On the triviality of inhomogeneous deformations of $\mathfrak{osp}(1|2n)$

Abstract

We analyze the triviality of inhomogeneous $\gamma$-deformations of the oscillator Lie superalgebra $B(0,n) = \mathfrak{osp}(1|2n)$. As the main theorem, we show that for $n \geq 2$, the $\gamma$-deformation is trivial if and only if all deformation parameters vanish. The proof is based on the explicit construction of $2n$ certificates (left null space vectors $c$ satisfying $c^\top A_\mu = 0$ and $c^\top L_\mu \neq 0$) for the structure constant matrices $A_\mu$ of the coboundary operator. We provide a unified construction of certificates classified into three Families, and in particular clarify the geometric meaning of the coefficient $1 + \delta_{n,2}$ that appears in the Family~III certificate. We also discuss the contrast with the exceptional case of $B(0,1) = \mathfrak{osp}(1|2)$ (where all deformations are trivial). As an appendix, we outline the computational verification performed using exact rational arithmetic over $\mathbb{Q}$.