Semiassociative Algebras over a Field

Central simple algebras are the building blocks of the associative structure theory. One of their key properties is that by extending the scalars, they become matrix algebras. Although nothing in algebra remains the same if we drop the associativity assumption, there is strong motivation (which I will discuss briefly) to try to recover this theory in the nonassociative realm. For this idea to have some hope, we must assume that the algebra contains a maximal subfield in the nucleus (the part which does associate with everything). Indeed, by extending the scalars, these algebras, termed "semiassociative'', become "skew matrix algebras''. I will show how this new structure leads to a nonassociative generalization of the Brauer group, in a way that should be understandable even if you momentarily forgot what the classical Brauer group is.

Based on joint work with Eliyahu Matzri, our students Guy Blachar and Edan Rein, and Darrell Haile.