Real division algebras, bounded quiver representations, and Euclidean configurations.

Time period:

1 January – 31 December 2009

Project leader:

Ernst Dieterich

Funder:

Swedish Research Council

Type of award:

Project grant

Total funding:

27,000 SEK

A (finite dimensional) real division algebra is a finite dimensional non-zero real vector space, endowed with a bilinear multiplication such that both left multiplication by any fixed non-zero vector and right multiplication by any fixed non-zero vector are bijective linear operators. A theorem of Hopf (1940), Kervaire, Bott and Milnor (1958) states that every real division algebra has dimension 1, 2, 4, or 8. The problem of classifying all real division algebras up to isomorphism is trivial in dimension 1, non-trivial but solved (by myself and others) in dimension 2, but only partially solved in the dimensions 4 and 8. The methods applied (by myself and others) in these partial solutions vary strongly, and the produced partially classifying lists appear often incomprehensible. Yet recently I discovered a common pattern underlying many of these approaches: For suitable full subcategories C of the category of all real division algebras there exists a subcategory B of the module category over an associative algebra, such that B and C are equivalent. Moreover, there exists a category A of configurations in a Euclidean space such that A and B are equivalent. Any such context A --> B --> C provides an unforeseen deep connection between the diverse theories of non-associative algebras, modules over associative algebras, and Euclidean geometry. The common pattern both unifies the already existing approaches and provides a source of intuition for future research in this area.