On Isoclinism \& Baer's theorem for Lie superalgebras

Abstract

In this paper we define isoclinism for Lie superalgebras and using the concept of isoclinism, we give the structure of all covers of Lie superalgebras when their Schur multipliers are finite dimensional. It has been shown that that maximal stem extensions of Lie superalgebras are precisely same as the stem covers. Furthermore, we have defined stem Lie superalgebra and prove that each isoclinic family $\mathcal{C}$ contain a stem Lie superalgebra $T$ and it is the one having minimum even and odd dimension. Finally we have proved a converse of Schur's theorem and have given bound for stem Lie superalgebra. Then we state and prove Baer's theorem( generalisation of Schur's theorem) and a converse of it.