Graded-division algebras over arbitrary fields

Abstract

A graded-division algebra is an algebra graded by a group such that all nonzero homogeneous elements are invertible. This includes division algebras equipped with an arbitrary group grading (including the trivial grading). We show that a classification of finite-dimensional graded-central graded-division algebras over an arbitrary field $\mathbb{F}$ can be reduced to the following three classifications, for each finite Galois extension $\mathbb{L}$ of $\mathbb{F}$: (1) finite-dimensional central division algebras over $\mathbb{L}$, up to isomorphism; (2) twisted group algebras of finite groups over $\mathbb{L}$, up to graded-isomorphism; (3) $\mathbb{F}$-forms of certain graded matrix algebras with coefficients in $\Delta\otimes_{\mathbb{L}}\mathcal{C}$ where $\Delta$ is as in (1) and $\mathcal{C}$ is as in (2). As an application, we classify, up to graded-isomorphism, the finite-dimensional graded-division algebras over the field of real numbers (or any real closed field) with an abelian grading group. We also discuss group gradings on fields.