Oscillator versus prefundamental representations

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.

26 pages, LaTeX2e

References in corpus (4)

• Baxter's Q-operators for supersymmetric spin chains
• The Baxter's Q-operator for the W-algebra $W_N$
• On the universal R-matrix for the Izergin-Korepin model
• Affinization of category O for quantum groups

Cited by in corpus (14)

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• Separation of variables bases for integrable $g{l}_{\mathcal{M}|\mathcal{N}}$ and Hubbard models
• Quantum groups, Verma modules and $q$-oscillators: General linear case
• Vertex Models and Spin Chains in Formulas and Pictures
• Evaluation of the operatorial Q-system for non-compact super spin chains
• Quantum groups and functional relations for arbitrary rank
• Engineering 3D $\mathcal{N}=2$ theories using the quantum affine $\mathfrak{sl}(2)$ algebra
• $\ell$-weights and factorization of transfer operators
• Affinization of $q$-oscillator representations of $U_q({\mathfrak{gl}}_n)$
• Unipotent quantum coordinate ring and prefundamental representations for types $A_n^{(1)}$ and $D_n^{(1)}$
• Highest $\ell$-Weight Representations and Functional Relations