On abelian group actions and Galois quantizations

Quantizations of actions of finite abelian groups GG are explicitly described by elements in the tensor square of the group algebra of GG. Over algebraically closed fields of characteristic 0 these are in one to one correspondence with the second cohomology group of the dual of GG. With certain adjustments this result is applied to group actions over any field of characteristic 0. In particular we consider the quantizations of Galois extensions, which are quantized by “deforming” the multiplication. For the splitting fields of products of quadratic polynomials this produces quantized Galois extensions that all are Clifford type algebras.