Twisting of quantum spaces and twisted coHom objects

Abstract

Twisting process for quantum linear spaces is defined. It consists in a particular kind of globally defined deformations on finitely generated algebras. Given a quantum space (A_1,A), a multiplicative cosimplicial quasicomplex C[A_1] in the category Grp is associated to A_1, in such a way that for every n a subclass of linear automorphisms of A^{\otimes n} is obtained from the groups C^n[A_1]. 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, constructed in a previous paper (cf. math.QA/0112233), can be described as twisting of the proper coHom objects, enabling us in turn to generalize the mentioned construction. The quasicomplexes C[V], V a vector space, 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].