Hadamard Decompositions of Nearly Commutative Algebras

The notion of Hadamard decomposition of a semisimple associative finite-dimensional complex algebra generalizes the notion of the classical Hadamard matrix, which corresponds to the case of a commutative algebra. Algebras admitting Hadamard decompositions are said to be Hadamard. The paper considers the structure of Hadamard decompositions of algebras all of whose irreducible characters are of degree $1$ except one character of degree $2$. In particular, it is shown how to construct an Hadamard matrix of order $n$ by using the Hadamard decomposition of such an algebra of dimension $n$.