As is well-known, the crux of the classical Brauer group theory of a field lies in the classification of central division algebras over the field .
On the other hand, we feel that the theory of division algebras in monoidal categories should be of independent interest and it merits a separate consideration.
It is known that there exist topological division algebras
which are not topologically isomorphic to C (see, for example, , pp.
Among the topics are the functoriality of Rieffel's generalized fixed-point algebras for proper actions, division algebras
and supersymmetry, Riemann-Roch and index formulae in twisted K-theory, noncommutative Yang-Mills theory for quantum Heisenberg manifolds, distances between matrix algebras that converge to coadjoint orbits, and geometric and topological structures related to M-branes.
The primary result of this paper is a construction of noncrossed product division algebras whose centers are rational function fields k(t) and Laurent series fields k((t)) over number fields k.
In the early days, all division algebras were constructed as crossed products, starting with Hamilton's real quaternions in 1843.
The seven chapters in the volume are dedicated to nonlinear elliptic equations, division algebras
, exceptional lie groups, and calibrations, Jordan algebras and the Cartan isoparametric cubics, solutions from trialities and isoparametric forms, cubic minimal cones, and singular solutions in calibrated geometrics.
Finally, the article by Erik Darpo presents interesting new aspects and results pertaining to the problem of classification of finite-dimensional real division algebras
with deep connections to geometry and invariants.
Using (2.1) one can obtain estimates for the u-invariants of division algebras
with involution over a [C.sub.i]-field.
Despite the simple definition, division algebras
are highly non-trivial objects.
Boelaert, Medts, and Stavrova show that every structurable division algebra
gives rise to a Moufang set, It turns out, they say, that every known proper Moufang set-that is, is not sharply doubly transitive-arises from a structural division algebra
, provided that the root groups do not contain elements of order two or three.
Next, we consider the case of the Witt group of a quaternion division algebra
endowed with its canonical involution.