Second cohomology group of the finite-dimensional simple Jordan superalgebra ${\mathcal{D}}_t$, $t\mathrm{\ne }0$

The second cohomology group (SCG) of the Jordan superalgebra ${\mathcal{D}}_t$, $t\mathrm{\ne }0$, is calculated by using the coefficients which appear in the regular superbimodule $\mathrm{R}\mathrm{e}\mathrm{g}{\mathcal{D}}_t$. Contrary to the case of algebras, this group is nontrivial thanks to the non-splitting caused by the Wedderburn Decomposition Theorem \cite{Faber1}. First, to calculate the SCG of a Jordan superalgebra we use split-null extension of the Jordan superalgebra and the Jordan superalgebra representation. We prove conditions that satisfy the bilinear forms $h$ that determine the SCG in Jordan superalgebras. We use these to calculate the SCG for the Jordan superalgebra ${\mathcal{D}}_t$ , $t\mathrm{\ne }0$. Finally, we prove that $$\mathcal{H}$^2($\mathcal{D}${t}, $\text{Reg}$$\mathcal{D}${t})=0\oplus$\mathbb{F}$^2$,$t\neq 0$.