On the Homogeneous Space of a Pair of Associative Composition Algebras

The relationship between associative composition algebras of dimensions 2 and 4 within the context of homogeneous spaces, with a particular focus on Hamiltonian quaternions, is explored. In the special case of Hamiltonian quaternions, the equivariant cohomology rings of the homogeneous spaces are computed to gain a deeper understanding of their topological structure. In the general case, equivariant K-theory is utilized to examine the categories of vector bundles on these spaces. Taking this one step further, the Grothendieck ring of the category of locally free modules on the variety of singular matrices of size $n$ with entries from an associative composition algebra is determined. As a natural extension of these ideas, to define and study determinantal varieties of matrices with entries from a composition algebra, a notion of rank with respect to the base composition algebra is introduced, and criteria are established for the spanning set of a finite set of matrices with entries from the associative composition algebra to contain elements of certain ranks.