Contents

# Contents

## 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.

## Examples

###### Example

(real structures on the complex line)
The standard real structure on the complex line $\mathbb{C}$ is complex conjugation $\overline{x + \mathrm{i} y} \,\equiv\, x - \mathrm{i}y$:

$\array{ \mathbb{C} &\longrightarrow& \mathbb{C} \\ z &\mapsto& \overline{z} \mathrlap{\,.} }$

The fixed locus of this real structure are the real numbers under their defining embedding into the complex numbers:

$\mathbb{C}^{\overline{(\text{-})}} \;\simeq\; \mathbb{R} \hookrightarrow \mathbb{R} \oplus \mathrm{i} \mathbb{R} \,\equiv\, \mathbb{C}$

with

$\array{ \mathbb{R} \otimes_{{}_{\mathbb{R}}} \mathbb{C} &\xrightarrow{ \;\sim\; }& \mathbb{C} \\ (x', x + \mathrm{i} y) &\mapsto& x' x + \mathrm{i} x' y \mathrlap{\,.} }$

Restricting the canonical complex-bilinear form

(1)$\array{ \mathbb{C} \otimes_{\mathbb{C}} \mathbb{C} &\longrightarrow& \mathbb{C} \\ (z_1, z_2) &\mapsto& z_1 z_2 }$

along this inclusion yields the canonical $\mathbb{R}$-bilinear form on $\mathbb{R}$:

$\array{ \mathbb{R} \otimes_{\mathbb{R}} \mathbb{R} &\hookrightarrow& \mathbb{C} \otimes_{\mathbb{C}} \mathbb{C} &\longrightarrow& \mathbb{C} \\ (x_1, x_2) &\mapsto& (x_1, x_2) &\mapsto& x_1 x_2 \mathrlap{\,.} }$

Another real structure on $\mathbb{C}$ is:

$\array{ \mathbb{C} &\longrightarrow& \mathbb{C} \\ z &\mapsto& -\overline{z} \mathrlap{\,.} }$

Now the fixed locus is the imaginary numbers $\mathrm{i} \mathbb{R}$

$\mathbb{C}^{-\overline{(\text{-})}} \;\simeq\; \mathrm{i}\mathbb{R} \hookrightarrow \mathbb{R} \oplus \mathrm{i} \mathbb{R} \,\equiv\, \mathbb{C}$

with

$\array{ \mathrm{i}\mathbb{R} \otimes_{{}_{\mathbb{R}}} \mathbb{C} &\xrightarrow{ \;\sim\; }& \mathbb{C} \\ (\mathrm{i} x', x + \mathrm{i} y) &\mapsto& - x' y + \mathrm{i} x' x \mathrlap{\,.} }$

Restricting the canonical complex bilinear form (1) along this inclusion now yields minus the canonical real bilinear form

$\array{ \mathbb{R} \otimes_{\mathbb{R}} \mathbb{R} &\hookrightarrow& \mathbb{C} \otimes_{\mathbb{C}} \mathbb{C} &\longrightarrow& \mathbb{C} \\ (x_1 ,\, x_2) &\mapsto& (\mathrm{i} x_1 ,\, \mathrm{i}x_2) &\mapsto& - x_1 x_2 \mathrlap{\,.} }$

###### Example

(real structures on the complex plane)
On the complex plane $\mathbb{C}^2 \,\equiv\, \mathbb{C} \oplus \mathbb{C}$ we have first of all the real structures inherited from $\mathbb{C}$ (Ex. ).

$\array{ \mathbb{C}^2 &\longrightarrow& \mathbb{C}^2 \\ (z_1, z_2) &\mapsto& (\overline{z_1}, \overline{z_2}) }$

and

$\array{ \mathbb{C}^2 &\longrightarrow& \mathbb{C}^2 \\ (z_1, z_2) &\mapsto& -(\overline{z_1}, \overline{z_2}) \,. }$

In addition, there is for instance

$\array{ \mathbb{C}^2 &\longrightarrow& \mathbb{C}^2 \\ \left[ \array{ z_1 \\ z_2 } \right] &\mapsto& \left[ \array{ +\overline{z_1} \\ -\overline{z_2} } \right] \,. }$

whose fixed locus is

$\array{ \mathbb{R}^2 &\hookrightarrow& \mathbb{C}^2 \\ \left[ \array{ x_1 \\ x_2 } \right] &\mapsto& \left[ \array{ x_1 \\ \mathrm{i} x_2 } \right] \mathrlap{\,,} }$

the restriction to which of the canonical complex bilinear form on the complex plane

(2)$\array{ \mathbb{C}^2 \otimes_{\mathbb{C}} \mathbb{C}^2 &\longrightarrow& \mathbb{C} \\ \left( \left[ \array{ z_1 \\ z_2 } \right] ,\, \left[ \array{ z'_1 \\ z'_2 } \right] \right) &\mapsto& z_1 z'_1 + z_2 z'_2 }$

is the hyperbolic form on the real plane:

$\array{ \mathbb{R}^2 \otimes_{\mathbb{R}} \mathbb{R}^2 &\hookrightarrow& \mathbb{C}^2 \otimes_{\mathbb{C}} \mathbb{C}^2 &\longrightarrow& \mathbb{C} \\ \left( \left[ \array{ x_1 \\ x_2 } \right] ,\, \left[ \array{ x'_1 \\ x'_2 } \right] \right) &\mapsto& \left( \left[ \array{ x_1 \\ \mathrm{i}x_2 } \right] ,\, \left[ \array{ x'_1 \\ \mathrm{i} x'_2 } \right] \right) &\mapsto& x_1 x'_1 - x_2 x'_2 \mathrlap{\,.} }$

