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DEMAZURE MODULES OF LEVEL TWO AND PRIME REPRESENTATIONS OF QUANTUM AFFINE $\mathfrak{sl}_{n+1}$

Part of: Linear algebraic groups and related topics Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  09 November 2015

Matheus Brito ,
Vyjayanthi Chari  and
Adriano Moura
Matheus Brito
Affiliation:
Departamento de Matemática, Unicamp, Campinas - SP - 13083-859, Brazil ([email protected]; [email protected])
Vyjayanthi Chari
Affiliation:
Department of Mathematics, University of California, Riverside, CA 92521, USA ([email protected])
Adriano Moura
Affiliation:
Departamento de Matemática, Unicamp, Campinas - SP - 13083-859, Brazil ([email protected]; [email protected])
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Abstract

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We study the classical limit of a family of irreducible representations of the quantum affine algebra associated to $\mathfrak{sl}_{n+1}$. After a suitable twist, the limit is a module for $\mathfrak{sl}_{n+1}[t]$, i.e., for the maximal standard parabolic subalgebra of the affine Lie algebra. Our first result is about the family of prime representations introduced in Hernandez and Leclerc (Duke Math. J.154 (2010), 265–341; Symmetries, Integrable Systems and Representations, Springer Proceedings in Mathematics & Statitics, Volume 40, pp. 175–193 (2013)), in the context of a monoidal categorification of cluster algebras. We show that these representations specialize (after twisting) to $\mathfrak{sl}_{n+1}[t]$-stable prime Demazure modules in level-two integrable highest-weight representations of the classical affine Lie algebra. It was proved in Chari et al. (arXiv:1408.4090) that a stable Demazure module is isomorphic to the fusion product of stable prime Demazure modules. Our next result proves that such a fusion product is the limit of the tensor product of the corresponding irreducible prime representations of quantum affine $\mathfrak{sl}_{n+1}$.

Keywords

Demazure modulesquantum affine algebrasprime representationsgraded limits

MSC classification

Primary: 17B10: Representations, algebraic theory (weights) 17B70: Graded Lie (super)algebras 20G42: Quantum groups (quantized function algebras) and their representations
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 17 , Issue 1 , February 2018 , pp. 75 - 105
Copyright
© Cambridge University Press 2015 

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