CENTRAL EXTENSIONS OF THE FAMILIES OF QUASI-UNITARY LIE-ALGEBRAS

The most general possible central extensions of two whole families of Lie algebras, which can be obtained by contracting the special pseudo- unitary algebras su(p, q) of the Cartan series Al and the pseudo-unita ry algebras u(p, q), are completely determined and classified for arbi trary p and q. In addition to the su(p, q) and u(p, q) algebras, whose second cohomology group is well known to be trivial, each family incl udes many non-semisimple algebras; their central extensions, which are explicitly given, can be classified into three types as far as their properties under contraction are involved. A closed expression for the dimension of the second cohomology group of any member of these famil ies of algebras is given.