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 VV is an antilinear map σ:VV\sigma \colon V \to V which is an involution.

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

Here W=Eig(σ,1)VW = Eig(\sigma,1) \hookrightarrow V is the eigenspace of σ\sigma for eigenvalue 1 and W{i}=Eig(σ,1)VW \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. See the history of this page for a list of all contributions to it.