Twisting of quantum spaces and twisted coHom objects

Twisting process for quantum linear spaces is defined. It consists in a particular kind of globally defined deformations on finitely generated al- gebras. Given a quantum space A = (A1, A), a multiplicative cosimplicial quasicomplex C • (A1) in the category Grp is associated to A1, in such a way that for every n ∈ N a subclass of linear automorphisms of A ⊗n is obtained from the groups C n (A1). Among the elements of this subclass, the counital 2-cocycles are those which define the twist transformations. In these terms, the twisted internal coHom objects hom � (B, A), constructed in (7), can be described as twisting of the proper coHom objects hom(B, A). Moreover, sym- metric twisted tensor products A ◦� B, in terms of which above objects were built up, can be seen as particular 2-cocycle twisting of A ◦ B, enabling us to generalize the mentioned construction. The quasicomplexes C • (V) are studied in detail, showing for instance that, when V is a coalgebra, the quasicomplexes related to Drinfeld twisting, corresponding to bialgebras generated by V, are subobjects of C• (V).