Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 072, 36 pages      arXiv:2108.13883      https://doi.org/10.3842/SIGMA.2022.072

Quadratic Relations of the Deformed $W$-Algebra for the Twisted Affine Lie Algebra of Type $A_{2N}^{(2)}$

Takeo Kojima
Department of Mathematics and Physics, Faculty of Engineering, Yamagata University, Jonan 4-3-16, Yonezawa 992-8510, Japan

Received December 15, 2021, in final form September 09, 2022; Published online October 04, 2022

Abstract
We revisit the free field construction of the deformed $W$-algebra by Frenkel and Reshetikhin [Comm. Math. Phys. 197 (1998), 1-32], where the basic $W$-current has been identified. Herein, we establish a free field construction of higher $W$-currents of the deformed $W$-algebra associated with the twisted affine Lie algebra $A_{2N}^{(2)}$. We obtain a closed set of quadratic relations and duality, which allows us to define deformed $W$-algebra ${\mathcal W}_{x,r}\big(A_{2N}^{(2)}\big)$ using generators and relations.

Key words: deformed $W$-algebra; twisted affine algebra; quadratic relation; free field construction; exactly solvable model.

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