A uniform description of compact symmetric spaces as Grassmannians using the magic square
The authors derive a representation for symmetric spaces of the compact type which extends the ``magic square'' representation of compact semisimple Lie groups due to Freudenthal and Tits. In the classical magic square, the Lie algebras of compact semisimple Lie groups are described by means of \((\mathbb A\otimes\mathbb B)^n\), where \(\mathbb A\) and \(\mathbb B\) are normed division algebras. The authors use the notation \(\{ P \subset V \}\) to denote the Grassmannian of planes of type \(P\) (given by dimension and certain metric conditions) in the vector space \(V\). For example, \(\{\mathbb R^k\subset\mathbb R^n\}\) denotes the Grassmannian \(O(n)/O(k)\times O(n-k)\) of \(k\)-planes in \(\mathbb R^n\), and \(\{\mathbb R^n\subset\mathbb C^n\}\) denotes the Grassmannian \(U(n)/O(n)\) of real Lagrangian subspaces in \(\mathbb C^n\). For a division algebra \(\mathbb A=\mathbb C,\mathbb H,\mathbb O\) let \(\frac{\mathbb A}{2}\) denote \(\mathbb R,\mathbb C,\mathbb H\), respectively. With this notation the main result of the paper is a description of symmetric spaces of compact type as one of the following types of Grassmannians: {\parindent=0.7cm\begin{itemize}\item[(I)]Isotropic Grassmannians: \(\{(\mathbb A\otimes\mathbb B)^n\subset (\mathbb A\otimes\mathbb B)^n\oplus(\mathbb A\otimes\mathbb B)^n\}^{\sigma}\), where \(\sigma\) is an involution for an indefinite symmetric bilinear form of neutral signature.\item[(II)] Grassmannians: \(\{(\mathbb A\otimes\mathbb B)^k\subset (\mathbb A\otimes\mathbb B)^n\}\).\item[(III)] Lagrangian Grassmannians: \(\{(\frac{\mathbb A}{2}\otimes\mathbb B)^n\subset(\mathbb A\otimes\mathbb B)^n\}\). \item[(IV)] Double Lagrangian Grassmannians: \(\{(\frac{\mathbb A}{2}\otimes\frac{\mathbb B}{2})^n\oplus(j_1\frac{\mathbb A}{2}\otimes j_2\frac{\mathbb B}{2})^n \subset(\mathbb A\otimes\mathbb B)^n\}\), where \(j_1\), \(j_2\) are one of \(i\in\mathbb C\), \(j\in\mathbb H\) or \(e^3\in\mathbb O\). \end{itemize}}
scientific article; zbMATH DE number 5898900