A general approach to finite-dimensional division algebras

Ernst Dieterich. "A general approach to finite-dimensional division algebras." Colloquium Mathematicae 126.1 (2012): 73-86. <http://eudml.org/doc/284339>.

@article{ErnstDieterich2012,
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 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.},
author = {Ernst Dieterich},
journal = {Colloquium Mathematicae},
keywords = {division algebra; groupoid; irreducible morphism; double sign decomposition; description of blocks},
language = {eng},
number = {1},
pages = {73-86},
title = {A general approach to finite-dimensional division algebras},
url = {http://eudml.org/doc/284339},
volume = {126},
year = {2012},
}

TY - JOUR
AU - Ernst Dieterich
TI - A general approach to finite-dimensional division algebras
JO - Colloquium Mathematicae
PY - 2012
VL - 126
IS - 1
SP - 73
EP - 86
AB - 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.
LA - eng
KW - division algebra; groupoid; irreducible morphism; double sign decomposition; description of blocks
UR - http://eudml.org/doc/284339
ER -