Three-vertex prime graphs and reality of trees

Abstract

We continue the study of prime simple modules for quantum affine algebras from the perspective of q-fatorization graphs. In this paper we establish several properties related to simple modules whose q-factorization graphs are afforded by trees. The two most important of them are proved for type A. The first completes the classification of the prime simple modules with three q-factors by giving a precise criterion for the primality of a 3-vertex line which is not totally ordered. Using a very special case of this criterion, we then show that a simple module whose q-factorization graph is afforded by an arbitrary tree is real. Indeed, the proof of the latter works for all types, provided the aforementioned special case is settled in general.

KEYWORD:

• Quantum affine algebra
• representation theory
• simple prime modules
• simple real modules
• tensor product factorization

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 17B10
• 17B37
• 20G42
• 81R10
• 05C05

Notes

1 Here we are using $p_{-}$ to shorten notation without making any assumption on the sign of $m-m\prime$.

Additional information

Funding

This work was developed as part of the Ph.D. project of the second author, which was supported by a PICME grant. The work of the first author was partially supported by CNPq grants 304261/2017-3 and 402449/2021-5, and Fapesp grant 2018/23690-6. We thank the anonymous referee for useful suggestions.