  Let \G be a (weak) quasi-Hopf algebra. Using a two-sided \G-coaction on an
algebra \M, we construct what we call the diagonal crossed product as a new
associative algebra structure on \M\otimes \dG, where \dG is the dual of \G.
This construction is largely motivated by the special case \M = \G, for which
we obtain an explicit definition of the quantum double \D(\G) for quasi-Hopf
algebras. Applications of our formalism include the field algebra construction
of Mack and Schomerus as well as the formulation of Hopf Spin chains or lattice
current algebras based on truncated quantum groups at roots of unity. A
complete proof that \D(\G) is even a (weak) quasi-triangular quasi-Hopf algebra
will be given in a separate paper.

Quantum Algebra;High Energy Physics - Lattice;Quantum Algebra