Representations of the quantum doubles of finite group algebras and spectral parameter dependent solutions of the Yang–Baxter equation

Quantum doubles of finite group algebras form a class of quasi-triangularHopf algebras which algebraically solve the Yang--Baxter equation. Eachrepresentation of the quantum double then gives a matrix solution of theYang--Baxter equation. Such solutions do not depend on a spectral parameter,and to date there has been little investigation into extending these solutionssuch that they do depend on a spectral parameter. Here we first explicitlyconstruct the matrix elements of the generators for all irreduciblerepresentations of quantum doubles of the dihedral groups $D_n$. These resultsmay be used to determine constant solutions of the Yang--Baxter equation. Wethen discuss Baxterisation ans\"atze to obtain solutions of the Yang--Baxterequation with spectral parameter and give several examples, including a new21-vertex model. We also describe this approach in terms of minimal-dimensionalrepresentations of the quantum doubles of the alternating group $A_4$ and thesymmetric group $S_4$.Comment: 19 pages, no figures, changed introduction, added reference