Hermitian form

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+-- {: .toc .clickDown tabindex="0"}
### Context
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
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#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Definition

+-- {: .num_defn #HermitianForm}
###### Definition
**([[Hermitian form]] and [[Hermitian space]])**

Let $V$ be a [[real vector space]] equipped with a [[complex structure]] $J\colon V \to V$. Then a _[[Hermitian form]]_ on $V$ is

* a complex-valued real-[[bilinear form]]

  $$
    h \;\colon\; V \otimes V \longrightarrow \mathbb{C}
  $$

such that this is _symmetric sesquilinear_, in that:

1. $h$ is complex-linear in the first argument;

1. $h(w,v) = \left(h(v,w) \right)^\ast$ for all $v,w \in V$

where $(-)^\ast$ denotes [[complex conjugation]].

A Hermitian form is _positive definite_ (often assumed by default) if for all $v \in V$

1. $h(v,v) \geq 0$

1. $h(v,v) = 0 \phantom{AA} \Leftrightarrow \phantom{AA} v = 0$.

A [[complex vector space]] $(V,J)$ equipped with a (positive definite) Hermitian form $h$ is called a (positive definite) _[[Hermitian space]]_.

=--

\begin{remark}
  A positive-definite and complete Hermitian vector space is called a *[[Hilbert space]]*.
\end{remark}

## Properties

### General properties

+-- {: .num_prop #BasicPropertiesOfHermitianForms}
###### Proposition
**(basic properties of [[Hermitian forms]])**

Let $((V,J),h)$ be a positive definite [[Hermitian space]] (def. \ref{HermitianForm}). Then:

1. the [[real part]] of the [[Hermitian form]]

   $$
     g(-,-) \;\coloneqq\; Re(h(-,-))
   $$

   is a [[Riemannian metric]], hence a symmetric positive-definite real-[[bilinear form]]

   $$
     g \;\colon\; V \otimes V \to \mathbb{R}
   $$

1. the [[imaginary part]] of the [[Hermitian form]]

   $$
     \omega(-,-) \;\coloneqq\; -Im(h(-,-))
   $$

   is a [[symplectic form]], hence a non-degenerate skew-symmetric real-[[bilinear form]]

   $$
     \omega \;\colon\; V \wedge V \to \mathbb{R}
     \,.
   $$

hence

$$
  h = g - i \omega
  \,.
$$

The two components are related by

$$
  \label{RelationBetweennOmegaAndgOnHermitianSpace}
  \omega(v,w)
  \;=\;
  g(J(v),w)
  \phantom{AAAAA}
  g(v,w)
  \;=\;
  \omega(v,J(v))
  \,.
$$

Finally

$$
  h(J(-),J(-)) = h(-,-)
$$

and so the  Riemannian metrics $g$ on $V$ appearing from (and fully determining) Hermitian forms $h$ via $h = g - i \omega$ are precisely those for which

$$
  \label{HermitianMetric}
  g(J(-),J(-)) = g(-,-)
  \,.
$$

These are called the _[[Hermitian metrics]]_.

=--

(eg. [Huybrechts 2004, Lem. 1.2.15](#Huybrechts04))

+-- {: .proof}
###### Proof

The positive-definiteness of $g$ is immediate from that of $h$. The symmetry of $g$ follows from the symmetric sesquilinearity of $h$:

$$
  \begin{aligned}
    g(w,v)
    & \coloneqq
    Re(h(w,v))
    \\
    & =
    Re\left( h(v,w)^\ast\right)
    \\
    & =
    Re(h(v,w))
    \\
    & =
    g(v,w)
    \,.
  \end{aligned}
$$

That $h$ is invariant under $J$ follows from its sesquilinarity

$$
  \begin{aligned}
    h(J(v),J(w))
    &=
    i h(v,J(w))
    \\
    & =
    i (h(J(w),v))^\ast
    \\
    & =
    i (-i) (h(w,v))^\ast
    \\
    & =
    h(v,w)
  \end{aligned}
$$

and this immediately implies the corresponding invariance of $g$ and $\omega$.

