Zobel network

In mathematics, a reality structure on a complex vector space V is a decomposition of V into two real subspaces, called the real and imaginary parts of V:

$V=V_{\mathbb{ℝ}}\oplus i{V}_{\mathbb{ℝ}}.$

Here V_R is a real subspace of V, i.e. a subspace of V considered as a vector space over the real numbers. If V has complex dimension n (real dimension 2n), then V_R must have real dimension n.

The standard reality structure on the vector space ${\mathbb{ℂ}}^n$ is the decomposition

${\mathbb{ℂ}}^n={\mathbb{ℝ}}^n\oplus i{\mathbb{ℝ}}^n.$

In the presence of a reality structure, every vector in V has a real part and an imaginary part, each of which is a vector in V_R:

$v=\mathrm{\Re }\{v\}+i\mathrm{\Im }\{v\}$

In this case, the complex conjugate of a vector v is defined as follows:

$\bar{v}=\mathrm{\Re }\{v\}-i\mathrm{\Im }\{v\}$

This map $v\mapsto \bar{v}$ is an antilinear involution, i.e.

$\overline{\stackrel{‾}{v}}=v,\overline{v+w}=\bar{v}+\bar{w},\text{and}\overline{\alpha v}=\bar{\alpha }\bar{v}.$

Conversely, given an antilinear involution $v\mapsto c(v)$ on a complex vector space V, it is possible to define a reality structure on V as follows. Let

$\mathrm{\Re }\{v\}=\frac{1}{2}(v+c(v)),$

and define

$V_{\mathbb{ℝ}}=\{\mathrm{\Re }\{v\}\mid v\in V\}.$

Then

$V=V_{\mathbb{ℝ}}\oplus i{V}_{\mathbb{ℝ}}.$

This is actually the decomposition of V as the eigenspaces of the real linear operator c. The eigenvalues of c are +1 and −1, with eigenspaces V_R and $i$ V_R, respectively. Typically, the operator c itself, rather than the eigenspace decomposition it entails, is referred to as the reality structure on V.

See also

• Linear complex structure
• Complexification

References

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