Isomorphisms of Tits--Kantor--Koecher Lie algebras of JB*-triples
María Cueto-Avellaneda, Lina Oliveira
TL;DR
This work investigates the isomorphisms of Tits–Kantor–Koecher (TKK) Lie algebras arising from JB*-triples by establishing a strong functorial correspondence between JB*-triples and their TKK algebras, yielding an equivalence of categories. It shows that isomorphisms between such TK Liegorithms are precisely surjective linear isometries, linking algebraic structure to metric properties. The authors develop a Lie-algebraic analogue of tripotent theory, introducing orthogonality and a partial order on tripotents in TKK algebras and proving extension results (notably Theorem ['t_atomic']) that recover real-linear graded isomorphisms from order-preserving, involution-commuting maps in the atomic case. Together, these results illuminate the structural interplay between JB*-triples and their TK Lie algebras and have implications for the symmetry and geometry of associated bounded symmetric domains.
Abstract
We characterise the isomorphisms of Tits--Kantor--Koecher Lie algebras of JB*-triples as a class of surjective linear isometries and show how these algebras form a category equivalent to that of JB*-triples. We introduce the concepts of tripotent, and orthogonality and order amongst tripotents for Tits--Kantor--Koecher Lie algebras. This leads to showing that a graded or negatively graded order isomorphism between certain subsets of tripotents of two Tits--Kantor--Koecher Lie algebras of atomic JB*-triples, which commutes with involutions, preserves orthogonality and is continuous at a non-zero tripotent of a specific type, can be extended as a real-linear isomorphism between the algebras.