On compact realifications of exceptional simple Kantor triple systems
Let \(F\) be a field of characteristic zero and let \(U\) be a finite dimensional vector space over \(F\) endowed with a trilinear map \(B: U\times U\times U \to U\). The pair \((B,U)\) is said to be a triple system. A triple system \((B,U)\) is said to be a Generalized Jordan triple system (GJTS for short) if \([L_{x,y},L_{u,v}]=L_{L_{x,y}(u),v}-L_{u,L_{y,x}(v)}\) for any \(x,y,u,v\in U\), where \(L_{x,y}(z)=B(x,y,z)\). In [Tr. Sem. Vektor. Tenzor. Analizu 16, 407--499 (1972; Zbl 0272.17001)] \textit{I. L. Kantor} associated to any GJTS \((B,U)\) a graded Lie algebra \({\mathcal L}(B)=\bigoplus_{l=-\infty}^\infty U_l\) such that \(U_{-1}=U\). A GJTS \((B,U)\) is said to be a Kantor Triple system (KTS for short) if \({\mathcal L}(B)=\bigoplus_{l=-2}^2 U_l\). B. N. Allison associated to any structurable algebra \((A,-)\) a 5-graded Lie algebra \({\mathcal K}(A)=\bigoplus_{l=-2}^2 K_l\) [Commun. Algebra 7, 1835--1875 (1979; Zbl 0422.17006)] which turns out to coincide with the Kantor's Lie algebra \({\mathcal L}(B_A)\) when we consider the triple system \(B_A(x,y,z)=(x\overline{y})z+(z\overline{y})x-(z\overline{x})y\) for any \(x,y,z\in A\) [see \textit{H. Asano} and \textit{S. Kaneyuki}, Tokyo J. Math. 11, No. 1, 105--118 (1988; Zbl 0665.17003) and Tokyo J. Math. 15, 483--484 (1992; Zbl 0774.17002)]. This paper presents a unified construction to obtain models of all realifications of complex compact exceptional simple KTS and therefore of all realifications of complex exceptional simple Lie algebras which are presented as Kantor Lie algebras of the corresponding KTS. The author first considers \({\mathcal M}(H_3(\mathbb{A^\mathbb{C}}))\) the \(2 \times 2\) matrix algebra defined by the Jordan algebra \(H_3(\mathbb{A^\mathbb{C}})\) of hermitian \(3\times 3\) matrices over a complex division algebra \(\mathbb{A^\mathbb{C}}\in\{ \mathbb{R}^\mathbb{ C }, \mathbb{C}^\mathbb{C},\;\mathbb{H}^\mathbb{C}, \;\mathbb{O}^\mathbb{C}\}\) where \(\mathbb{A^\mathbb{C}}\) denotes any of the division composition algebras \(\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}\) regarded as an algebra over \(\mathbb{C}\). It is showed that over \(\mathbb{C}\) the algebras \((A,-)={\mathcal M}(H_3(\mathbb{A^\mathbb{C}}))\), where \(-\) denotes the standard involution, are simple structurable of skew-dimension one. One of the main results of this paper is that the realifications \({\mathcal M}(H_3(\mathbb{A^\mathbb{C}}))_{\mathbb{R}}\) of the algebra \({\mathcal M}(H_3(\mathbb{A^\mathbb{C}})) \) can be endowed with trilinear maps \(\phi\) in such a way that each \((\phi,{\mathcal M}(H_3(\mathbb{A^\mathbb{C}}))_{\mathbb{R}})\) becomes a compact simple KTS. Then since by a result of \textit{S. Kaneyuki} and \textit{H. Asano} [Nagoya Math. J. 112, 81--115 (1988; Zbl 0699.17021)] to classify all compact simple KTS's it suffices to find one such model for each 5-grading of each real simple Lie algebra, making use of the isomorphism between Kantor's and Allison's constructions and also by a dimensional argument, the author shows that the Kantor algebras corresponding to the compact simple exceptional KTS's \((\phi,{\mathcal M}(H_3(\mathbb{A^\mathbb{C}}))_{\mathbb{R}})\) where \(\mathbb{A^\mathbb{C}}\in\{ \mathbb{R}^\mathbb{C}, \mathbb{C}^\mathbb{C},\;\mathbb{H}^\mathbb{C}, \;\mathbb{O}^\mathbb{C}\}\) are, respectively, \(F^\mathbb{C}_{4,\mathbb{R}}\), \(E^\mathbb{C}_{6,\mathbb{R}}\), \(E^\mathbb{C}_{7,\mathbb{R}}\) and \(E^\mathbb{C}_{8,\mathbb{R}}\). Next, for \(F\in \{\mathbb{R}, \mathbb{R}^\mathbb{C}=\mathbb{C}\}\), the author considers the \(2\times 2\) matrix algebra \({\mathcal M}(F)\) determined by \(F\) [see \textit{K. McCrimmon}, Trans. Am. Math. Soc. 139, 495--510 (1969; Zbl 0175.02703)]. Denoting by \({\mathcal M}(\mathbb{C})_\mathbb{R}\) the realification of \({\mathcal M}(\mathbb{C}) \), trilinear maps can be defined such that \((\phi,{\mathcal M}(\mathbb{C})_\mathbb{R})\) and \((\phi, {\mathcal M}(\mathbb{R}))\) are compact simple KTS's. Finally, by a similar argument as the one used before, it is proved that the Kantor algebras corresponding to \((\phi,{\mathcal M}(\mathbb{C})_\mathbb{R})\) and \((\phi, {\mathcal M}(\mathbb{R}))\) are the exceptional simple Lie algebras \(G^\mathbb{C}_{2,\mathbb{R}}\) and \(G_2\) respectively.
scientific article; zbMATH DE number 5125110