Yet another real structure is

$\array{ \mathbb{C}^2 &\longrightarrow& \mathbb{C}^2 \\ \left[ \array{ z_1 \\ z_2 } \right] &\mapsto& \left[ \array{ \overline{z_2} \\ \overline{z_1} } \right] \,. }$

whose fixed locus is

$\array{ \mathbb{R}^2 &\hookrightarrow& \mathbb{C}^2 \\ \left[ \array{ x_1 \\ x_2 } \right] &\mapsto& \left[ \array{ x_1 + \mathrm{i} x_2 \\ x_1 - \mathrm{i} x_2 } \right] \,, }$

the restriction to which of the canonical complex bilinear form (2) is twice the hyperbolic form on $\mathbb{R}^2$

$\array{ \mathbb{R}^2 \otimes_{\mathbb{R}} \mathbb{R}^2 &\hookrightarrow& \mathbb{C}^2 \otimes_{\mathbb{C}} \mathbb{C}^2 &\longrightarrow& \mathbb{C} \\ \left( \left[ \array{ x_1 \\ x_2 } \right] ,\, \left[ \array{ x'_1 \\ x'_2 } \right] \right) &\mapsto& \left( \left[ \array{ x_1 + \mathrm{i}x_2 \\ x_1 - \mathrm{i}x_2 } \right] ,\, \left[ \array{ x'_1 + \mathrm{i}x'_2 \\ x'_1 - \mathrm{i}x'_2 } \right] \right) &\mapsto& 2 x_1 x'_1 - 2 x_2 x'_2 \mathrlap{\,.} }$

###### Example

(real structure as dagger-self-dual Hermitian structure)
Let $\mathscr{V}$ be a finite-dimensional complex vector space equipped with both

1. a non-degenerate sesquilinear (Hermitian) inner product ${\langle - \vert - \rangle}$,

2. a non-degenerate symmetric bilinear inner product ${(-\vert-)}$

such that

• the Hermitian-adjoint ${(-\vert-)}^\dagger$ of the bilinear pairing is the coevaluation map that exhibits $\mathscr{V}$ as a self-dual object,

then this induces a real structure on $\mathscr{V}$.

Moreover, for $\mathscr{W}$ another finite-dimensional complex vector space equipped with a compatible pair of such structures, then the linear maps

$\mathscr{V} \longrightarrow \mathscr{W}$

which preserve both structures also preserve that real structure, hence come from $\mathbb{R}$-linear maps of underlying real vector spaces.

###### Proof

1. the non-degenerate sesquilinear form ${\langle -\vert -\rangle}$ is equivalently given by an antilinear map
$a \colon \mathscr{V} \to \mathscr{V}^\ast$ to the dual linear space, via

${\langle - \vert -\rangle} \,=\, a(-)(-)$
2. the non-degenerate bilinear form ${(- \vert -)}$ is equivalently given by a complex linear map $l \,\colon\, \mathscr{V} \to \mathscr{V}^\ast$, via

${(- \vert -)} \,=\, l(-)(-)$

(where the second argument means evaluation).

In bra-ket notation this is suggestively written as:

${( v \vert} \;\;\coloneqq\;\; l {\vert v )} \,\;\;\;\;\; {\langle w \vert} \;\;\coloneqq\;\; a {\vert w \rangle} \,.$

For example, the Hermitian adjoint $f^\dagger \,\colon\, W \to \mathscr{V}$ of any linear map $f \,\colon\, \mathscr{V} \to \mathscr{W}$

is

$f^\dagger {\vert w \rangle} \;\coloneqq\; a^{-1}\big( {\langle w \vert} f\big) \;=\; a^{-1} \circ f^\ast \circ a {\vert w \rangle} \,.$

Moreover, if $\big\{ {\vert w \rangle} \big\}_{w \in W}$ is any orthonormal basis of $\mathscr{V}$ with respect to $\langle - \vert - \rangle$, then the coevaluation map which exhibits $\mathscr{V}^\ast \,\coloneqq\, Map_{\mathbb{C}}(\mathscr{V}, \mathbb{C})$ as the linear dual space to $\mathscr{V}$ may be written as

and hence equivalently as
which we use at the very end below.

Now regarding the concrete proof:

The composite

$\tau \;\coloneqq\; a^{-1} \circ l \;\colon\; \mathscr{V} \to \mathscr{V} \,,$

is an antilinear endomorphism of $\mathscr{V}$, hence it will be sufficient to show that it is an involution (and hence the sought-after real structure). For that, in turn, it is clearly sufficient that

(3)$a^{-1} \circ l \,=\, l^{-1} \circ a \,.$

because then

$\tau \circ \tau \;=\; \big( a^{-1} \circ l \big) \circ \big( a^{-1} \circ l \big) \;=\; a^{-1} \circ l \circ l^{-1} \circ a \;=\; id_{\mathscr{V}} \,.$

We now show that this condition (3) is equivalent to the assumption that $(-\vert-)^\dagger$ is the coevaluation map.

Namely, with

we have

which by the zig-zag identity for $ev$ & $coev$ equals

On the other hand, the coevaluation which exhibits $(-\vert-)$ as a self-duality is clearly

which by the above discussion equals

This being equal to $(-\vert-)^\dagger$ above is clearly equivalent to (3).

## 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 November 9, 2023 at 06:06:48. See the history of this page for a list of all contributions to it.