Invariants for equivalence of central simple \(G\)-algebras. (Q1043830)

Clifford classes are equivalence classes of central simple \(G\)-algebras over fields, and they can be used to describe the Clifford theory of finite groups. The equivalence between two central simple \(G\)-algebras is defined in a way similar to the classical definition of the Brauer group, see for example [\textit{A. Turull}, J. Algebra 170, No. 2, 661-677 (1994; Zbl 0813.20010)] where this equivalence was introduced. The paper surveys recent results on the problem of finding invariants of these \(G\)-algebras which characterize their equivalence classes, and provides some improvements. The study of these invariants involves some elements in second cohomology groups, some elements in Brauer groups, as well as some Galois descent.