Universal measuring coalgebras and R - transformation algebras

Universal measuring coalgebras provide an enrichment of the category of algebras over the category of coalgebras. By considering the special case of the tensor algebra on a vector space V, the category of linear spaces itself becomes enriched over coalgebras, and the universal measuring coalgebra is the dual coalgebra of the tensor algebra T(V tensor V*). Given a braiding R on V the universal measuring coalgebra P_R(V) which preserves the grading is naturally dual to the Fadeev-Takhtadjhan-Reshitikin bialgebra A(R) and therefore provides a representation of the quantized universal enveloping algebra as an algebra of transformations. The action of P_R(V) descends to actions on quotients of the tensor algebra, whenever the kernel of the quotient map is preserved by the action of a generating subcoalgebra of P_R(V). This allows representations of quantized enveloping algebras as transformation groups of suitably quantized spaces.