On oriented quantum algebras derived from representations of the quantum double of a finite-dimensional Hopf algebra.
Twist oriented quantum algebras were introduced by \textit{L. Kauffman} and \textit{D. E. Radford} [J. Knot Theory Ramifications 10, No. 7, 1047-1084 (2001; Zbl 1008.57010)] in order to construct regular isotopy invariants of oriented knots and links. They are given by an algebra \(A\) together with a tuple of elements and endomorphisms satisfying technical requirements. The basic example of a twist oriented quantum algebra is the Drinfeld double \(D(H)\) of a Hopf algebra \(H\), together with: the canonical \(R\)-matrix; the square inverse \(S^{-2}\) of its antipode \(S\); the inverse of the Drinfeld element \(u\). Given a twist oriented quantum algebra \(A\) and a finite dimensional \(A\)-module \(M\), the author provides a condition for \(\text{End}(M)\) to be a twist oriented quantum algebra. Applying this result he gives an explicit description of the oriented quantum algebra structures on \(\text{End}(M)\) when \(M\) is a simple \(D(H)\)-module and \(H\) is a Taft algebra \(H_{\omega,n}\) or a close relative \(H_{\omega,n,N}\). Since a Yetter-Drinfeld module for a Hopf algebra \(H\) is just a \(D(H)\)-module, the above description is achieved through an analysis of simple Yetter-Drinfeld modules in general and through a characterization of simple Yetter-Drinfeld modules when \(H\) is graded with commutative cocommutative bottom term. The reader could compare the description of simple Yetter-Drinfeld modules of the Taft algebras considered, with a different approach, by \textit{H. Chen} [J. Algebra 225, No. 1, 391-409 (2000; Zbl 0952.16032)].
scientific article; zbMATH DE number 2037082
scientific article