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

Abstract

The second cohomology group (SCG) of the Jordan superalgebra ${\mathcal{D}}_t$, $t\ne 0$, 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 ${\mathcal{D}}_t$ , $t\ne 0$. Finally, we prove that ${\mathcal{H}}^2({\mathcal{D}}_t,\text{Reg}{\mathcal{D}}_t)=0\oplus {\mathbb{F}}^2$, $t\ne 0$.

10 pages