# Deformations of W-algebras associated to simple Lie algebras

@article{Frenkel1997DeformationsOW, title={Deformations of W-algebras associated to simple Lie algebras}, author={Edward Vladimir Frenkel and Nicolai Reshetikhin}, journal={Communications in Mathematical Physics}, year={1997}, volume={197}, pages={1-32} }

Deformed W-algebra Wq,t(g) associated to an arbitrary simple Lie alge- bra g is defined together with its free field realizations and the screening operators. Explicit formulas are given for generators of Wq,t(g) when g is of classical type. These formulas exhibit a deep connection between Wq,t(g) and the analytic Bethe Ansatz in integrable models associated to quantum affine algebrasUq(b) and Ut( L b). The scaling limit of Wq,t(g) is closely related to affine Toda field theories.

## 107 Citations

On Deformed W-algebras and Quantum Affine Algebras

- Mathematics
- 1998

We discuss some aspects of the deformed W-algebras W_{q,t}[g]. In particular, we derive an explicit formula for the Kac determinant, and discuss the center when t^2 is a primitive k-th root of unity.…

And Quantum Affine Algebras

- Mathematics
- 1998

We discuss some aspects of the deformed W-algebras Wq,t[g]. In particular, we derive an explicit formula for the Kac determinant, and discuss the center when t is a primitive k-th root of unity. The…

Quadratic relations of the deformed $W$-algebra for the twisted affine algebra $A_{2N}^{(2)}$

- Mathematics
- 2021

We revisit the free field construction of the deformed W -algebra by Frenkel and Reshetikhin, Commun. Math. Phys. 197, 1-31 (1998), where the basic W -current has been identified. Herein, we…

Quadratic relations of the deformed W-superalgebra Wq,tA(M,N)

- Mathematics
- 2021

We find the free field construction of the basic W-current and screening currents for the deformed W-superalgebra Wq,tA(M,N) associated with Lie superalgebra of type A(M, N). Using this free field…

The q-characters of representations of quantum affine algebras and deformations of W-algebras

- Mathematics
- 1998

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…

Quiver W-algebras

- Mathematics
- 2015

For a quiver with weighted arrows, we define gauge-theory K-theoretic W-algebra generalizing the definition of Shiraishi et al. and Frenkel and Reshetikhin. In particular, we show that the…

Quantum $q$-Langlands Correspondence

- MathematicsTransactions of the Moscow Mathematical Society
- 2018

We formulate a two-parameter generalization of the geometric Langlands correspondence, which we prove for all simply-laced Lie algebras. It identifies the q-conformal blocks of the quantum affine…

Quadratic relations of the deformed W-superalgebra

- Mathematics
- 2020

This paper is a continuation of the study by Ding and Feigin, Contemp.Math. 248, 83 (1998). We find a bosonization of the deformed W -superalgebras Wq,t(sl(2|1)) that commute up to the total…

Quiver Wε1,ε2 algebras of 4d N = 2 gauge theories

- Mathematics
- 2020

We construct an ϵ-deformation of W algebras, corresponding to the additive version of quiver W q , t − 1 algebras which feature prominently in the 5D version of the BPS/CFT correspondence and refined…

Double quantization of Seiberg–Witten
geometry and W-algebras

- MathematicsProceedings of Symposia in Pure Mathematics
- 2018

We show that the double quantization of Seiberg-Witten spectral curve for $\Gamma$-quiver gauge theory defines the generating current of W$(\Gamma)$-algebra in the free field realization. We also…

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