CROSSED-PRODUCTS BY A COALGEBRA

We introduce the notion of a crossed product of an algebra by a coalge bra C, which generalises the notion of a crossed product by a bialgebr a well-studied in the theory of Hoof algebras. The result of such a cr ossed product is an algebra which is also a right C-comodule. We find the necessary and sufficient conditions for two coalgebra crossed prod ucts be equivalent. We show that the two-dimensional quantum Euclidean group is a coalgebra crossed product. The paper is completed with an appendix describing the dualisation of construction of coalgebra cross ed products.