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

Abstract

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 ansätze 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 $S_4$.

19 pages, no figures, changed introduction, added references

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• Integrable Hamiltonians with $D(D_n)$ symmetry from the Fateev-Zamolodchikov model
• Bethe ansatz solution of an integrable, non-Abelian anyon chain with D(D_3) symmetry
• $C^{\ast }$-index of observable algebra in the field algebra determined by a normal group
• Universal Baxterization for $\mathbb{Z}$-graded Hopf algebras
• Jones type basic construction on field algebras of $G$-spin models
• The construction of observable algebra in field algebra of $G$-spin models determined by a normal subgroup
• Polya Theory for Orbiquotient Sets