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Abstract

We present a short and rather self-contained introduction to the theory of finite-dimensional division algebras, setting out from the basic definitions and leading up to recent results and current directions of research. In Sections 2–3 we develop the general theory over an arbitrary ground field $k$, with emphasis on the trichotomy of fields imposed by the dimensions in which a division algebra exists, the groupoid structure of the level subcategories $\mathscr{D}_n(k)$, and the role played by the irreducible morphisms. Sections 4–5 deal with the classical case of real division algebras, emphasizing the double sign decomposition of the level subcategories $\mathscr{D}_n(\mathbb R)$ for $n \in \{ 2,4,8\}$ and the problem of describing their blocks, along with an account of known partial solutions to this problem.