Higher order Kirillov-Reshetikhin modules for \(\mathbf{U}_q(A_{n}^{(1)})\), imaginary modules and monoidal categorification. (English) Zbl 1534.17012
In this paper the authors study higher order versions of Kirillov-Reshetikhin modules (KR-modules) for a quantum loop algebra associated to a simple Lie algebra of type \(A_n\). They also study the relations between higher order KR-modules and the work on monoidal categorification of cluster algebras of D. Hernandez and B. Leclerc [Duke Math. J. 154, No. 2, 265–341 (2010; Zbl 1284.17010)]. The results obtained are applied to give a systematic way to construct imaginary modules, where ’imaginary’, for a finite-dimensional irreducible module, means that its tensor square is not irreducible. In fact, it is shown that the tensor product of a (higher order) KR-module with its dual always contains an imaginary module in its Jordan-Hölder series.
Reviewer: Sonia Natale (Córdoba)
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |
Citations:
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