Magic Square and
Classification of Symmetric Spaces
Mr. Yongdong
Huang
Department of Mathematics
The
Abstract
In
this article we introduce and study the (i) Grassmannian, (ii) Lagrangian Grassmannian and (iii) double Lagrangian
Grassmannian of subspaces in (A Ä B)n, where A and B are normed division algebras, that is R, C, H or O.
We
show that every compact symmetric space X
must be one of these Grassmannian spaces (up to a
finite cover) or a compact Lie group. Furthermore, its noncompact
dual symmetric space is the open submanifold of X consisting of spacelike
linear subspaces.
This
gives a simple and uniform description of all symmetric spaces. This is similar
to Tits magic square description for simple Lie algebras.
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Room 517, Meng Wah Complex, HKU |
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