we construct a functor from the derived
category of graded A-modules to the derived category of graded modules over
the quadratic dual A^! of A. This functor is an equivalence of certain
bounded subcategories if and only if the algebra A is Koszul. In the latter
case the functor gives the classical Koszul duality. The approach I will
talk about uses the category of linear complexes of projective A-modules.
Its advantage is that the Koszul duality functor is given in a nice and
explicit way for computational applications. The applications I am going to
discuss are Koszul dualities between certain functors on the regular block
of the category O, which lead to connections between different
categorifications of certain knot invariants. (Joint work with S.Ovsienko
and C.Stroppel.)