Orthogonal Bases of Involution in Hadamard Algebras

The notion of a Hadamard decomposition of a semisimple associative finite-dimensional complex algebra generalizes the notion of classical Hadamard matrices, which correspond to the case of commutative algebras. Algebras admitting a Hadamard decomposition are said to be Hadamard. Images of orthogonal bases of involution in Hadamard algebras under the canonical projections of these algebras onto their simple components are studied. Using a technique related to the study of central primitive idempotents of Hadamard algebras, we obtain a necessary condition for a family of involutory matrices of fixed order to be such an image. It is also shown that this necessary condition is not sufficient. We also present new proofs of results proved earlier.