Reality structure

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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.,

1. c² = Id, (i.e., c is an involution).
2. 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

which, by analogy with the real part of a complex number, manifests c as a generalized conjugation operator on V.

[edit] See also

• Linear complex structure
• Complexification