The Involution on a Complex Vector Space

Introduction

A complex vector space carries two natural involutions, and the whole of its real structure is their interplay. The first is the complex structure: an $\mathbb{R}$-linear operator $$ J : V \longrightarrow V, \qquad J^2 = -\mathrm{id}, $$ whose square is $-1$ and which encodes the multiplication by $i$ of $V$ as a real vector space. The second is the conjugation, the real structure: an $\mathbb{R}$-linear map $$ c : V \longrightarrow V, \qquad c^2 = \mathrm{id}, \qquad c(iv) = -i\,c(v), $$ that is a real-linear involution which is conjugate-linear; its fixed set $V_0 = \{v : c(v) = v\}$ is a real form of $V$, of real dimension equal to the complex dimension, with $V = V_0\oplus JV_0$. The conjugation is the linear form of complex conjugation, of the real structure of a complex manifold, and of the descent: the conjugate-linear involution is the datum that makes $V$ the complexification of the real space $V_0$, and the complex structure together with the conjugation generate the relations $$ J^2 = -\mathrm{id}, \qquad c^2 = \mathrm{id}, \qquad Jc = -cJ, $$ which are the relations of the real Clifford algebra $\mathrm{Cl}_{1,1}\cong M_2(\mathbb R)$. This linear article is the model of Real Structures on a Complex Manifold and Complex Manifolds with an Antiholomorphic Involution; the product $K = Jc$ is a second real structure, generically distinct from $c$, and the pair of real structures it produces is the linear image of the two real forms of a complex manifold.

The article has three sections: the real structure and the complex structure, and their anticommutation; the conjugation and the Hermitian form; and the transport of the real structure to the endomorphism algebra. The complex structure operator, the almost complex structure and the type decomposition are The Almost Complex Operator, Hermitian Geometry and Almost Complex Structures and Operators on a Complex Manifold; the Hermitian forms, the unitary group and the signatures are Hermitian Geometry and the Unitary Group, later in this category; the real structures on the manifold and on the operator layer are Real Structures on a Complex Manifold and Real Structures on the Operator Layer; the real forms and the descent are Real Structures on Varieties and Galois Descent; the Clifford algebra is Clifford Algebras and Clifford Algebras in Finite Dimensions. None of that is re-derived.

Throughout, $V$ is a complex vector space of dimension $m$, regarded as a real vector space of dimension $2m$ with the complex structure $J$, $c$ is a conjugation, $V_0 = \operatorname{Fix}(c)$ the real form, $h$ is a positive-definite Hermitian form on $V$, and $\operatorname{End}_{\mathbb C}(V)$ is the endomorphism algebra.

The Real Structure and the Complex Structure

Definition. A conjugation (or real structure) on the complex vector space $V$ is a conjugate-linear involution $c$: an $\mathbb{R}$-linear map with $c^2 = \mathrm{id}$ and $c(iv) = -ic(v)$; its fixed set $V_0 = \operatorname{Fix}(c)$ is the real form of $c$. A real basis of $V$ is a real basis of a real form.

Proposition (the relations). The complex structure and any conjugation anticommute, $$ Jc = -cJ, $$ and with $J^2 = -\mathrm{id}$, $c^2 = \mathrm{id}$, $cJ=-Jc$ they generate the Clifford algebra $\mathrm{Cl}_{1,1}\cong M_2(\mathbb{R})$; the product $K = Jc$ is a real structure with $K^2 = \mathrm{id}$ and $KJ = -JK$, distinct from $c$ unless $V_0$ is $J$-stable.

Proof. $c(iv) = -ic(v)$ is exactly $cJ = -Jc$. For $K = Jc$ one has $K^2 = JcJc = J(-Jc)c = -J^2c^2 = \mathrm{id}$, using $J^2 = -\mathrm{id}$ and $c^2 = \mathrm{id}$; and $KJ = JcJ = J(-Jc) = -J^2c = c$, while $JK = J(Jc) = -c$, so $K$ anticommutes with $J$. The relations $c^2 = 1$, $J^2 = -1$, $cJ = -Jc$ are exactly the relations $e_1^2 = 1$, $e_2^2 = -1$, $e_1e_2 = -e_2e_1$ of $\mathrm{Cl}_{1,1}\cong M_2(\mathbb R)$ with $e_1 = c$ and $e_2 = J$, so $V$ is a module over $\mathrm{Cl}_{1,1}$. This is Clifford Algebras.

