Reality structure

In mathematics, particularly in representation theory, a reality structure on a complex vector space "V" of dimension "n" provides a means for identifying a real subspace "V"_R of "V" so that "V" itself splits as a direct sum into real and imaginary parts: "V" = "V"_R ⊕ "i" "V"_R.

A reality structure is often defined implicitly by a complex antilinear involution "c" : "V" → "V", i.e.,

# "c"² = "Id", (i.e., "c" is an involution).
# "c" is real linear, but complex antilinear:::, for all "z" ∈ C and "v" ∈ "V".

If "c" satisfies these two properties, then the eigenvalues of "c" are ±1. "V"_R is the eigenspace corresponding to the eigenvalue +1, and "i" "V"_R is the eigenspace for the eigenvalue -1. There is a real-linear projection operator into "V"_R given by:$vmapsto\; frac\{1\}\{2\}(v+c(v))$which, by analogy with the real part of a complex number, manifests "c" as a generalized conjugation operator on "V".

ee also

*Linear complex structure
*Complexification