Oscillator versus prefundamental representations

TL;DR

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Contribution

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Abstract

For the case of quantum loop algebras $\mathrm U_q(\mathcal L(\mathfrak{sl}_{l + 1}))$ with $l = 1, 2$ we find the $\ell$-weights and the corresponding $\ell$-weight vectors for the representations obtained via Jimbo's homomorphism, known also as evaluation representations. Then we find the $\ell$-weights and the $\ell$-weight vectors for the $q$-oscillator representations of Borel subalgebras of the same quantum loop algebras. This allows, in particular, to relate $q$-oscillator and prefundamental representations.