Involutions on rectangular Jordan pairs
Let \(R\) be a unital associative algebra, \(M=(M^+_R,\; _RM^-)\) be a pair of unital \(R\)-modules (a right and a left), and \(f\colon M^+\times M^-\rightarrow R\) be an \(R\)-bilinear form. The associative pair \(A(R,M,f)\) was defined by \textit{O. Loos} [Jordan pairs. Lecture Notes Mathematics 460. Berlin etc.: Springer-Verlag (1975; Zbl 0301.17003)] as the pair \((M^+, M^-)\) with the triple multiplications from \(M^+\times M^-\times M^+\) to \(M^+\) and from \(M^-\times M^+\times M^-\) to \(M^-\): \[ \langle x^+,y^-,z^+\rangle=f(x^+,y^-)z^+,\quad \langle y^-,x^+,w^-\rangle=y^-f(x^+,w^-). \] The induced Jordan pair \(A(R,M,f)^J\) has quadratic products \(Q_x(y)=\langle x,y,x\rangle\), \[ Q_{x^+}(y^-)=f(x^+,y^-)x^+,\;\;Q_{y^-}(x^+)=y^-f(x^+,y^-). \] The author calls this pair \textit{rectangular} since the typical example is rectangular matrices. The objective of the paper is to describe all involutions on nondegenerate Jordan pairs (that is, where \(Q_x=0\) implies \(x=0\)) of rectangular type \(A(R,M,f)^J\) having a simple Artinian coordinate algebra \(R\) or, more generally, a simple unital coordinate algebra \(R\) such that the form \(f\) is unital-valued: \(f(u,v)=1\) for some \(u\in M^+, v\in M^-\). The involutions are of ``hermitian'' type determined by an involution (anti-isomorphism \(\sigma\) with \(\sigma^2=1\)) of \(R\), ``automorphism'' type determined by an automorphism \(\sigma\) of \(R\) with \(\sigma^2\) inner, or of ``isomorphism'' type determined by an isomorphism \(\sigma\) of \(R\) (which is necessarily non-Artinian) onto a proper subring.
scientific article; zbMATH DE number 1429474