Quadratic Relations of the Deformed 𝑊-Algebra for the Twisted Affine Lie Algebra of Type 𝐴⁽²⁾₂N

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⁽²⁾₂N. We obtain a closed set of quadratic relations and duality, which allows us to define the deformed 𝑊-algebra 𝒲ₓ,ᵣ(𝐴⁽²⁾₂N) using generators and relations.