Menichetti's nonassociative $G$-crossed product algebras

Abstract: We construct unital central nonassociative algebras over a field $F$ which have either an abelian Galois extensions $K/F$ or a central simple algebra over a separable extension of $F$ in their nucleus. We give conditions when these algebras are division algebras. Our constructions generalize algebras studied by Menichetti over finite fields. The algebras are examples of non-trivial semiassociative algebras and thus relevant for the semiassociative Brauer monoid recently defined by Blachar, Haile, Matzri, Rein, and Vishne. When ${\rm Gal}(K/F)=G$ the algebras of the first type can be viewed as nonassociative $G$-crossed product algebras.