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arXiv:math/9805089 · Journal of Physics A Mathematical and General

Difference equations and highest-weight modules of

A. Zapletal

Published Nov 27, 1998• 4 citations
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Abstract

Based on the trigonometric R -matrix of a family of q -deformed discrete Knizhnik-Zamolodchikov equations is formulated. Solutions to them are constructed by using the generalization of the algebraic Bethe ansatz developed to solve difference equations. These solutions are shown to be of highest weight with respect to the underlying quantum group structure. Using the variant of the nested Bethe ansatz method these results are extended to the higher-ranked symmetry of .

DOI
10.1088/0305-4470/31/47/018

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Based on the trigonometric R -matrix of a family of q -deformed discrete Knizhnik-Zamolodchikov equations is formulated. Solutions to them are constructed by using the generalization of the algebraic Bethe ansatz developed to solve difference equations. These solutions are shown to be of highest weight with respect to the underlying quantum group structure. Using the variant of the nested Bethe ansatz method these results are extended to the higher-ranked symmetry of .
A system of O(N)-matrix difference equations is solved by means of the off-shell version of the nested algebraic Bethe ansatz. In the nesting process a new object, the ΠΠΠ-matrix, is introduced to overcome the complexities of the O(N) group structure. The proof of the main theorem is presented in detail. In particular, the cancellation of all unwanted terms is shown explicitly. The highest weight property of the solutions is proved.
The system of SU(N)- and U(N)-matrix difference equations are solved by means of a nested version of a generalized Bethe ansatz, also the called `off shell' Bethe ansatz (Babujian H M 1990 Correlation functions in WZNW model as a Bethe wavefunction for the Gaudin magnets Proc. XXIVth Int. Symp. (Ahrenshoop, Zeuthen)). To solve this new Bethe ansatz in the algebraic language analogous to the conventional case, a new type of monodromy matrices is introduced. They fulfil a new type of Yang - Baxter equations which simplify the proofs. Using a similar approach as for the conventional nested Bethe ansatz the problem is solved iteratively. The vanishing of the `unwanted terms' of the first level ansatz is equivalent to a set of second level difference equations. The solutions are obtained as sums over `off-shell' Bethe vectors. These sums are `Jackson-type integrals'. The highest weight property of the solutions is proved. The solutions are calculated explicitly for several examples of SU(N)- and U(N)-representations.