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Symmetric Spaces

读物

• J.-H. Eschenburg. Lecture Notes on Symmetric Spaces This includes discussion of many examples, notably the complex structures on {$\mathbb{R}^n$}.
• Tobias Strubel. Basics on Hermitian Symmetric Spaces.
• Lizhen Ji. From symmetric spaces to buildings, curve complexes and outer spaces
• 11. Homogeneous and Symmetric Spaces
• Symplectic structure
• Generalized complex structure Generalizes complex structures and symplectic structures by considering their direct sum.
• Math Overflow. Examples of quotients of Lie group actions.
• Quotient space

Symmetric spaces and Bott perioidicity

• John Baez. Symmetric Spaces and the Tenfold Way. (Slides)
  • Symmetric Spaces and the Tenfold Way

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.

A symmetric space has, at each point, a (global) isometry that (locally) inverts each tangent vector. Compare that with symmetry where we reflect across the center of a space.

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

• Symmetric spaces - inversion is the symmetry of choice - like the symmetry in the binomial theorem.
• Symmetric spaces - those that undo any perturbation, thus make for stability.

Examples of Symmetric Spaces {$S$}

Grassmannian

• Grassmannian can be thought as defined on center at x as it moves in a manifold and yields tangent spaces.
• Freudenthal magic square organizes not just Lie groups but also symmetric spaces. Yong Dong Huang. Thesis.

Complex structures on {$\mathbb{R}^n$}

Linear complex structure

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

Klein geometry

If {$H↪G$} is an inclusion of Lie groups then the quotient {$G/H$} is called a Klein geometry.

Grassmannians

• Amplituhedron
• Lauren Williams - Combinatorics of the amplituhedron

Grassmannian G(4,2) has five Plucker coordinates (4 choose 2) minus 1. Because it is up to scaling, which gives an extra dimension. Four points on a circle - if two chords cross, they can be untangled in two ways. Cluster algebra and "mutations" by which one product is the sum of two products. Ptolemy theorem.

Classifying space

• Classifying space for U(n) is the complex Grassmannian.
• Classifying space for O(n) is the real Grassmannian.