  In general, if M is a moduli space of stable sheaves on X, there is a unique
alpha in the Brauer group of M such that a pi_M^* alpha^{-1}-twisted universal
sheaf exists on X times M. In this paper we study the situation when X and M
are K3 surfaces, and we identify alpha in terms of Mukai's map between the
cohomology of X and of M (defined by means of a quasi-universal sheaf). We
prove that the derived category of sheaves on X and the derived category of
alpha-twisted sheaves on M are equivalent.
  This suggests a conjecture which describes, in terms of Hodge isometries of
lattices, when derived categories of twisted sheaves on two K3 surfaces are
equivalent. If proven true, this conjecture would generalize a theorem of Orlov
and a recent result of Donagi-Pantev.
