Quantum current algebra bf U(frakgln[t]): canonical bases and polynomial representations
Abstract
We introduce a quantum deformation U(gln[t]) of the universal enveloping algebra of the current algebra gln[t], realized as a Hopf parabolic subalgebra of quantum affine gln. Unlike the Yangian---the standard quantization of the current algebra---our algebra admits a canonical basis. We give a BLM-type realization of U(gln[t]) via certain subalgebras of affine quantum Schur algebras, and then construct canonical bases for the modified quantum current algebra U̇(gln[t]) and for its finite dimensional irreducible graded modules. Moreover, we prove a rigidity theorem: every finite dimensional polynomial irreducible module for quantum affine gln remains irreducible when restricted to Uv(frakgln[t]) (the specialization of U(frakgln[t]) at a non-root-of-unity complex number v); conversely, every finite dimensional affine polynomial irreducible Uv(frakgln[t])-module extends uniquely to a polynomial irreducible module for quantum affine gln. More generally, we classify all finite dimensional polynomial irreducible Uv(frakgln[t])-modules by 𝒬(n)×Λ+(n). As an application, we establish a bijection between the isomorphism classes of finite dimensional polynomial irreducible modules of bf Uv(frakgln[t]) and those of the Yangian Y(gln). Thus bf U(gln[t]) admits canonical bases for its modified form and irreducible graded modules, and its finite dimensional polynomial irreducible modules are in correspondence with those of the Yangian.
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BibTeX
@misc{fu2026quantum,
title = {{Quantum current algebra $\bf U(\frakgl_n[t])$: canonical bases and polynomial representations}},
author = {Qiang Fu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.08937},
eprint = {2606.08937},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.08937},
}
RIS
TY - UNPB
TI - Quantum current algebra bf U(frakgl_n[t]): canonical bases and polynomial representations
AU - Qiang Fu
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2606.08937
UR - https://arxiv.org/abs/2606.08937
AB - We introduce a quantum deformation U(gl_n[t]) of the universal enveloping algebra of the current algebra gl_n[t], realized as a Hopf parabolic subalgebra of quantum affine gl_n. Unlike the Yangian---the standard quantization of the current algebra---our algebra admits a canonical basis. We give a BLM-type realization of U(gl_n[t]) via certain subalgebras of affine quantum Schur algebras, and then construct canonical bases for the modified quantum current algebra U̇(gl_n[t]) and for its finite dimensional irreducible graded modules. Moreover, we prove a rigidity theorem: every finite dimensional polynomial irreducible module for quantum affine gl_n remains irreducible when restricted to U_v(frakgl_n[t]) (the specialization of U(frakgl_n[t]) at a non-root-of-unity complex number v); conversely, every finite dimensional affine polynomial irreducible U_v(frakgl_n[t])-module extends uniquely to a polynomial irreducible module for quantum affine gl_n. More generally, we classify all finite dimensional polynomial irreducible U_v(frakgl_n[t])-modules by 𝒬(n)×Λ^+(n). As an application, we establish a bijection between the isomorphism classes of finite dimensional polynomial irreducible modules of bf U_v(frakgl_n[t]) and those of the Yangian Y(gl_n). Thus bf U(gl_n[t]) admits canonical bases for its modified form and irreducible graded modules, and its finite dimensional polynomial irreducible modules are in correspondence with those of the Yangian.
ER -