Analogously it follows that $\omega$ is skew symmetric:

$$
  \begin{aligned}
    \omega(w,v)
    & \coloneqq
    -Im(h(w,v))
    \\
    & =
    -Im\left( h(v,w)^\ast\right)
    \\
    & =
    Im(h(v,w))
    \\
    & = - \omega(v,w)
    \,,
  \end{aligned}
$$

and the relation between the two components:

$$
  \begin{aligned}
    \omega(v,w)
    & =
    - Im(h(v,w))
    \\
    & =
    Re(i h(v,w))
    \\
    & =
    Re(h(J(v),w))
    \\
    & =
    g(J(v),w)
  \end{aligned}
$$

as well as

$$
  \begin{aligned}
    g(v,w)
    & =
    Re(h(v,w)
    \\
    & =
    Im(i h(v,w))
    \\
    & =
    Im(h(J(v),w))
    \\
    & =
    Im(h(J^2(v),J(w)))
    \\
    & =
    - Im(h(v,J(w)))
    \\
    & = \omega(v,J(w))
    \,.
  \end{aligned}
$$

=--

Saying the same again in other words:

\begin{remark}\label{HermitianFormsViaHermitianMetrics}
**(Hermitian forms via Hermitian metrics)**
\linebreak
The decomposition of a Hermitian form into its [[real part]] and [[imaginary part]] is traditionally written with the symbols
$$
  \langle-\vert-\rangle
  \;=\;
  g(-,-)
  +
  \mathrm{i} \omega(-,-)
$$
or similar, where
$$
  g(-,-)
  \;\coloneqq\;
  \Re \langle-\vert-\rangle
  \,,
$$
$$
  \omega(-,-)
  \;\coloneqq\;
  \Im \langle-\vert-\rangle
  \,.
$$
are real [[bilinear forms]] on the [[underlying]] [[real vector space]].

In fact, the real part fully determines the imaginary part and hence the Hermitian form:
$$
  \begin{array}{l}
    \omega(-,-)
    & =
    \Im \langle-\vert-\rangle
    \\
    & =
    \Re \big(-\mathrm{i} \langle-\vert-\rangle\big)
    \\
    & =
    \Re \langle \mathrm{i} - \vert - \rangle
    \\
    & =
    \Re \langle J(-) \vert - \rangle
    \\
    & =
    g\big(J(-), -\big)
    \,,
  \end{array}
$$
where in the last line we introduced the [[complex structure]] $J$ on the [[underlying]] real vector space which promotes it back to the original complex vector space.

Notice that the complex structure $J$ is an [[orthogonal map]] with respect to the inner product $g$, equivalently $g$ is a [[Hermitian metric]] with respect to $J$, in that

$$
  \begin{array}{l}
    g\big(J(-), J(-)\big)
    \\
    \;=\;
    \Re
    \big\langle
      \mathrm{i}(-)
    \big\vert
      \mathrm{i}(-)
    \big\rangle
    \\
    \;=\;
    \Re
    \big(
      -\mathrm{i} \cdot \mathrm{i}
      \langle - \vert - \rangle
    \big)
    \\
    \;=\;
    \Re
      \langle - \vert - \rangle
    \\
    \;=\;
    g(-,-)
    \,.
  \end{array}
$$

Conversely, for a real vector space equipped with a [[complex structure]] $J$ and a [[Hermitian metric]] $g$, in that $g\big(J(-), J(-)\big) \,=\, g(-,-,)$, the formula
$$
  \langle-\vert-\rangle
  \;=\;
  g(-,-)
  +
  \mathrm{i}
  g\big(J(-), -\big)
$$
defines a Hermitian form on the corresponding complex vector space:

The linearity in the second argument is clear. The [[anti-linear map|antilinearity]] in the first argument follows by:
$$
  \begin{array}{l}
    g(\mathrm{i}-,-)
    +
    \mathrm{i}
    g\big(J(\mathrm{i}-), -\big)
    \\
    \;=\;
    g\big(J(-),-\big)
    +
    \mathrm{i}
    g\big(J\circ J (-), - \big)
    \\
    \;=\;
    g\big(J(-),-\big)
    -
    \mathrm{i}
    g\big(-, - \big)
    \\
    \;=\;
    -\mathrm{i}
    \Big(
      g(-,-)
      +
      \mathrm{i} g\big( J(-), - \big)
    \Big)
    \,.
  \end{array}
$$
and Hermiticity follows by:
$$
  \begin{array}{l}
    g(w,v)
      +
    \mathrm{i}
    g\big( J(w), v \big)
    \\
    \;=\;
    g(w,v)
      +
    \mathrm{i}
    g\big( J \circ J(w), J(v) \big)
    \\
    \;=\;
    g(w,v)
      -
    \mathrm{i}
    g\big( w, J(v) \big)
    \\
    \;=\;
    g(v,w)
      -
    \mathrm{i}
    g\big( J(v),  w\big)
    \,.
  \end{array}
$$
\end{remark}

### Relation to Kähler spaces

+-- {: .num_prop #RelationBetweenKaehlerVectorSpacesAndHermitianSpaces}
###### Proposition
**(relation between [[Kähler vector spaces]] and [[Hermitian spaces]])**

