  We introduce and study kernel algebras, i.e., algebras in the category of
sheaves on a square of a scheme, where the latter category is equipped with a
monoidal structure via a natural convolution operation. We show that many
interesting categories, such as D-modules, equivariant sheaves and their
twisted versions, arise as categories of modules over kernel algebras. We
develop the techniques of constructing derived equivalences between these
module categories. As one application we generalize the results of
math.AG/9901009 concerning modules over algebras of twisted differential
operators on abelian varieties. As another application we recover and
generalize the results of Laumon in alg-geom/9603004 concerning an analog of
the Fourier transform for derived categories of quasicoherent sheaves on a dual
pair of generalized 1-motives.
