Representations of the quantum doubles of finite group algebras and solutions of the Yang--Baxter equation

Quantum doubles of finite group algebras form a class of quasi-triangular Hopf algebras which algebraically solve the Yang–Baxter equation. Each representation of the quantum double then gives a matrix solution of the Yang–Baxter equation. Such solutions do not depend on a spectral parameter, and to date there has been little investigation into extending these solutions such that they do depend on a spectral parameter. Here we first explicitly construct the matrix elements of the generators for all irreducible representations of quantum doubles of the dihedral groups $D_n$. These results may be used to determine constant solutions of the Yang–Baxter equation. We then discuss Baxterisation ans"atze to obtain solutions of the Yang–Baxter equation with spectral parameter and give several examples, including a new 21-vertex model. We also describe this approach in terms of minimal-dimensional representations of the quantum doubles of the alternating group $A_4$ and the symmetric group $S_4$.