Second cohomology group of the finite-dimensional simple Jordan Superalgebras D_t, t \= 0 (2022)
The second cohomology group (SCG) of the Jordan superalgebra Dt, t ≠ 0, is calculated by using the coefficients which appear in the regular superbimodule Reg Dt. Contrary to the case of algebras, this group is nontrivial thanks to the non-splitting caused by the Wedderburn Decomposition Theorem. 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 Dt , t ≠ 0. Finally, we prove that H2 (Dt, RegDt) = 0 ⊕ F2, t ≠ 0.