Epistemology
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Bott periodicity, Lie group embeddings, Linear complex structure, Clifford algebras, Topological invariants, Super division algebras Relate linear complex structures with symmetric spaces. Symmetric Spaces
读物
Symmetric spaces and Bott perioidicity Symmetric spaces A symmetric space has, at each point, an isometry that maps each geodesic to the reflected geodesic. An isometry of a manifold is any (smooth) mapping of that manifold into itself, or into another manifold that preserves the notion of distance between points. The definition of an isometry requires the notion of a metric on the manifold. The isometry expresses that we have all manner of twin choices. Symmetric spaces include some quotients of Lie groups, however, not as groups but as manifolds. Ideas
Examples of Symmetric Spaces {$S$} Grassmannian
Complex structures on {$\mathbb{R}^n$} Quaternionic structures on {$\mathbb{C}^{2m}$} Real structures on {$\mathbb{C}^n$} Complex structures on {$\mathbb{H}^n$} Orthogonal group Compact Lie group Grassmannians: Projection model Grassmannians: Reflection model If {$H↪G$} is an inclusion of Lie groups then the quotient {$G/H$} is called a Klein geometry. |