Multipliers and covers of perfect diassociative algebras

Abstract

The paper concerns perfect diassociative algebras and their implications to the theory of central extensions. It is first established that perfect diassociative algebras have strong ties with universal central extensions. Then, using a known characterization of the multiplier in terms of a free presentation, we obtain a special cover for perfect diassociative algebras, as well as some of its properties. The subsequent results connect and build on the previous topics. For the final theorem, we invoke an extended Hochschild–Serre-type spectral sequence to show that, for a perfect diassociative algebra, its cover is perfect and has trivial multiplier. As an important consequence, we obtain the entire theory for associative algebras as a special case of diassociative algebras. We conclude with a concrete example.