nLab
real structure
Contents
Context
Linear algebra
homotopy theory , (∞,1)-category theory , homotopy type theory

flavors: stable , equivariant , rational , p-adic , proper , geometric , cohesive , directed …

models: topological , simplicial , localic , …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents
Definition
A real structure on a complex vector space $V$ is an antilinear map $\sigma \colon V \to V$ which is an involution .

Equivalently this is a real vector space $W$ and an isomorphism $V \simeq W \otimes_{\mathbb{R}} \mathbb{C}$ of $V$ with its complexification .

Here $W = Eig(\sigma,1) \hookrightarrow V$ is the eigenspace of $\sigma$ for eigenvalue 1 and $W \otimes \{i\} = Eig(\sigma,-1) \hookrightarrow V$ is the eigenspace for eigenvalue -1.

References
More generally in spectral geometry (via spectral triples ) and KR-theory :

Alain Connes , definition 3 of Noncommutative geometry and reality , J. Math. Phys. 36 (11), 1995 (pdf )
Last revised on May 9, 2018 at 12:01:24.
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