Fundamental fields in the deformed $W$-algebras

TL;DR

This paper reformulates the definition of deformed W-algebras for simple Lie algebras, proves an algorithm's well-definedness, constructs specific elements, and verifies a conjecture for certain types, advancing explicit study of these algebras.

Contribution

It introduces a new framework for deformed W-algebras, proves an algorithm's correctness, and explicitly constructs elements to verify a key conjecture.

Findings

Proved the well-definedness of the algorithm for deformed W-algebras.

Constructed explicit elements in the algebra for types B and C.

Verified a conjecture of Frenkel and Reshetikhin for certain algebra types.

Abstract

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 $B_\ell$, $C_\ell$, and for some nodes in other types. This framework opens up new possibilities for studying explicitly fields in the deformed $W$-algebra $\mathbf{W}_{q,t}(\mathfrak{g})$.