Notes on $WGL_n$-Algebras and Quantum Miura Transformation

TL;DR

This paper derives explicit quantum L-operator formulas and the action of $W_l$ currents for $W$-algebras associated with $GL(n)$, using deformation of pseudodifferential symbols that preserve Adler's trace.

Contribution

It provides a clear formula for the quantum L-operator and the action of $W_l$ currents, extending the quantum Miura transformation framework for $GL(n)$.

Findings

Explicit quantum L-operator formula derived

Action of $W_l$ currents on degenerated representations characterized

Deformation of pseudodifferential symbols preserves Adler's trace

Abstract

We start from the quantum Miura transformation [7] for the $W$-algebra associated with $GL(n)$ group and find an evident formula for quantum L-operator as well as for the action of $W_l$ currents (l=1,..,n) on elements of the completely degenerated n-dimensional representation. Quantum formulae are obtained through the deformation of the pseudodifferential symbols. This deformation is independent of $n$ and preserves Adler's trace. Our main instrument of the proof is the notation of pseudodifferential symbol with right action which has no counterpart in classical theory.