Oscillator versus prefundamental representations II. Arbitrary higher ranks

Abstract

We find the $\ell$-weights and the $\ell$-weight vectors for the highest $\ell$-weight $q$-oscillator representations of the positive Borel subalgebra of the quantum loop algebra $U_q(\mathcal{L}({\mathfrak{sl}}_{l+1}))$ for arbitrary values of $l$. Having this, we establish the explicit relationship between the $q$-oscillator and prefundamental representations. Our consideration allows us to conclude that the prefundamental representations can be obtained by tensoring $q$-oscillator representations.

26 pages, LaTeX2e; typos corrected, references with comments added. arXiv admin note: text overlap with arXiv:1512.04446, arXiv:1702.08710

References in corpus (6)

• A Shortcut to the Q-Operator
• Baxter Q-Operators and Representations of Yangians
• The Baxter's Q-operator for the W-algebra $W_N$
• Oscillator versus prefundamental representations
• Affinization of category O for quantum groups
• Quantum groups, Verma modules and $q$-oscillators: General linear case

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• Quantum groups and functional relations for arbitrary rank
• Unipotent quantum coordinate ring and prefundamental representations for types $A_n^{(1)}$ and $D_n^{(1)}$
• Affinization of $q$-oscillator representations of $U_q({\mathfrak{gl}}_n)$
• Highest $\ell$-Weight Representations and Functional Relations
• Algebraic Bethe ansatz for Q-operators of the open XXX Heisenberg chain with arbitrary spin