2022/11/17 by Edward Frenkel, Frenkel, Edward, David Hernandez +1 · 2 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Statistical Mechanics (cond-mat.stat-mech)
paper · pdf · doi:10.48550/arxiv.2211.09779
openalex publication_date 2022/11/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We define an action of the Weyl group W of a simple Lie algebra g on a completion of the ring Y, which is the codomain of the q-character homomorphism of the corresponding quantum affine algebra Uq(g^). We prove that the subring of W-invariants of Y is precisely the ring of q-characters, which is isomorphic to the Grothendieck ring of the category of finite-dimensional representations of Uq(g^). This resolves an old puzzle in the theory of q-characters. We also identify the screening operators, which were previously used to describe the ring of q-characters, as the subleading terms of simple reflections from W in a certain limit. Our results have already found applications to the study of the category O of representations of the Borel subalgebra of Uq(g^) in arXiv:2312.13256 and to the categorification of cluster algebras in arXiv:2401.04616.