Centre and representations of small quantum superalgebras at roots of unity

Abstract

When the deformation parameter is a root of unity, the centre of a quantum group can be described by a set of generators and non trivial relations. In the case ofUq(sl(N)), these relations simply derive from the expressions of the deformed Casimir operators. In the case ofUq(osp(1|2)), the relation is simple if we use an operator which anticommutes with the fermionic generators and whose square is the quadratic Casimir. This operator also simplifies the classification of finite dimensional irreducible representations. In the case ofUq(sl(1|2)), the relations derive from the (infinite set of) standard Casimir operators.