Quadratic relations of the deformed $W$-algebra

The deformed $W$-algebra is a quantum deformation of the $W$-algebra ${\mathcal{W}}_{\beta }(\mathfrak{g})$ in conformal field theory. Using the free field construction, we obtain a closed set of quadratic relations of the $W$-currents of the deformed $W$-algebra. This allows us to define the deformed $W$-algebra by generators and relations. In this review, we study two types of deformed $W$-algebra. One is the deformed $W$-algebra ${\mathcal{W}}_{x,r}\left(A_{2N}^{(2)}\right)$, and the other is the $q$-deformed corner vertex algebra $q$-$Y_{L_1,L_2,L_3}$ that is a generalization of the deformed $W$-algebra ${\mathcal{W}}_{x,r}(A(M,N{)}^{(1)})$ via the quantum toroidal algebra.