2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/54305The complete structure of the $WG_2$ algebra is obtained from an explicit realization by an abstract Virasoro algebra and a free boson field. We then construct its BRST operator and find a seven-parameter family of nilpotnt BRST operators. These free parameters are related to the canonical transformations of the ghost, antighost fields which leave the total stress-energy tensor and the antighost field $b$ invariant.14 pages, LaTeX (elsart)High Energy Physics - Theorytext