Given a [[real vector space]] $V$ with a [[linear complex structure]] $J$, then the following are equivalent:

1. $\omega \in \wedge^2 V^\ast$ is a [[linear Kähler structure]] (def. \ref{KaehlerVectorSpace});

1. $g \in V \otimes V \to  \mathbb{R}$ is a [[Hermitian metric]] (eq:HermitianMetric)

where $\omega$ and $g$ are related by (eq:RelationBetweennOmegaAndgOnHermitianSpace)

$$
  \omega(v,w)
  \;=\;
  g(J(v),w)
  \phantom{AAAAA}
  g(v,w)
  \;=\;
  \omega(v,J(v))
  \,.
$$

=--

### As Real modules
 {#AsRealComplexModules}

While a non-degenerate [[inner product]] $(-\vert-)$ on a [[finite-dimensional vector space|finite-dimensional]] [[real vector space]] $V$ is equivalently a [[linear isomorphism]] to its [[dual vector space]]

$$
  \array{
    V
      &\overset{\sim}{\longrightarrow}&
    V^\ast
      &\overset{\sim}{\longrightarrow}&
    V
    \\
    v &\mapsto& (v\vert-) &\mapsto& v
  }
$$

the analogous statement for  Hermitian complex inner products $\langle - \vert - \rangle$ fails, since the corresponding maps

\[
  \label{AntiLinearIsomorphismWithDualVectorSpace}
  \array{
    H
      &\overset{\sim}{\longrightarrow}&
    H^\ast
      &\overset{\sim}{\longrightarrow}&
    H
    \\
    \vert \psi \rangle
    &\mapsto&
    \langle \psi \vert
    &\mapsto&
    \vert \psi \rangle
  }
\]

are now complex *[[anti-linear map|anti-linear]]* and hence not [[morphisms]] in the [[category]] of [[complex vector spaces]].

(What one does get is a complex-linear isomorphism to the [[anti-dual linear space]], eg. [Karoubi & Villamayor 1973, p. 58 (3 of 31)](#KaroubiVillamayor73); [Karoubi 2010, §1](#Karoubi10).)

But one may absorb this anti-linearity into an ambient category so that Hermitian/unitary structure regarded *[[internalization|internally]]* to that category looks just like Euclidean/orthogonal structure. Since this uses basic structures known from Atiyah's *[[Real K-theory]]* --- with capital "R", [[KR-theory]] --- it is suggestive and convenient to refer to "Real" structures, as follows:

\begin{definition}
Write
\[
  \label{CategoryOfRealvectorBundlesWithInvolution}
  Mod_{\mathbb{R}}^{\mathbf{B}\mathbb{Z}_2}
  \;\;\;
    \coloneqq
  \;\;\;
  Func\big(
    \mathbf{B}\mathbb{Z}_2
    ,\,
    Mod_{\mathbb{R}}
  \big)
\]
for the category of [[real vector spaces]] equipped with a [[linear map|linear]] [[involution]] (equivalently the [[functor category]] from the [[delooping groupoid]] of the [[cyclic group of order two]] to the category [[Vect|$Mod_{\mathbb{R}}$]] of [[real vector spaces]]).

This becomes a [[monoidal category]] whose modoidal stucture $\otimes$ is the [[tensor product of vector spaces|tensor product]] of the [[underlying]] [[vector spaces]] equipped with the tensor product of their involutions.
\end{definition}
\begin{example}\label{RealComplexNumbers}
**(Real complex numbers)**
  The [[complex numbers]] regarded as a [[real vector space]] equipped with its real-linear involution given by [[complex conjugation]] is an object of (eq:CategoryOfRealvectorBundlesWithInvolution). Moreover, since complex-conjugation is actually an [[algebra homomorphism]] on the complex numbers, the usual product operation makes this a [[monoid object]] [[internalization|internal]] to (eq:CategoryOfRealvectorBundlesWithInvolution), which we will refer to as the *Real complex numbers*:

\[
  \label{RealComplexNumbersObject}
  \mathbb{Z}_2
  \curvearrowright
  \mathbb{C}
  \;\in\;
  Mon\big(
    Mod_{\mathbb{R}}^{\mathbf{B}\mathbb{Z}_2}
  \big)
  \,.
\]

