Derivations, invariant forms and the second homology group of orthosymplectic Lie superalgebras

We develop the description of the derivation algebras of orthosymplectic Lie super-algebras over supercommutative, associative superrings containing ½ and determine conditions under which the derivation algebra can be written as a semidirect product of the inner and the outer derivations. We then describe the supersymmetric invariant forms of the elementary orthosymplectic Lie superalgebra and determine the outer derivations which are skew with respect to a given supersymmetric invariant form. Finally, we describe the universal central extension and its centre, the second homology group, of the elementary orthosymplectic Lie superalgebra. The original motivation for this comes from the theory of extended affine Lie algebras. Specialized to the orthosymplectic Lie superalgebras representing the centreless cores of extended affine Lie algebras of type B and D, the above descriptions are the necessary and sufficient building blocks for the construction of an extended affine Lie algebra of type B and D from its centreless core.