We suggest two explicit descriptions of the Poisson q-W algebras which are Poisson algebras of regular functions on certain algebraic group analogues of the Slodowy transversal slices to adjoint orbits in a complex semisimple Lie algebra g. To obtain the first description we introduce certain projection operators which are analogous to the quasi-classical versions of the so-called Zhelobenko and extremal projection operators. As a byproduct we obtain some new formulas for natural coordinates on Bruhat cells in algebraic groups.
Sevostyanov, A. (2020). THE STRUCTURE OF Q-W ALGEBRAS. Transformation Groups, 25(1), 279–304. https://doi.org/10.1007/s00031-019-09533-8