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

Abstract

The second cohomology group (SCG) of the Jordan superalgebra $\mathcal{D}_{t}$, $t\neq 0$, is calculated by using the coefficients which appear in the regular superbimodule $\mathrm{Reg}\mathcal{D}_t$. 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 $\mathcal{D}_{t}$ , $t\neq 0$. Finally, we prove that $\mathcal{H}^2(\mathcal{D}_{t}, \textrm{Reg}\mathcal{D}_{t})=0\oplus\mathbb{F}^2$, $t\neq 0$.