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Weyl group symmetry of q-characters

Edward Frenkel, David Hernandez

TL;DR

The paper constructs a genuine Weyl group action of $W$ on a completed target $\Pi$ for the $q$-character setting, so that the $W$-invariants in $\mathcal{Y}$ equal the ring of $q$-characters, isomorphic to the Grothendieck ring of finite-dimensional $\mathcal{U}_q(\widehat{\mathfrak{g}})$-modules. It introduces automorphisms $\Theta_i$ defined via infinite-series data $\Sigma_{i,a}$ that extend to $\Pi$ and generate a $W$-action, with the $q\to1$ limit recovering the classical Weyl reflections. The work identifies screening operators as the subleading terms in deformations of $\Theta_i$ and proves braid relations by a rank-2 reduction and explicit $TQ$-relation analysis, including simply-laced and non-simply-laced types. It also relates these structures to existing symmetries, including a $q$-analogue of a rational-function ring and the category $\mathcal{O}$ framework, and provides expansions in types $A_2$, $B_2$, and $G_2$ to illustrate the braid-combinatorics explicitly. Overall, the results give a conceptual and technical bridge between $q$-characters, Weyl-group symmetries, screening operators, and categorical/cluster-algebra connections.

Abstract

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 U_q(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 U_q(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 U_q(g^) in arXiv:2312.13256 and to the categorification of cluster algebras in arXiv:2401.04616.

Weyl group symmetry of q-characters

TL;DR

The paper constructs a genuine Weyl group action of on a completed target for the -character setting, so that the -invariants in equal the ring of -characters, isomorphic to the Grothendieck ring of finite-dimensional -modules. It introduces automorphisms defined via infinite-series data that extend to and generate a -action, with the limit recovering the classical Weyl reflections. The work identifies screening operators as the subleading terms in deformations of and proves braid relations by a rank-2 reduction and explicit -relation analysis, including simply-laced and non-simply-laced types. It also relates these structures to existing symmetries, including a -analogue of a rational-function ring and the category framework, and provides expansions in types , , and to illustrate the braid-combinatorics explicitly. Overall, the results give a conceptual and technical bridge between -characters, Weyl-group symmetries, screening operators, and categorical/cluster-algebra connections.

Abstract

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 U_q(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 U_q(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 U_q(g^) in arXiv:2312.13256 and to the categorification of cluster algebras in arXiv:2401.04616.
Paper Structure (27 sections, 23 theorems, 187 equations)

This paper contains 27 sections, 23 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Completions
  3. Lie algebra and Weyl group
  4. Laurent polynomials
  5. Completions by formal series
  6. Uniqueness of solutions of $q$-difference equations
  7. The ring $\Pi$
  8. The Weyl group action
  9. Braid group relations
  10. Some invariant elements
  11. Reduction to the rank $2$ Lie algebras
  12. Type $A_1\times A_1$
  13. Image of the $\Sigma_{j,a}$
  14. Simply-laced types
  15. General types
  16. ...and 12 more sections

Key Result

Lemma 2.2

$\widetilde{\mathcal{Y}}^w$ is a complete topological ring with respect to the natural topology induced by the partial ordering on the sets of weights of the form omm, and it is a completion of the ring $\mathcal{Y}$ with respect to this topology.

Theorems & Definitions (48)

  • Definition 2.1
  • Lemma 2.2
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • Definition 2.7
  • Lemma 2.8
  • Definition 3.1
  • ...and 38 more