Level 0 and the Bethe ansatz
First I will review the mechanics for finding eigenvalues and eigenvectors of the Murphy elements in the group algebra of the symmetric group as a model method for finding the eigenvalues and eigenvectors of the transfer matrices that appear in the Algebraic Bethe ansatz. Second I will review the connection between transfer matrices and pseudoquasitriangular Hopf (psqtH) algebras. Third I will visit the 3 favourite families of pqtH-algebras and hint at their relation to affine Lie algebras. Finally, I will explain, via an illustrative (9-dimensional) example for sl_3, the construction of the eigenvalues and eigenvectors in a level 0 representation.
Professor Arun Ram, University of Melbourne