Second cohomology group of the finite-dimensional simple Jordan superalgebra Dt, t ≠ 0

The second cohomology group (SCG) of the Jordan superalgebra D-t, t not equal 0, over an algebraically closed field F of characteristic zero is calculated by using the coefficients which appear in the regular superbimodule Reg D-t. Contrary to the case of algebras, this group is nontrivial thanks to the non-splitting caused by the Wedderburn Decomposition Theorem [F. A. Gomez-Gonzalez, Wedderburn principal theorem for Jordan superalgebras I, J. Algebra 505 (2018) 1-32]. 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 D-t, t not equal 0. Finally, we prove that H-2 (D-t, RegD(t)) = 0 +F-2, t not equal 0.