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| In [[mathematics]], a '''real structure''' on a [[complex number|complex]] [[vector space]] is a way to decompose the complex vector space in the [[direct sum of vector spaces|direct sum]] of two [[real number|real]] vector spaces. The prototype of such a structure is the field of complex numbers itself, considered as a complex vector space over itself and with the conjugation [[function (mathematics)|map]] <math>\sigma: {\mathbb C} \to {\mathbb C}\,</math>, with <math>\sigma (z)={\bar z}</math>, giving the "canonical" '''real structure''' on <math>{\mathbb C}\,</math>, that is <math>{\mathbb C}={\mathbb R}\oplus i{\mathbb R}\,</math>.
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| The conjugation map is [[antilinear]]: <math>\sigma (\lambda z)={\bar \lambda}\sigma(z)\,</math> and <math>\sigma (z_1+z_2)=\sigma(z_1)+\sigma(z_2)\,</math>.
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| ==Vector space==
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| A '''real structure''' on a [[complex vector space]] ''V'' is an [[antilinear]] [[Involution (mathematics)|involution]] <math>\sigma: V \to V</math>. A real structure defines a real subspace <math>V_{\mathbb{R}} \subset V</math>, its fix locus, and the natural map
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| :<math> V_{\mathbb R} \otimes_{\mathbb{R}} {\mathbb C} \to V </math>
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| is an isomorphism. Conversely any vector space that is the [[complexification]]
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| of a real vector space has a natural real structure.
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| One first notes that every complex space ''V'' has a [[real form]] obtained by taking the same vectors as in the original set and restricting the scalars to be real. If <math>t\in V\,</math> and <math>t\neq 0</math> then the vectors <math>t\,</math> and <math>it\,</math> are [[linear independence|linear independent]] in the real form of ''V''. Hence:
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| :<math> \dim_{\mathbb R}V = 2\dim_{\mathbb C}V </math>
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| Naturally, one would wish to represent ''V'' as the direct sum of two real vector spaces, the "real and imaginary parts of ''V''". There is no canonical way of doing this: such a splitting is an additional '''real structure''' in ''V''. It may be introduced as follows.<ref>Budinich, P. and Trautman, A. ''The Spinorial Chessboard''. Spinger-Verlag, 1988, p. 29.</ref> Let <math>\sigma: V \to V\,</math> be an [[antilinear map]] such that <math>\sigma\circ\sigma=id_{V}\,</math>, that is an antilinear involution of the complex space ''V''.
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| Any vector <math>t\in V\,</math> can be written <math>{t = t^{+} + t^{-}}\,</math>,
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| where <math>t^+ ={1\over {2}}(t+\sigma t)</math> and <math>t^- ={1\over {2}}(t-\sigma t)\,</math>.
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| Therefore, one gets a [[direct sum]] of vector spaces <math>V=V^{+}\oplus V^{-}\,</math> where:
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| :<math>V^{+}=\{t\in V | \sigma t = t\}</math> and <math>V^{-}=\{t\in V | \sigma t = -t\}\,</math>. | |
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| Both sets <math>V^+\,</math> and <math>V^-\,</math> are real [[vector space]]s. The linear map <math>K: V^+ \to V^-\,</math>, where <math>K(t)=it\,</math>, is an isomorphism of real vector spaces, whence:
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| :<math> \dim_{\mathbb R}V^+ = \dim_{\mathbb R}V^- = \dim_{\mathbb C}V\,</math>.
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| The first factor <math>V^+\,</math> is also denoted by <math>V_{\mathbb{R}}\,</math> and is left invariant by <math>\sigma\,</math>, that is <math>\sigma(V_{\mathbb{R}})\subset V_{\mathbb{R}}\,</math>. The second factor <math>V^-\,</math> is
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| usually denoted by <math>iV_{\mathbb{R}}\,</math>. The direct sum <math>V=V^{+}\oplus V^{-}\,</math> reads now as:
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| :<math>V=V_{\mathbb{R}} \oplus iV_{\mathbb{R}}\,</math>,
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| i.e. as the direct sum of the "real" <math>V_{\mathbb{R}}\,</math> and "imaginary" <math>iV_{\mathbb{R}}\,</math> parts of ''V''. This construction strongly depends on the choice of an [[antilinear]] [[involution]] of the complex vector space ''V''. The [[complexification]] of the real vector space <math>V_{\mathbb{R}}\,</math>, i.e.,
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| <math>V^{\mathbb{C}}= V_{\mathbb R} \otimes_{\mathbb{R}} \mathbb{C}\,</math> admits
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| a natural '''real structure''' and hence is canonically isomorphic to the direct sum of two copies of <math>V_{\mathbb R}\,</math>:
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| :<math>V_{\mathbb R} \otimes_{\mathbb{R}} \mathbb{C}= V_{\mathbb{R}} \oplus iV_{\mathbb{R}}\,</math>.
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| It follows a natural linear isomorphism <math> V_{\mathbb R} \otimes_{\mathbb{R}} \mathbb{C} \to V\,</math> between complex vector spaces with a given real structure.
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| A '''real structure''' on a complex vector space ''V'', that is an antilinear involution <math>\sigma: V \to V\,</math>, may be equivalently described in terms of the [[linear map]] <math>\hat \sigma:V\to\bar V\,</math> from the vector space <math>V\,</math> to the [[complex conjugate vector space]] <math>\bar V\,</math> defined by
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| :<math>v \mapsto \hat\sigma (v):=\overline{\sigma(v)}\,</math>.<ref>Budinich, P. and Trautman, A. ''The Spinorial Chessboard''. Spinger-Verlag, 1988, p. 29.</ref>
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| ==Algebraic variety==
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| For an [[algebraic variety]] defined over a [[subfield]] of the [[real numbers]],
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| the real structure is the complex conjugation acting on the points of the variety in complex projective or affine space.
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| Its fixed locus are the space of real points of the variety (which may be empty).
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| ==Scheme==
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| For a scheme defined over a subfield of the real numbers, complex conjugation
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| is in a natural way a member of the [[Galois group]] of the [[algebraic closure]] of the basefield.
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| The real structure is the Galois action of this conjugation on the extension of the
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| scheme over the algebraic closure of the base field.
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| The real points are the points whose residue field is fixed (which may be empty).
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| == See also ==
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| *[[Antilinear map]]
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| *[[Linear map]]
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| *[[Canonical complex conjugation map]]
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| *[[Complex conjugate]]
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| *[[Complex conjugation]]
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| *[[Complex conjugate vector space]]
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| *[[Conjugate linear maps]]
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| *[[Complexification]]
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| *[[Linear complex structure]]
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| *[[Sesquilinear form]]
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| *[[Spinor calculus]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * Horn and Johnson, ''Matrix Analysis,'' Cambridge University Press, 1985. ISBN 0-521-38632-2. (antilinear maps are discussed in section 4.6).
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| * Budinich, P. and Trautman, A. ''The Spinorial Chessboard''. Spinger-Verlag, 1988. ISBN 0-387-19078-3. (antilinear maps are discussed in section 3.3).
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| [[Category:Structures on manifolds]]
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