Universal Central Extension of the Tensor Algebra of a Lie Superalgebra and a Commutative Associative Algebra

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  are representations of the central extensions of. 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 the universal one (if it exists). In 1984, universal central extensions of the algebras of the form  for a unital commutative associative algebra  and a simple finite dimensional Lie algebra , were identified. Then in 2011, the case when  is a basic classical simple Lie superalgebra was studied by K. Iohara and Y. Koga.  They first study the case for Lie superalgebras  of rank 1; then they study -forms of  and prove the existence of a Chevalley base type for  using its structure as a basic classical simple Lie superalgebra. This in particular helps them to define an even nondegenerate symmetric invariant bilinear form on 

In this work, we study universal central extensions of Lie superalgebras of the form ,  where  is a finite dimensional  perfect  Lie superalgebra equipped with a nondegenerate homogeneous invariant supersymmetric bilinear form which is invariant under all  derivations and   is a unital commutative associative algebra. Our techniques are totally different from the ones done before; in fact to get our results we use the materials of the previous work of the author (joint with Karl-Hermann Neeb) regarding central extensions of .

We find the universal central extensions of Lie superalgebras of the form ,  where  is a finite dimensional  perfect  Lie superalgebra equipped with a nondegenerate homogeneous invariant supersymmetric bilinear form which is invariant under all derivations and   is a unital commutative associative algebra. 

Universal central extensions of Lie superalgebras of the form A ⊗  as above are identified. Our main result covers the results of the previous works in this regard and moreover, since odd nondegenerate invariant bilinear forms on   are allowed, we get something more, e.g., the uinversal central extension of A ⊗  for the queer Lie superalgebra   = (n) is also covered by our main theorem.