A celebrated theorem of Hopf, Bott, Milnor, and Kervaire [11],[1],[12] states that every finite-dimensional real division algebra has dimension 1, 2, 4, or 8. While the real division algebras of dimension 1 or 2 and the real quadratic division algebras of dimension 4 have been classified [6],[3],[9], the problem of classifying all 8-dimensional real quadratic division algebras is still open. We...

A celebrated theorem of Hopf, Bott, Milnor, and Kervaire [11],[1],[12] states that every finite-dimensional real division algebra has dimension 1, 2, 4, or 8. While the real division algebras of dimension 1 or 2 and the real quadratic division algebras of dimension 4 have been classified [6],[3],[9], the problem of classifying all 8-dimensional real quadratic division algebras is still open. We...

In this paper we consider Selberg-type square matrices integrals with focus on Kummer-beta types I & II integrals. For generality of the results for real normed division algebras, the generalized matrix variate Kummer-beta types I & II are defined under the abstract algebra. Then Selberg-type integrals are calculated under orthogonal transformations.

in this paper we consider selberg-type square matrices integrals with focus on kummer-beta types i & ii integrals. for generality of the results for real normed division algebras, the generalized matrix variate kummer-beta types i & ii are defined under the abstract algebra. then selberg-type integrals are calculated under orthogonal transformations.