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Quantum W-algebras and Elliptic Algebras

Boris Feigin, Edward Frenkel

TL;DR

The paper develops a two-parameter deformation ${ m W}_{q,p}( rak{sl}_N)$ of quantum ${ m W}$-algebras, extending the ${ m W}_q( rak{sl}_N)$ framework and SKAO’s ${ m W}_{q,p}( rak{sl}_2)$ to higher rank. It provides a free-field realization via a quantum Miura map, introduces elliptic exchange relations through generating currents $oldsymbol{ ext Lambda}_i(z)$ and currents $T_i(z)$, and constructs screening currents that commute with ${ m W}_{q,p}( rak{sl}_N)$ up to total ${ m D}_q$-differences, enabling intertwiners and potential singular vectors connected to Macdonald functions. The elliptic structure is encoded by functions like $oldsymbol{ extvarphi}_N(w/z)$ and theta-functions, yielding elliptic deformations of quantum Serre relations and linking to elliptic quantum groups. In key limits, the construction recovers the ordinary ${ m W}$-algebras, the Poisson ${ m W}_q$ structures, and known Wakimoto realizations, highlighting the significance of these algebras in representation theory and integrable models, with implications for quantum DS reductions and Macdonald theory.

Abstract

We define quantum W-algebras generalizing the results of Reshetikhin and the second author, and Shiraishi-Kubo-Awata-Odake. The quantum W-algebra associated to sl_N is an associative algebra depending on two parameters. For special values of parameters it becomes the ordinary W-algebra of sl_N, or the q-deformed classical W-algebra of sl_N. We construct free field realizations of the quantum W-algebras and the screening currents. We also point out some interesting elliptic structures arising in these algebras. In particular, we show that the screening currents satisfy elliptic analogues of the Drinfeld relations in U_q(n^).

Quantum W-algebras and Elliptic Algebras

TL;DR

The paper develops a two-parameter deformation of quantum -algebras, extending the framework and SKAO’s to higher rank. It provides a free-field realization via a quantum Miura map, introduces elliptic exchange relations through generating currents and currents , and constructs screening currents that commute with up to total -differences, enabling intertwiners and potential singular vectors connected to Macdonald functions. The elliptic structure is encoded by functions like and theta-functions, yielding elliptic deformations of quantum Serre relations and linking to elliptic quantum groups. In key limits, the construction recovers the ordinary -algebras, the Poisson structures, and known Wakimoto realizations, highlighting the significance of these algebras in representation theory and integrable models, with implications for quantum DS reductions and Macdonald theory.

Abstract

We define quantum W-algebras generalizing the results of Reshetikhin and the second author, and Shiraishi-Kubo-Awata-Odake. The quantum W-algebra associated to sl_N is an associative algebra depending on two parameters. For special values of parameters it becomes the ordinary W-algebra of sl_N, or the q-deformed classical W-algebra of sl_N. We construct free field realizations of the quantum W-algebras and the screening currents. We also point out some interesting elliptic structures arising in these algebras. In particular, we show that the screening currents satisfy elliptic analogues of the Drinfeld relations in U_q(n^).

Paper Structure

This paper contains 18 sections, 4 theorems, 146 equations.

Table of Contents

  1. Introduction.
  2. Poisson algebras ${\cal W}_q(\goth{sl}_N)$.
  3. The algebra ${\cal W}_{q,p}(\goth{sl}_2)$.
  4. The algebra ${\cal W}_{q,p}(\goth{sl}_N)$.
  5. Heisenberg algebra.
  6. Definition of the quantum ${\cal W}$--algebra.
  7. Screening currents for ${\cal W}_{q,p}(\goth{sl}_N)$.
  8. Relations in ${\cal W}_{q,p}(\goth{sl}_N)$.
  9. Relations between $T_1(z)$ and $T_m(w)$.
  10. Relations in ${\cal W}_{q,p}(\goth{sl}_2)$.
  11. Relations in ${\cal W}_{q,p}(\goth{sl}_3)$.
  12. Relations in elliptic form.
  13. The case of $\goth{sl}_2$.
  14. The case of $\goth{sl}_N$.
  15. Elliptic relations for the screening currents.
  16. ...and 3 more sections

Key Result

Lemma 1

Let $b[n], n\in{\Bbb Z}$, and $c[n], n\in{\Bbb Z}$, satisfy commutation relations: where $|t|<1$. Then for $|z| > \underset{i,j}{\operatorname{max}} \left\{ |\alpha_i|,|\beta_j|\right\} |w|$ the composition acting on each module $\pi_\mu$ exists and is equal to

Theorems & Definitions (11)

  • Remark 1
  • Remark 2
  • Lemma 1
  • Remark 3
  • Remark 4
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Remark 5
  • Remark 6
  • ...and 1 more