Universal measuring coalgebras and R - transformation algebras

TL;DR

This paper explores universal measuring coalgebras and their role in representing quantized universal enveloping algebras, linking algebraic structures with coalgebra enrichments and braiding.

Contribution

It introduces the universal measuring coalgebra P_R(V) associated with a braiding R, connecting it to the Fadeev-Takhtadjhan-Reshitikin bialgebra and quantized enveloping algebras.

Findings

Universal measuring coalgebras enrich algebra categories over coalgebras.

P_R(V) is dual to the A(R) bialgebra and models quantized enveloping algebras.

Actions of P_R(V) can be extended to quotients of tensor algebras.

Abstract

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.