On compact generalized Jordan triple systems of the second kind (Q1115950)

Let U be a finite dimensional generalized Jordan triple system (GJTS) over a field of characteristic 0, that is, U is a triple system with product \((xyz):=L(x,y)z:=R(y,z)x\) satisfying [L(u,v), L(x,y)]\(=L(L(u,v)x,y)-L(x,L(v,u)y)\). It was shown that U induces a graded Lie algebra \({\mathcal L}(U)=\sum U_ i\) with \(U_{-1}=U\), by \textit{I. L. Kantor} [Tr. Semin. Vektorn. Tenzorn. Anal. Prilozh. Geom. Mekh. Fiz. 16, 407-499 (1972; Zbl 0272.17001)]. U is called of the second kind if \({\mathcal L}(U)\) is expressed as \(\sum^{2}_{i=-2}U_ i.\) Let \(\gamma\) be a bilinear form defined as \(\gamma (x,y)=(1/2)Trace(2R(x,y)+2R(y,x)-L(x,y)-L(y,x))\) which was considered by the reviewer. The main purpose of this paper is to study the GJTS of the second kind such that \(\gamma\) is nondegenerate. It is shown that if U is simple, then \(\gamma\) is nondegenerate, which implies \({\mathcal L}(U)\) is semisimple. A simple GJTS of the second kind satisfies the condition (A): if \(B_ a(x,y):=(xay)=0\) for all x, y in U, then \(a=0\). If \(\gamma\) is nondegenerate, then U satisfies the condition (A). Assuming \(\gamma\) is nondegenerate, an explicit relation between \(\gamma\) and the Killing form of \({\mathcal L}(U)\) is given. A real GJTS of the second kind U is said to be compact if \(\gamma\) is positive definite. Assume U is compact, then U is simple if and only if \({\mathcal L}(U)\) is simple. If U satisfies the condition (A), then there is a grade-reversing involutive automorphism \(\tau\) of \({\mathcal L}(U)\) such that \(\tau (a)=B_ a\) for \(a\in U\). The authors obtain that U is compact if and only if \(\tau\) is a Cartan involution.