  We discuss a general construction of a deformation of a smash product algebra
coming from an action of a particular Hopf algebra. This Hopf algebra is
generated by skew-primitive and group-like elements, and depends on a complex
parameter. The smash product algebra is defined on the quantum symmetric
algebra of a finite-dimensional vector space and a group. In particular, an
application of this result has enabled us to find a deformation of such a smash
product algebra which is, to the best of our knowledge, the first known example
of a deformation in which the new relations in the deformed algebra involve
elements of the original vector space. Using Hochschild cohomology, we show
that the resulting deformations are nontrivial by giving the precise
characterization of the infinitesimal.
