On Isoclinism \& Baer's theorem for Lie superalgebras

TL;DR

This paper introduces the concept of isoclinism for Lie superalgebras, explores their structural properties, and generalizes Baer's theorem, providing new insights into their extensions and classifications.

Contribution

It defines isoclinism for Lie superalgebras, characterizes their covers, and proves a generalized Baer's theorem with related bounds and converses.

Findings

Maximal stem extensions are equivalent to stem covers.

Each isoclinic family contains a minimal-dimension stem Lie superalgebra.

A generalized Baer's theorem and its converse are established.

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.