Universal Central Extension of Current Superalgebras

Representation as well as central extension are two of the most important concepts in the theory of Lie (super)algebras. Apart from the interest of mathematicians, the attention of physicist are also drawn to these two subjects because of the significant amount of their applications in Physics. In fact for physicists, the study of projective representations of Lie (super)algebras  are very important.      Projective representations of a Lie superalgebra L are representations of the central extensions of  L. So the study of projective representations has two steps; at first, one needs to know the central extensions and then to study their representations.      The first question in the study of central extensions is finding a universal one (if it exists). In 1984, universal central extensions of algebras of the form g=A⨂k, for a unital commutative associative algebra A and a simple finite dimensional Lie algebra k, was obtained. Then in 2001, the case when k is a basic classical simple Lie superalgebra was studied. In this work, we study universal central extensions of a class of Lie superalgebras of the form A⨂k; this class contains the above mentioned Lie (super)algebras. Our techniques are totally different from the ones done before and moreover, our results cover the previous ones.