  We introduce a family of algebras which are multiplicative analogues of
preprojective algebras, and their deformations, as introduced by M. P. Holland
and the first author. We show that these algebras provide a natural setting for
the 'middle convolution' operation introduced by N. M. Katz in his book 'Rigid
local systems', and put in an algebraic setting by M. Dettweiler and S. Reiter,
and by H. Volklein. We prove a homological formula relating the dimensions of
Hom and Ext spaces, study varieties of representations of multiplicative
preprojective algebras, and use these results to study simple representations.
We apply this work to the Deligne-Simpson problem, obtaining a sufficient (and
conjecturally necessary) condition for the existence of an irreducible solution
to the equation $A_1 A_2 ... A_k = 1$ with the $A_i$ in prescribed conjugacy
classes in $GL_n(C)$.
