New approaches to the octonions and their maximal orders

Depies, Connor Matthew

New approaches to the octonions and their maximal orders

File

Depies_iastate_0097E_21335.pdf (784.99 KB)

Date

2024-05

Authors

Depies, Connor Matthew

Advisor

Smith, Jonathan D.H.

Basak, Tathagata

Hartwig, Jonas

Slutsky, Konstantin

Weber, Eric

Altmetrics

Abstract

The associative Cayley-Dickson algebras over the field of real numbers are also Clifford algebras. The alternative but nonassociative real Cayley-Dickson algebras, notably the octonions and split octonions, share with Clifford algebras an involutary anti-automorphism and a set of mutually anticommutative generators. On the basis of these similarities, we introduce Kingdon algebras, written: K(V,B) for vector space V and symmetric bilinear form B. This construction produces the octonions and split octonions in a Clifford like way, and grants them a natural universal property akin to that of Clifford algebras. It also allows the construction of an dimension 8 alternative algebra similar in many ways to the exterior algebra. This algebra generalizes to higher dimensions, and the resulting algebras are called Medenwald algebras. This construction requires the addition of identities of the form (xy)z-z(yx)=0, for x, y and z elements of the generating vector space to those of the form xy+yx-B(x,y). This allows us to expand the set of algebras under consideration by replacing the trilinear (xy)z-z(yx) with (xy)z-z(yx)-T(x,y,z), for T a trilinear form. These algebras are called T-Kingdon algebras, and written K(V,B,T) for formed vector space (V,B) and trilinear form T. This extended construction is examined over vector spaces of 1, 2, 3 and 4 dimensions. These algebras are shown to have dimension 2, 4, 8 and 16 respectively. Particular interest is taken in the case where B and T are uniformly 0, the previously mentioned Medenwald algebras. In addition, in the 4-dimensional case, all K(V,B,T) are shown to be non-quadratic. This construction can be related to maximal orders of the octavians. In the reals, complex numbers and quaternions there is exactly one maximal order of rational elements. In the octonions there are seven. Since this exhausts the normed real division algebras, it is unusual that only one of them has more than one rational maximal order. This problem is minimized by using the natural (ℤ /2)2-grading on the octonions to specify a copy of the octavians which coheres with this grading. Then a (ℤ /2)3-grading of the octonions is examined which connects them with the Kingdon construction of the octonions by taking each basis element i, j, and k to have grading (1,0,0), (0,1,0) and (0,0,1).

Type

dissertation