# Quantum Group Duality in Vertex Models and other Results in the Theory of Quasitriangular Hopf Algebras

## Abstract

The Hopf algebra duality between observables and states in the author’s non-commutative geometric approach to quantum mechanics on curved spacetimes, is transferred to the context of vertex models. Among the results, for general invertible solution *R* of the QYBE we obtain from the bialgebra *A(R)* a quantum group *Ǎ(R)* and a dual quantum group *Ǔ(R)*, with *Ǔ(R)* quasi triangular. Previously this was known only for specific examples such as *SL* _{ q } *(2)* and *U* _{ q } *(sl* _{ 2 } *)*. The pairing between *Ǎ(R)* and *Ǔ(R)* leads to a direct expression for the partition function of associated exactly solvable vertex models in terms of the quantum group structures, as well as to a general variant of an ansatz of Kulish and Reshetikhin. A 5-vertex model is given as a simple example. We also obtain a general category-theoretic rank for the representation theory of quasitriangular Hopf algebras, generalizing the known “quantum dimension” *(q* ^{2j+1} *— q* ^{ - } ^{(2j+1)} */(q — q* ^{-1} *)* for the spin *j* representation of *U* _{ q } *(su(2))*.

## Keywords

Partition Function Hopf Algebra Quantum Group Monoidal Category Tensor Category## Preview

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