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Quantum Affine Algebras and Deformations of the Virasoro and W-algebras

Edward Frenkel, Nikolai Reshetikhin

TL;DR

This work extends Feigin–Frenkel-type center results from affine algebras at the critical level to quantum affine algebras by constructing a Wakimoto-based, $q$-deformed Miura map from the center to a Heisenberg-Poisson algebra. It provides explicit realizations in the $\mathfrak{sl}_2$ case, including the $q$-Sugawara operators and the $q$-Miura transformation, and then lifts these constructions to $U_q(\widehat{\goth{sl}}_N)$ with a full $q$-deformed ${\cal W}$-algebra ${\cal W}_h(\goth{sl}_N)$ whose generators $s_i(z)$ encode spectra of transfer matrices. The center $Z_h(\widehat{\goth{sl}}_N)$ at the critical level is shown to be isomorphic to ${\cal W}_h(\goth{sl}_N)$ as Poisson algebras, via the Wakimoto realization and the $q$-Miura map, providing a quantum deformation of the classical Miura correspondence and a new perspective on Bethe Ansatz-related spectra. The framework extends to all classical quantum affine algebras, with explicit constructions for $A_n^{(1)}$, $B_n^{(1)}$, $C_n^{(1)}$, and $D_n^{(1)}$, and suggests a direct link between quantum centers, $q$-difference operators, and integrable-model transfer matrices.

Abstract

We generalize some results concerning affine algebras at the critical level to the corresponding quantum algebras. In particular, we show that the Wakimoto realization provides a homomorphism of Poisson algebras from the center of a quantum affine algebra to a Heisenberg-Poisson algebra. This homomorphism is a q-deformation of the Miura transformation. It is given by the same formulas as the spectra of transfer-matrices of the corresponding quantum integrable models. The image of the center in the Heisenberg-Poisson algebra is a Poisson subalgebra of the latter, which is a q-deformation of the corresponding classical W-algebra. The relations in this Poisson algebra are explicitly computed in the case of sl(n)^.

Quantum Affine Algebras and Deformations of the Virasoro and W-algebras

TL;DR

This work extends Feigin–Frenkel-type center results from affine algebras at the critical level to quantum affine algebras by constructing a Wakimoto-based, -deformed Miura map from the center to a Heisenberg-Poisson algebra. It provides explicit realizations in the case, including the -Sugawara operators and the -Miura transformation, and then lifts these constructions to with a full -deformed -algebra whose generators encode spectra of transfer matrices. The center at the critical level is shown to be isomorphic to as Poisson algebras, via the Wakimoto realization and the -Miura map, providing a quantum deformation of the classical Miura correspondence and a new perspective on Bethe Ansatz-related spectra. The framework extends to all classical quantum affine algebras, with explicit constructions for , , , and , and suggests a direct link between quantum centers, -difference operators, and integrable-model transfer matrices.

Abstract

We generalize some results concerning affine algebras at the critical level to the corresponding quantum algebras. In particular, we show that the Wakimoto realization provides a homomorphism of Poisson algebras from the center of a quantum affine algebra to a Heisenberg-Poisson algebra. This homomorphism is a q-deformation of the Miura transformation. It is given by the same formulas as the spectra of transfer-matrices of the corresponding quantum integrable models. The image of the center in the Heisenberg-Poisson algebra is a Poisson subalgebra of the latter, which is a q-deformation of the corresponding classical W-algebra. The relations in this Poisson algebra are explicitly computed in the case of sl(n)^.

Paper Structure

This paper contains 20 sections, 9 theorems, 158 equations.

Table of Contents

  1. Introduction.
  2. The center of $\widetilde{U}(\widehat{\goth{sl}}_2)$ at the critical level.
  3. The structure of the center.
  4. Wakimoto modules and Miura transformation.
  5. The Drinfeld realization.
  6. The RS realization.
  7. The isomorphism of two realizations.
  8. The $q$--analogues of the Sugawara operators.
  9. The Wakimoto realization of $U_h(\widehat{\goth{sl}}_2)_k$.
  10. Deformation of the Miura transformation.
  11. Poisson bracket.
  12. Generalization to $U_q(\widehat{\goth{sl}}_N)$.
  13. A $q$--deformation of the Heisenberg-Poisson algebra.
  14. A $q$--deformation of the ${\cal W}$--algebra.
  15. The center of $\widetilde{U}_h(\widehat{\goth{sl}}_N)$ at the critical level.
  16. ...and 5 more sections

Key Result

Lemma 1

The Fourier coefficients of the power series $k^\pm_1(z) k^\pm_2(zq^{-2})-1$ are central elements of $U_h(\widehat{\goth{g}\goth{l}}_2)_k$.

Theorems & Definitions (20)

  • Remark 1
  • Lemma 1
  • Proposition 1
  • Remark 2
  • Proposition 2: DF
  • Lemma 2
  • Corollary 1
  • Proposition 3
  • Proposition 4
  • Remark 3
  • ...and 10 more