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)).
Work supported by SERC Research Assistantship
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Majid, S. (1990). Quantum Group Duality in Vertex Models and other Results in the Theory of Quasitriangular Hopf Algebras. In: Chau, LL., Nahm, W. (eds) Differential Geometric Methods in Theoretical Physics. NATO ASI Series, vol 245. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-9148-7_38
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DOI: https://doi.org/10.1007/978-1-4684-9148-7_38
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