$T^∗$-Extension of Lie Supertriple Systems

Abstract

In this article, we study the Lie supertriple system (LSTS) $T$ over a field $\mathbb{K}$ admitting a nondegenerate invariant supersymmetric bilinear form (call such a $T$ metrisable). We give the definition of $T^∗_ω$-extension of an LSTS $T$, prove a necessary and sufficient condition for a metrised LSTS ($T$, $ϕ$) to be isometric to a $T^∗$-extension of some LSTS, and determine when two $T^∗$-extensions of an LSTS are "same", i.e., they are equivalent or isometrically equivalent.