Irreducibility criterion for a finite-dimensional highest weight representation of the sl(2) loop algebra and the dimensions of reducible representations

We present a necessary and sufficient condition for a finite-dimensional highest weight representation of the $sl_2$ loop algebra to be irreducible. In particular, for a highest weight representation with degenerate parameters of the highest weight, we can explicitly determine whether it is irreducible or not. We also present an algorithm for constructing finite-dimensional highest weight representations with a given highest weight. We give a conjecture that all the highest weight representations with the same highest weight can be constructed by the algorithm. For some examples we show the conjecture explicitly. The result should be useful in analyzing the spectra of integrable lattice models related to roots of unity representations of quantum groups, in particular, the spectral degeneracy of the XXZ spin chain at roots of unity associated with the $sl_2$ loop algebra.Comment: 32 pages with no figure; with corrections on the published versio

Lattice fermion models with supersymmetry

We investigate a family of lattice models with manifest N=2 supersymmetry. The models describe fermions on a 1D lattice, subject to the constraint that no more than k consecutive lattice sites may be occupied. We discuss the special properties arising from the supersymmetry, and present Bethe ansatz solutions of the simplest models. We display the connections of the k=1 model with the spin-1/2 antiferromagnetic XXZ chain at \Delta=-1/2, and the k=2 model with both the su(2|1)-symmetric tJ model in the ferromagnetic regime and the integrable spin-1 XXZ chain at \Delta=-1/\sqrt{2}. We argue that these models include critical points described by the superconformal minimal models.Comment: 28 pages. v2: added new result on mapping to XXZ chai