nLab real structure

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.

Related concepts

• complex structure

• quaternionic structure

• real space

• real oriented cohomology theory

  • KR-theory

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)