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

Abstract

The second cohomology group (SCG) of the Jordan superalgebra [Formula: see text], [Formula: see text], over an algebraically closed field [Formula: see text] of characteristic zero is calculated by using the coefficients which appear in the regular superbimodule Reg [Formula: see text]. Contrary to the case of algebras, this group is nontrivial thanks to the non-splitting caused by the Wedderburn Decomposition Theorem [F. A. Gómez-González, 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 [Formula: see text] that determine the SCG in Jordan superalgebras. We use these to calculate the SCG for the Jordan superalgebra [Formula: see text], [Formula: see text]. Finally, we prove that [Formula: see text], [Formula: see text].