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

Reversible skew Laurent polynomial rings and deformations of Poisson automorphisms