Fundamental fields in the deformed $W$-algebras

Let $\mathfrak{g}$ be a simple Lie algebra. Frenkel and Reshetikhin introduced the deformed $W$-algebra ${\mathbf{W}}_{qt}(\mathfrak{g})$. In this work, we propose a formal reformulation of this definition in a different context. In this framework, we reformulate and prove the well-definedness of an algorithm (arxiv:2103.15247, arxiv:2205.08312) inspired by the Frenkel-Mukhin algorithm (arXiv:math/9911112) which, starting from a given dominant monomial $m$ satisfying some degree conditions, produces elements of the deformed $W$-algebra. Then, we apply this algorithm to construct explicitly some specific elements of ${\mathbf{W}}_{q,t}(\mathfrak{g})$. In particular, we apply this to prove a conjecture of Frenkel and Reshetikhin in arXiv:q-alg/9708006 in types $B_{\mathrm{\ell }}$, $C_{\mathrm{\ell }}$, and for some nodes in other types. This framework opens up new possibilities for studying explicitly fields in the deformed $W$-algebra ${\mathbf{W}}_{q,t}(\mathfrak{g})$.