On the Teichmüller map and a class of nonassociative algebras
This is joint work with Yuval Ginosar. Let K/F be a finite Galois extension with Galois group G. The Teichmüller map is a function that associates to every central simple K-algebra B normal over F an element of H^3(G, K*). The value of the function is trivial precisely when the class of B is restricted from F. The classical definition of this map involves the use of a crossed-product algebra over B. The associativity of this algebra is also equivalent to the class of B being restricted from F. The aim of this lecture is to elucidate the nature of the nonassociative algebras that arise when B is normal but not restricted. It turns out that the resulting theory is remarkably similar to the theory of associative algebras arising from the noninvertible cohomology of a Galois extension L/F such that L contains K, and I want to explain that relationship.