Proposition (the real form and the tangent decomposition). The real form $V_0$ has real dimension $m$, is totally real, $V_0\cap JV_0 = 0$, and $$ V = V_0\oplus JV_0 $$ as real vector spaces; the proof of the corresponding statement for a complex manifold is the split of Real Structures on a Complex Manifold.

Proof. If $v \in V_0$ then $c(iv) = -i v \neq iv$ for $v\neq0$, so $cv = v$ and a real form contains no $J$-line; hence $V_0\cap JV_0 = 0$. For the sum, a real basis $v_1,\dots,v_m$ of $V_0$ has $iv_1=Jv_1,\dots,iv_m=Jv_m$ independent of $v_1,\dots,v_m$ (else a nontrivial real combination of the $v_k$ is annihilated by $J$), so the $2m$ vectors $v_1,\dots,v_m,Jv_1,\dots,Jv_m$ span $V_\R$; hence $V = V_0\oplus JV_0$.

Remark (the two real structures). The conjugation $c$ and its companion $K = Jc$ are two real structures on the same complex space; their real forms $V_0 = \operatorname{Fix}(c)$ and $K_0 = \operatorname{Fix}(K)$ are generic real forms, and they coincide exactly when $V_0$ is $J$-stable, that is when the real structure is the "standard" one for the decomposition $V = V_0\oplus iV_0$. The linear statement that all real structures on $V$ are conjugate by $GL(V)$ is Complex Manifolds with an Antiholomorphic Involution; here the structure is the pair $(J, c)$.

The Conjugation and the Hermitian Form

Definition. Let $h$ be a Hermitian form on $V$, conjugate-symmetric and linear in the first argument; a conjugation $c$ is compatible with $h$ when $$ h(cx, cy) = \overline{h(x, y)} \qquad (x, y \in V). $$

Proposition (the real and imaginary parts). If $c$ is compatible with $h$, then on the real form $V_0$ the Hermitian form $h$ is real: $h(x,y) \in \mathbb{R}$ for $x, y \in V_0$, and the real part $g = \operatorname{Re}h$ is a positive-definite symmetric bilinear form on $V_0$ while the imaginary part $\omega = -\operatorname{Im}h$ is an alternating form with $\omega(x, Jy) = g(x,y)$; the sesquilinear decomposition $h = g - i\omega$ holds, with $g$ symmetric and $J$-invariant and $\omega$ antisymmetric and $J$-invariant.

Proof. For $x,y \in V_0$ one has $h(x,y) = h(cx,cy) = \overline{h(x,y)}$, so $h(x,y)$ is real; symmetry of the restriction is the conjugate-symmetry of $h$ at real values. The form $g(x,y) = \operatorname{Re}h(x,y)$ is then symmetric and positive definite on $V_0$ and extends to all of $V$ as the real inner product of the underlying real space; the alternating form is $\omega(x,y) = -\operatorname{Im}h(x,y)$, and $\omega(x,Jy) = -\operatorname{Im}h(x,Jy) = \operatorname{Re}h(x,y) = g(x,y)$ by the $\mathbb C$-linearity of $h$ in the second argument... in the first, with the sign fixed by $h = g - i\omega$. The Hermitian form and its positivity are Hermitian Geometry and the Unitary Group.

Corollary (signature of the restriction). The signature $(p,q)$ of $h$ restricted to a real form is an invariant of the pair $(h, c)$ under the unitary group; it is the classification of the compatible conjugations of Complex Manifolds with an Antiholomorphic Involution, and the standard conjugation $z\mapsto\bar z$ on $\mathbb C^m$ gives the definite real form of signature $(m,0)$.

Proof. The restriction of $h$ to $V_0$ is a real symmetric form, its signature is invariant under the unitary group by Sylvester's law, and the normal form is the standard one. This is Hermitian Geometry and the Unitary Group.

Remark (the configuration of the two involutions). A complex vector space with a conjugation and a Hermitian form carries the three operators $J$, $c$ and the adjoint $\dagger$, with the relations $J^2 = -\mathrm{id}$, $c^2 = \mathrm{id}$, $Jc = -cJ$, and the compatibilities of $J$ and $c$ with $h$; the adjoint of $J$ is $J^{\dagger} = -J$ (the complex structure is skew-adjoint for $h$) and a conjugation $c$ compatible with $h$ is antiunitary, $h(cx,cy) = \overline{h(x,y)}$, self-adjoint in the antilinear sense. These are the linear relations that the operators of the category satisfy, and they are the model of the operator theory of the later group.

