On abelian group actions and Galois quantizations (Q390849)

Quantizations of actions of a finite group \(G\) (not necessarily abelian) depend on the group only and are explicitely described by elements in the tensor square of group algebra, called quantizers. If \(G\) is finite abelian over an algebraically closed field \(\mathbb F\) with \(\mathrm{char}(\mathbb F)=0\) the quantizers are in one-to-one correspondence with the second cohomology group of the dual group \(\hat{G}\) with coefficients in the group of units of \(\mathbb F\). NEWLINENEWLINENEWLINEWith certain adjustments this result is applied to finite abelian group actions over any field \(\mathbb F\) with \(\mathrm{char}(\mathbb F)=0\). The quantizations of splitting fields of (products of) quadratic polynomials yield quantized Galois extensions that are Clifford-type algebras.