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

  • Shahn Majid
Part of the NATO ASI Series book series (NSSB, volume 245)


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)).


Partition Function Hopf Algebra Quantum Group Monoidal Category Tensor Category 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • Shahn Majid
    • 1
  1. 1.Department of Mathematics & CSUniversity College SwanseaSwanseaUK

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