Open Access Publications from the University of California

## Self-Extensions and Prime Factorizations of Quantum Affine Algebras

• Author(s): Lunde, Mathew Arthur
• et al.
Abstract

It is well known that the category of finite dimensional representations of a quantum affine algebra is not semi-simple.

Moreover, the tensor product of irreducible representations remains irreducible generically. This observation leads naturally

to the definition of prime objects and the factorization of irreducible objects into irreducible primes. We show that there is

an interesting connection between the prime objects and the homological properties of the category: an irreducible

representation $V$ of $\hat{\mathbf U}_q(\mathfrak sl_2)$ is a tensor product of $r$ prime representations if and only if

the dimension of the space of self-extensions of $V$ is $r$. In addition, in the case when $V$ is a tensor power of an irreducible

prime module, we give generators and relations for $V$, as well as classify all self-extensions of $V$ (up to equivalence) in

terms of polynomials.