On the cohomology of the Weyl algebra, the quantum plane, and the q-Weyl algebra

Deformation theory can be used to compute the cohomology of a deformed algebra with coefficients in itself from that of the original. Using the invariance of the Euler–Poincaré characteristic under deformation, it is applied here to compute the cohomology of the Weyl algebra, the algebra of the quantum plane, and the q-Weyl algebra. The behavior of the cohomology when q is a root of unity may encode some number theoretic information.