The geometric mean and B-loop quasigroups of the convex cone of positive definite matrices

In this paper we consider the convex cone of positive definite matrices as algebraic system equipped with geometric mean and B-loop from the standard matrix polar decomposition. Some algebraic structures of these quasigroups are investigated in the context of matrix theory. In particular, their autotopism groups are completely determined: they are isomorphic to the group of positive real numbers.