Jordan triples and Riemannian symmetric spaces (Q952433)

From the text: The author introduces a class of real Jordan triple systems, called JH-triples, and show, via the Tits-Kantor-Koecher construction of Lie algebras, that they correspond to a class of Riemannian symmetric spaces including the Hermitian symmetric spaces and the symmetric R-spaces. He shows in Section 2 that the category of normed Jordan triple systems with continuous left multiplication is equivalent to the category of quasi normed canonical Tits--Kantor--Koecher Lie algebras. He defines the symmetric part, so-called because of the correspondence with symmetric spaces, of these Lie algebras and determine when the Lie product on the symmetric part is norm continuous. In Section 3, he shows the correspondence between symmetric spaces and orthogonal involutive Lie algebras, where a Lie algebra \(\mathfrak g\) with involution \(\theta\) is called orthogonal if there is a positive definite quadratic form on \(p\) which is invariant under the isotropy representation of \(k\) on \(p\), with \(k\) and \(p\) being the \(1\) and \(-1\) eigenspace of \(\theta\), respectively. A real non-degenerate Jordan triple system \(V\) with Jordan triple product \(\{\cdot, \cdot, \cdot\}\) is called a JH-triple if it is a real Hilbert space with continuous left multiplication \((x, y) \in V^2\mapsto \{x, y, \cdot\}: V \to V\) and the inner product satisfies \(\langle\{x,y,z\},z\rangle=\langle z, \{y, x, z\}\rangle\). He gives examples of JH-triples and proves that a Jordan triple system \(V\) is a JH-triple if, and only if, the corresponding Tits--Kantor--Koecher Lie algebra \(L(V)\) admits an orthogonal symmetric part. This enables him to conclude the correspondence between Jordan symmetric spaces and JH-triples in Section 4.