We are interested in the category of finite--dimensional representations of quantum affine algebras. This is a tensor category and a prime representation is one which is not isomorphic to the tensor product of two non trivial representations in the category. Understanding the prime simple representations is an important problem and several important families of examples are known of such representations. But no unifying feature is known to connect these families. In joint work with Charles Young and Adriano Moura, we show that the notion of prime is closely connected with the homological properties of these representations. In this talk we shall give evidence for our conjecture: a simple finite--dimensional representation V is prime iff the space of self--extensions Ext(V,V) is one--dimensional. We shall also see that this feature is quite unusual and has no counterpart in the case of the affine Lie algebra for instance.