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1. CMB 2011 (vol 55 pp. 351)
Rational Homogeneous Algebras An algebra $A$ is homogeneous if the automorphism group of $A$
acts transitively on the one-dimensional subspaces of $A$. The existence of homogeneous algebras depends critically on the choice of the scalar field. We examine the case where the scalar field is the rationals. We prove that if $A$ is a rational homogeneous algebra with $\operatorname{dim} A>1$, then $A^{2}=0$.
Keywords:non-associative algebra, homogeneous, automorphism Categories:17D99, 17A36 |
2. CMB 2006 (vol 49 pp. 492)
Conjugacy Classes of Subalgebras of the Real Sedenions By applying the Cayley--Dickson process to the division algebra
of real octonions, one obtains a 16-dimensional real algebra
known as (real) sedenions. We denote this algebra by $\bA_4$.
It is a flexible quadratic algebra (with unit element 1) but
not a division algebra.
We classify the subalgebras of $\bA_4$ up to conjugacy (\emph{i.e.,}
up to the action of the automorphism group $G$ of $\bA_4$)
with one exception: we leave aside the more complicated case
of classifying the quaternion subalgebras.
Any nonzero subalgebra contains 1 and we show that there are
no proper subalgebras of dimension 5, 7 or $>8$.
The proper non-division subalgebras have dimensions
3, 6 and 8. We show that in each of these dimensions
there is exactly one conjugacy class of such subalgebras.
There are infinitely many conjugacy classes of subalgebras in
dimensions 2 and 4, but only 4 conjugacy classes in dimension 8.
Categories:17A45, 17A36, 17A20 |