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 𝒟ₙ(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 𝒟ₙ(ℝ) for n ∈ {2,4,8} and the problem of describing their blocks, along with an account of known partial solutions to this problem.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm126-1-4,
author = {Ernst Dieterich},
title = {A general approach to finite-dimensional division algebras},
journal = {Colloquium Mathematicae},
volume = {126},
year = {2012},
pages = {73-86},
zbl = {1268.17003},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm126-1-4}
}
Ernst Dieterich. A general approach to finite-dimensional division algebras. Colloquium Mathematicae, Tome 126 (2012) pp. 73-86. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm126-1-4/