  A skew Laurent polynomial ring R[x^{\pm 1};\alpha] is reversible if it has a
reversing automorphism, that is, an automorphism of period two that transposes
x and x^{-1} and restricts to an automorphism of R. We study invariants for
reversing automorphisms and apply our methods to determine the rings of
invariants of reversing automorphisms of two simple skew Laurent polynomial
rings, namely a localization of the enveloping algebra of the two-dimensional
non-abelian solvable Lie algebra and the coordinate ring of the quantum torus.
These two skew Laurent polynomial rings are deformations of Poisson algebras
and we interpret their reversing automorphisms and their invariants as
deformations of Poisson automorphisms and their invariants.
