New examples of simple Jordan superalgebras over an arbitrary field of characteristic 0 (Q2849190)

Let \(F\) be a field of characteristic \(0\). An example of a \(F\)-subsuperalgebra of a Jordan superalgebra of vector type \(J(\Gamma,D)\) which is not isomorphic to a superalgebra of this type is constructed. The ingredients of the construction are the polynomial algebra in two variables \(F[x,y]\) and the polynomial \(f(x,y)=x^2+y^4-1\). Consider also the derivation \(D=2y^2{{\partial}\over{\partial x}}-x{{\partial}\over{\partial y}}\). Then \(D\) induces a derivation in the quotient algebra \(\Gamma:=F[x,y]/f(x,y)F(x,y)\). Denoting by \(x\) and \(y\) the images of these elements in \(\Gamma\) we take \(A\) to be the subalgebra of \(\Gamma\) generated by \(1,y^2,xy\) and \(M=xA+yA\). Then \(A+\bar M\) is provided with a superalgebra structure by viewing it as a subsuperalgebra of \(J(\Gamma,D)\). It is proved that this algebra is simple and it is not a superalgebra of vector type.NEWLINENEWLINEThis generalizes some previous results: in a joint paper of the author with I.P. Shestakov they constructed a unital simple special Jordan superalgebra over the reals with the property of being a subsuperalgebra of a Jordan superalgebra of vector type \(J(\Gamma,D)\), but the subsuperalgebra itself was not isomorphic to a superalgebra of this type. Later, a similar example was constructed over fields of characteristic \(0\) in which the equation \(t^2+1=0\) is unsolvable.