Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 043, 31 pages      arXiv:1702.08710      https://doi.org/10.3842/SIGMA.2017.043
Contribution to the Special Issue on Recent Advances in Quantum Integrable Systems

Highest $\ell$-Weight Representations and Functional Relations

Khazret S. Nirov ab and Alexander V. Razumov c
a) Institute for Nuclear Research of the Russian Academy of Sciences, 60th October Ave. 7a, 117312 Moscow, Russia
b) Mathematics and Natural Sciences, University of Wuppertal, 42097 Wuppertal, Germany
c) Institute for High Energy Physics, NRC ''Kurchatov Institute'', 142281 Protvino, Moscow region, Russia

Received March 01, 2017, in final form June 06, 2017; Published online June 17, 2017; Misprints corrected August 17, 2017

Abstract
We discuss highest $\ell$-weight representations of quantum loop algebras and the corresponding functional relations between integrability objects. In particular, we compare the prefundamental and $q$-oscillator representations of the positive Borel subalgebras of the quantum group $\mathrm{U}_q(\mathcal L(\mathfrak{sl}_{l+1}))$ for arbitrary values of $l$. Our article has partially the nature of a short review, but it also contains new results. These are the expressions for the $L$-operators, and the exact relationship between different representations, as a byproduct resulting in certain conclusions about functional relations.

Key words: quantum loop algebras; Verma modules; highest $\ell$-weight representations; $q$-oscillators.

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