nLab real structure



Linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

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Basic facts




A real structure on a complex vector space VV is an antilinear map σ:VV\sigma \colon V \to V which is an involution.

Equivalently this is a real vector space WW and an isomorphism VW V \simeq W \otimes_{\mathbb{R}} \mathbb{C} of VV with its complexification.

Here W=Eig(σ,1)VW = Eig(\sigma,1) \hookrightarrow V is the eigenspace of σ\sigma for eigenvalue 1 and W{i}=Eig(σ,1)VW \otimes \{i\} = Eig(\sigma,-1) \hookrightarrow V is the eigenspace for eigenvalue -1.



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

z z¯. \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:

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


(x,x+iy) xx+ixy. \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) (z 1,z 2) z 1z 2 \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}:

(x 1,x 2) (x 1,x 2) x 1x 2. \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:

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

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

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


i (ix,x+iy) xy+ixx. \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

(x 1,x 2) (ix 1,ix 2) x 1x 2. \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{\,.} }


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

2 2 (z 1,z 2) (z 1¯,z 2¯) \array{ \mathbb{C}^2 &\longrightarrow& \mathbb{C}^2 \\ (z_1, z_2) &\mapsto& (\overline{z_1}, \overline{z_2}) }


2 2 (z 1,z 2) (z 1¯,z 2¯). \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

2 2 [z 1 z 2] [+z 1¯ z 2¯]. \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

2 2 [x 1 x 2] [x 1 ix 2], \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) 2 2 ([z 1 z 2],[z 1 z 2]) z 1z 1+z 2z 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:

2 2 2 2 ([x 1 x 2],[x 1 x 2]) ([x 1 ix 2],[x 1 ix 2]) x 1x 1x 2x 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 } \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

2 2 [z 1 z 2] [z 2¯ z 1¯]. \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

2 2 [x 1 x 2] [x 1+ix 2 x 1ix 2], \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 2\mathbb{R}^2

2 2 2 2 ([x 1 x 2],[x 1 x 2]) ([x 1+ix 2 x 1ix 2],[x 1+ix 2 x 1ix 2]) 2x 1x 12x 2x 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{\,.} }


(real structure as dagger-self-dual Hermitian structure)
Let 𝒱\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

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

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

𝒱𝒲 \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.


To start with, we note some generalities:

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

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

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

(where the second argument means evaluation).

In bra-ket notation this is suggestively written as:

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

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


f |wa 1(w|f)=a 1f *a|w. 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 {|w} wW\big\{ {\vert w \rangle} \big\}_{w \in W} is any orthonormal basis of 𝒱\mathscr{V} with respect to |\langle - \vert - \rangle, then the coevaluation map which exhibits 𝒱 *Map (𝒱,)\mathscr{V}^\ast \,\coloneqq\, Map_{\mathbb{C}}(\mathscr{V}, \mathbb{C}) as the linear dual space to 𝒱\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

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

is an antilinear endomorphism of 𝒱\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 1l=l 1a. a^{-1} \circ l \,=\, l^{-1} \circ a \,.

because then

ττ=(a 1l)(a 1l)=a 1ll 1a=id 𝒱. \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 evev & coevcoev 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).


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.