On Central Extensions of Associative Dialgebras

The concept of central extensions plays an important in constructing extensions of algebras. This technique has been successfully used in the classification problem of certain classes of algebras. In 1978 Skjelbred and Sund reduced the classification of nilpotent Lie algebras in a given dimension to the study of orbits under the action of automorphism group on the space of second degree cohomology of a smaller Lie algebra with coefficients in a trivial module. Then W. de Graaf applied the Skjelbred and Sund method to the classification problem of low-dimensional nilpotent Lie and associative algebras over some fields. The main purpose of this note is to establish elementary properties of central extensions of associative dialgebras and apply the above mentioned method to the classification of low dimensional nilpotent associative dialgebras. (paper)