[Paper Review] The q-characters of representations of quantum affine algebras and deformations of W-algebras

Edward Frenkel, Nicolai Reshetikhin|ArXiv.org|Oct 8, 1998

Algebraic structures and combinatorial modelsMathematics10 references135 citations

TL;DR

This paper introduces q-characters for finite-dimensional representations of quantum affine algebras, generalizing classical characters to the quantum setting. It constructs an injective homomorphism from the Grothendieck ring of representations to a ring of Laurent polynomials, conjectures that q-characters lie in the kernel of screening operators, and provides a geometric realization via q-difference operators, offering a combinatorial method to compute irreducible representations and their tensor products.

ABSTRACT

We propose the notion of q-characters for finite-dimensional representations of quantum affine algebras. It is motivated by our theory of deformed W-algebras. We show that the q-characters give rise to a homomorphism from the Grothendieck ring of representations of a quantum affine algebra to a polynomial ring. We conjecture that the image of this homomorphism is equal to the intersection of certain "screening operators". We also discuss the connection between q-characters and Bethe Ansatz.

Motivation & Objective

• To develop a q-character theory for finite-dimensional representations of quantum affine algebras, analogous to classical characters for Lie groups.
• To provide a combinatorial framework for classifying irreducible representations and their tensor products in the quantum affine setting.
• To relate q-characters to deformed W-algebras and the difference Drinfeld-Sokolov reduction, especially in the simply-laced case.
• To conjecture that the image of the q-character homomorphism equals the intersection of kernels of screening operators, enabling explicit computation of irreducible modules.
• To establish a geometric realization of q-characters via q-difference operators and orbit spaces of loop groups.

Proposed method

• Define the q-character homomorphism χq: Rep U_qĝ → Y = ℤ[Y_{i,a}^±1], mapping representations to Laurent polynomials in commuting variables indexed by i ∈ {1,…,ℓ} and a ∈ ℂ×.
• Construct screening operators S_i on the ring Y, conjectured to vanish precisely on the image of χq, thus characterizing irreducible representations as kernel elements.
• Use the universal R-matrix of U_qĝ to define χq, ensuring compatibility with tensor product structures and restriction to U_q𝔤.
• Realize q-characters geometrically via the q-gauge action on spaces of first-order difference operators, identifying orbits with fundamental representations.
• For simply-laced 𝔤, show that the q-character homomorphism coincides with the map induced by the projection μ_q from a space of operators to the orbit space M^J_{n,q}/LN.
• Use the formula t_i(s) = ∑_{j1<…<ji} λ_{j1}(s)λ_{j2}(sq^{-2})…λ_{ji}(sq^{-2i+2}) to match monomials in λ_i(s) with q-characters of fundamental representations.

Experimental results

Research questions

• RQ1 How can one generalize the notion of characters from classical Lie algebras to quantum affine algebras in a way that captures irreducible representations and their tensor products?
• RQ2 What is the algebraic structure of the image of the q-character homomorphism χq in the ring Y?
• RQ3 Can the q-characters of irreducible representations be characterized as solutions to a system of equations defined by screening operators S_i?
• RQ4 How does the q-character theory relate to deformed W-algebras and the difference Drinfeld-Sokolov reduction?
• RQ5 Can the q-character of a given highest weight representation be computed combinatorially via geometric data such as orbit spaces of q-difference operators?

Key findings

• The q-character homomorphism χq: Rep U_qĝ → Y is injective, providing a faithful representation of the Grothendieck ring in a Laurent polynomial ring.
• For 𝔤 = sl_2, the conjecture that the image of χq equals the intersection of the kernels of screening operators S_i is proven.
• The q-characters of fundamental representations of U_qĝ for 𝔤 = sl_N match the expressions t_i(s) = ∑_{j1<…<ji} λ_{j1}(s)λ_{j2}(sq^{-2})…λ_{ji}(sq^{-2i+2}) derived from the q-difference operator model.
• The geometric construction via the q-gauge action on M^J_{n,q} realizes the q-character homomorphism as the composition of embedding and projection μ_q, with image isomorphic to the orbit space M^J_{n,q}/LN.
• In the limit q→1, the q-character formulas reduce to classical characters related to the center of the completed enveloping algebra, linking to the Drinfeld-Sokolov reduction.
• The method provides a purely combinatorial algorithm to reconstruct the q-character of an irreducible representation from its dominant monomial, by finding minimal positive integral combinations in the image of μ_q.