Prime representations from a homological perspective

Chari, Vyjayanthi, Moura, Adriano and Young, Charles A. S. (2013) Prime representations from a homological perspective. Mathematische Zeitschrift, 274 (1-2). pp. 613-645. ISSN 0025-5874
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We explore the relation between self extensions of simple representations of quantum affine algebras and the property of a simple representation being prime. We show that every nontrivial simple representation has a nontrivial self extension. Conversely, we prove that if a simple representation has a unique nontrivial self extension up to isomorphism, then its Drinfeld polynomial is a power of the Drinfeld polynomial of a prime representation. It turns out that, in the sl 2 -case, a simple module is prime if and only if it has a unique nontrivial self extension up to isomorphism. It is tempting to conjecture that this is true in general and we present a large class of prime representations satisfying this homological property


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