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

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