  We define Poisson structures on certain transversal slices to conjugacy
classes in complex simple algebraic groups introduced in arXiv:0809.0205. These
slices are associated to the elements of the Weyl group, and the Poisson
structures on them are analogous to the Poisson structures introduced by J. de
Boer, T. Tjin and A. Premet in papers arXiv:hep-th/9211109 and
http://www.maths.man.ac.uk/DeptWeb/Homepages/aap/Reprints/Transverse.ps on the
Slodowy slices in complex simple Lie algebras. The quantum deformations of
these Poisson structures are known as W-algebras of finite type. As an
application of our definition we obtain some new Poisson structures on the
coordinate rings of simple Kleinian singularities.
