Centre and representations of at roots of unity

Abstract

Quantum groups at roots of unity have the property that their centre is enlarged. Polynomial equations relate the standard deformed Casimir operators and the new central elements. These relations are important from a physical point of view since they correspond to relations among quantum expectation values of observables that have to be satisfied on all physical states. In this paper, we establish these relations in the case of the quantum Lie superalgebra U_q(sl(2|1)). In the course of the argument, we find and use a set of representations such that any relation satisfied on all the representations of the set is true in U_q(sl(2|1)). This set is a subset of the set of all the finite dimensional irreducible representations of U_q(sl(2|1)), that we classify and describe explicitly.