Jordan superalgebras of vector type and projective modules (Q1760519)

The aim of the paper is to study connections between Jordan superalgebras of vector type and finitely generated projective modules of rank \(1\) over integral domains. Given an associative commutative \(F\)-algebra \(\Gamma\) with a nonzero derivation \(D\) we denote by \(\Gamma x\) a copy of \(\Gamma\) and define in \(\Gamma\oplus\Gamma x\) the product: \[ a\cdot b:=ab,\quad a\cdot bx:=(ab)x,\quad ax\cdot b:=(ab)x,\quad (ax)\cdot(bx)=D(a)b-aD(b), \] for \(a,b\in\Gamma\) and denoting by \(ab\) the product in \(\Gamma\). This new algebra \(\Gamma\oplus\Gamma x\) is denoted by \(J(\Gamma,D)\) and it turns out to be a Jordan superalgebra (of vector type) whose even part is \(A:=\Gamma\) and odd part \(M:=\Gamma x\). The superalgebra \(J(\Gamma,D)\) is simple if and only if \(\Gamma\) is \(D\)-simple. It is proved in the paper that if \(J=A\oplus M\) is a unital simple Jordan superalgebra such that the even part \(A\) is an associative algebra and the odd part \(M\) an associative \(A\)-module (and \(J\) is not a superalgebra of a nondegenerate bilinear superform) then \(M=Ax_1+\cdots Ax_n\) for some \(x_i\in M\) and the product in \(M\) is given by \[ ax_i\cdot bx_j=\gamma_{ij} ab+D_{ij}(a)b-aD_{ji}(b),\quad i,j=1,\dots,n, \] where \(\gamma_{ij}\in A\) and \(D_{ij}\) is a derivation of \(A\). The algebra \(A\) is differentiably simple with respect to the set of derivations \(\Delta=\{D_{ij}\}\) and \(M\) is a projective \(A\)-module if rank \(1\). Furthermore \(J\) is a subalgebra of \(J(\Gamma,D)\). The following is also proved in the work under review. Take \(A\) to be an integral domain and \(M=Ax_1+\cdots Ax_n\) a finitely generated projective \(A\)-module of rank \(1\). Consider a derivation \(A\to (M\otimes_A M)^*\) such that \(a\mapsto\bar a\). Take then a set of derivations \(\Delta=\{D_{ij}: i,j=1,\dots,n\}\) such that \(D_{ij}(a):=\bar a(x_i\otimes x_j)\). Fix elements \(\gamma_{ij}\in A\) such that \(\gamma_{ij}=-\gamma_{ji}\) and define in the vector space \(J(A,\Delta):=A\oplus M\) the multiplication \[ a\cdot b=ab,\quad a\cdot(bx_i)=(bx_i)\cdot a=(ab)x_i,\quad i=1,\dots n, \] \[ (ax_i)\cdot(bx_j)=\gamma_{ij}ab+D_{ij}(a)b-aD_{ij}(b),\quad i,j=1,\dots,n, \] Then if the derivations \(D_{ij}\) satisfy certain relations, \(J(A,\Delta)\) is a Jordan superalgebra with even part \(A\) and odd part \(M\). Define also \(L=AD_{11}+\cdots+AD_{nn}\), then the \(A\)-module \(L\) is a subalgebra of \(\text{Der}(A)\) and if \(L\) is a simple Lie algebra then \(J(A,\Delta)\) is a simple Jordan superalgebra. In Theorem 2 of the paper it is shown how to construct a subsuperalgebra of \(J(\Gamma,D_{11})\) which is simple if \(J(\Gamma,D_{11})\) is.