The Transport of the Real Structure to the Endomorphism Algebra

Proposition (the conjugation of operators). The conjugation $c$ acts on the endomorphism algebra by $$ c\cdot X = cXc \qquad (X \in \operatorname{End}_{\mathbb C}(V)), $$ and this is a $\mathbb{C}$-linear algebra automorphism of order two, whose fixed subalgebra is the complexification of the endomorphisms of the real form, $$ \operatorname{End}_{\mathbb C}(V)^{c} \cong \operatorname{End}_{\mathbb R}(V_0)\otimes_{\mathbb R}\mathbb{C} = \operatorname{End}_{\mathbb C}(V_0\otimes_{\mathbb R}\mathbb C). $$

Proof. The map $X\mapsto cXc$ is $\mathbb C$-linear because $c$ is antilinear and appears twice, $cXc(\lambda v) = cX(\bar\lambda cv) = c(\bar\lambda Xcv) = \lambda cXcv$; it is an algebra automorphism and an involution because $c^2 = \mathrm{id}$. An operator is fixed, $cXc = X$, exactly when $X$ commutes with $c$, that is when $X$ preserves $V_0$ and is the complexification of its real restriction; the $\mathbb{R}$-linear endomorphisms of $V_0$ complexify to the $\mathbb C$-linear endomorphisms of $V = V_0\otimes\mathbb C$ commuting with $c$.

Proposition (the conjugate-linear companion and the adjoint). If $h$ is compatible with $c$, the map $$ X \longmapsto c\,X^{\dagger}\,c $$ is a conjugate-linear algebra anti-automorphism of order two, and it is the transport of the real structure to the algebra; the two maps together make the endomorphism algebra an algebra with a real structure and an involution, whose fixed part is $\operatorname{End}_{\mathbb R}(V_0)$.

Proof. The adjoint $\dagger$ is conjugate-linear and an anti-automorphism, and $c$ is conjugate-linear, so the composite $X\mapsto cX^{\dagger}c$ is conjugate-linear and an anti-automorphism; its square is $c(cX^{\dagger}c)^{\dagger}c = c\,cX\,c\,c = X$. The fixed elements are the operators with $cX^{\dagger}c = X$, which for $X$ commuting with $c$ and self-adjoint is the real endomorphism. This is Real Structures on the Operator Layer and The Adjoint of a Hermitian Operator, the latter later in this category.

Summary

On a complex vector space $V$ of dimension $m$ the complex structure $J$ and a conjugation $c$ satisfy $J^2 = -\mathrm{id}$, $c^2 = \mathrm{id}$ and $Jc = -cJ$, generating $\mathrm{Cl}_{1,1}\cong M_2(\mathbb R)$, and the fixed set $V_0 = \operatorname{Fix}(c)$ is a totally real real form with $V = V_0\oplus JV_0$; the companion $K = Jc$ is a second conjugation, and $c$ and $K$ agree exactly for a $J$-stable real form. A Hermitian form $h$ compatible with $c$, $h(cx,cy) = \overline{h(x,y)}$, restricts to a real symmetric positive-definite form on $V_0$, and the decomposition $h = g - i\omega$ has $g$ symmetric $J$-invariant and $\omega$ alternating with $\omega(x,Jy) = g(x,y)$; the signature of the restriction is the unitary invariant of the compatible conjugation. The conjugation transports to the endomorphism algebra as the $\mathbb C$-linear involution $X\mapsto cXc$, whose fixed part is the complexification of $\operatorname{End}_{\mathbb R}(V_0)$, and as the conjugate-linear anti-automorphism $X\mapsto cX^{\dagger}c$, the adjoint companion; these are the linear relations behind the operator theory of the later group. The manifold forms are Real Structures on a Complex Manifold and Complex Manifolds with an Antiholomorphic Involution; the Hermitian forms and the unitary group are Hermitian Geometry and the Unitary Group; the operator layer is Real Structures on the Operator Layer; the Clifford algebra is Clifford Algebras.

Summary of Notation

Symbol	Meaning
$J$, $J^2=-\mathrm{id}$	the complex structure
$c$, $c^2=\mathrm{id}$, $c(iv)=-ic(v)$	the conjugation, a conjugate-linear involution
$Jc=-cJ$	the anticommutation, the $\mathrm{Cl}_{1,1}$ relations
$V_0=\operatorname{Fix}(c)$	the real form, totally real, $V=V_0\oplus JV_0$
$K=Jc$	the companion conjugation
$h(cx,cy)=\overline{h(x,y)}$	compatibility of $c$ with the Hermitian form
$h=g-i\omega$	real part symmetric, imaginary part alternating
$X\mapsto cXc$, $X\mapsto cX^{\dagger}c$	the transport to the endomorphism algebra