The family of quaternionic quasi-unitary Lie algebras and their central extensions

The family of quaternionic quasi-unitary (or quaternionic unitary Cayley-Kl ein algebras) is described in a unified setting. This family includes the s imple algebras sp(N + 1) and sp(p, q) in the Cartan series CN+1 as well as many non-semisimple real Lie algebras which can be obtained from these simp le algebras by particular contractions. The algebras in this family are rea lized here in relation with the groups of isometries of quaternionic Hermit ian spaces of constant holomorphic curvature. This common framework allows one to perform the study of many properties for all these Lie algebras simu ltaneously. In this paper the central extensions for all quasi-simple Lie a lgebras of the quaternionic unitary Cayley-Klein family are shown to be tri vial no matter their dimension.