Tits-Kantor-Koecher Lie algebras of JB*-triples
In this paper the authors study the connection between JB\(^*\)-triple systems and Lie algebras through a TKK-construction. In a purely algebraic setting, to any complex \textit{nondegenerate} (\(Q_a=0 \Rightarrow a=0\)) Jordan triple system \(V\) (conjugate linear in the middle variable), it is associated a complex \(3\)-graded Lie algebra \(\mathcal L(V) =\mathfrak g =\mathfrak g_{-1} \oplus \mathfrak g_0 \oplus \mathfrak g_1\) endowed with a conjugate linear involution \(\theta\) satisfying: \begin{itemize} \item[(i)] \([\mathfrak g_{-1}, \mathfrak g_1]=\mathfrak g_0\), \item[(ii)] \(\theta(\mathfrak g_i)=\mathfrak g_{-i}\), \(i=\pm 1\), \item[(iii)] under the triple product \(\{ a, b, c\}:=[[a, \theta(b)], c]\), \(\mathfrak g_{-1}\) is a Jordan triple isomorphic to \(V\). \item[(iv)] the Lie algebra \(\mathfrak g\) is nondegenerate: \(ad_a^2\mathfrak g=0 \Rightarrow a=0\) for all \(a \in \mathfrak g\). \end{itemize} And conversely, the \(-1\)-component of a pair \((\mathfrak g, \theta)\) as above is a nondegenerate Jordan triple system for the triple product given in (iii). If \(V\) is actually a JB\(^*\)-triple system (JB\(^*\)-triple systems are nondegenerate), then \(\mathfrak g\) becomes a normed Lie algebra and \(\theta\) is continuous. Among other characterizations given in \textbf{Theorem 3.4}, the authors prove that a complex Lie algebra \(\mathfrak g\) is the TKK-algebra of a JB\(^*\)-triple system if and only if \(\mathfrak g\) has a real form which is isomorphic to the reduced Lie algebra of a bounded symmetric domain. In the last section of the paper, matrix representations of the TKK-algebras of the JB\(^*\)-triple systems are given by reducing the problem to Cartan factors and then proceeding case by case.
scientific article; zbMATH DE number 6915745
scientific article