Abstract
A Lie superalgebra is called quasi-Frobenius if it admits a closed anti-symmetric non-degenerate bilinear form. We study the notion of double extensions of quasi-Frobenius Lie superalgebra when the form is either orthosymplectic or periplectic. We show that every quasi-Frobenius Lie superalgebra that satisfies certain conditions can be obtained as a double extension of a smaller quasi-Frobenius Lie superalgebra. We classify all 4-dimensional quasi-Frobenius Lie superalgebras, and show that such Lie superalgebras must be solvable. We study the notion of [Formula: see text]-extensions (or Lagrangian extensions) of Lie superalgebras, and show that they are classified by a certain cohomology space we introduce. Several examples are provided to illustrate our construction.
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