A general approach to finite-dimensional division algebras

Ernst Dieterich¹

¹ Department of Mathematics Uppsala University Box 480 SE-751 06 Uppsala, Sweden

Colloquium Mathematicum, Tome 126 (2012) no. 1, pp. 73-86.

Résumé

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.

Détail

DOI : 10.4064/cm126-1-4

Keywords: present short rather self contained introduction theory finite dimensional division algebras setting out basic definitions leading recent results current directions research sections develop general theory arbitrary ground field emphasis trichotomy fields imposed dimensions which division algebra exists groupoid structure level subcategories mathscr role played irreducible morphisms sections classical real division algebras emphasizing double sign decomposition level subcategories mathscr mathbb problem describing their blocks along account known partial solutions problem