\end{example}

\begin{definition}
 **(Real complex modules)**
\label{CategoryOfRealComplexModules}
 We say that the [[category of modules|category of]] [[module objects]] over the Real complex numbers (eq:RealComplexNumbersObject) [[internalization|internal]] to (eq:CategoryOfRealvectorBundlesWithInvolution) is the category of *Real complex modules*:
$$
  Mod_{
    \mathbb{Z}_2 \curvearrowright \mathbb{C}
  }
  \;\;\;\;
    \equiv
  \;\;\;\;
  Mod_{
    \mathbb{Z}_2 \curvearrowright \mathbb{C}
  }
  \big(
    Mod_{\mathbb{R}}^{\mathbf{B}\mathbb{Z}_2}
  \big)
  \,.
$$
\end{definition}
\begin{lemma}
\label{AntilinearInvolutionFromRealComplexModules}
  The category of Real complex modules (Def. \ref{CategoryOfRealComplexModules}) is [[equivalence of categories|equivalent]], as a [[symmetric monoidal category]], to the category of [[complex vector spaces]] equipped with *[[anti-linear map|anti-linear]]* [[involutions]]:
$$
  Mod_{
    \mathbb{Z}_2 \curvearrowright \mathbb{C}
  }
  \;\;
  \simeq
  \;\;
  \big(
    V \,\colon\, Mod_{\mathbb{C}}
  \big)
  \times
  \big(
    \sigma
    \,\colon\,
    V
    \xrightarrow{
      \mathbb{C} antilinear
    }
    V
  \big)
$$
\end{lemma}
\begin{proof}
By direct unwinding of the definitions, one finds that the module property enforces the anti-linearity of the involution:

\begin{imagefromfile}
    "file_name": "AntilinInvViaRealComplexModules-230905.jpg",
    "width": 300,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

Moreover, to see that the [[tensor products]] agree one can argue that the relevant [[coequalizers]] (see [here](Introduction+to+Stable+homotopy+theory+--+1-2#TensorProductOfModulesOverCommutativeMonoidObject)) in $Mod_{\mathbb{R}}^{\mathbf{B}\mathbb{Z}}$ are [[colimits]] in a [[category of presheaves]] which [[limits of presheaves are computed objectwise|are computed objectwise]] --- hence here over the single object of $\mathbf{B}\mathbb{Z}_2$, where they agree with the usual [[tensor product of vector spaces|tensor product]] of the [[underlying]] [[complex vector spaces]].
\end{proof}

The following construction is closely related to what is known as the *[[hyperbolic functor]]* with $\mathbb{Z}/2$-equivariance, establishing an equivalence between [[KR-theory]] and [[topological Hermitian K-theory]] (see the references [there](#ReferencesTopologicalHermitianKTheory)):

\linebreak

Recall the [[equivalence of categories]] between [[real vector spaces]] and [Real vector bundles](real+vector+bundle#InKRTheory) over the point, given by [[complexification]] equipped with the [[involution]] by [[complex conjugation]]:

\linebreak

\linebreak

\[\label{RModulesAsRealBundlesViaComplexification}\]
\begin{imagefromfile}
    "file_name": "RModulesAsRealBundles-231110.jpg",
    "width": 520,
    "unit": "px",
    "margin": {
        "top": -75,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

\begin{proposition}\label{HermitianFormsAreRealInnerProducts}
**(Hermitian forms are Real inner products)**
\linebreak
  Under the equivalence (eq:RModulesAsRealBundlesViaComplexification), a [[real vector space|real]] [[inner product space]] $\mathscr{V} \,\in\, Mod_{\mathbb{R}}$, $g \,\colon\, \mathscr{V} \otimes_{{}_{\mathbb{R}}} \mathscr{V} \to \mathbb{R}$  equipped with an [[isometry|isometric]] [[complex structure]] $J \,\colon\, \mathscr{V} \to \mathscr{V}$, $g\big(J(-), J(-)\big) = g(-,-)$ is sent to corresponding [[Hermitian form]] $\langle -\vert-\rangle \,\equiv\, g(-,-) + \mathrm{i}g\big(J(-),-\big)$, as follows:

\linebreak

\linebreak

\[\label{RealInnerProductsAsHermitianForms}\]
\begin{imagefromfile}
    "file_name": "RealInnerProductsAsHermitianForms-231110b.jpg",
    "width": 530,
    "unit": "px",
    "margin": {
        "top": -75,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

\end{proposition}
Here $\mathscr{V}_{\pm J}$ denotes the complex vector space whose underlying real vector space is $\mathscr{V}$ and with complex structure $\pm J$.
\begin{proof}
On the right we used the following $\mathbb{Z}_2 \curvearrowright \mathbb{C}$-module [[isomorphism]] from the plain complexification in (eq:RModulesAsRealBundlesViaComplexification):

\begin{imagefromfile}
    "file_name": "ComplexificationOfComplexStructure-231111.jpg",
    "width": 510,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

which serves to bring out the [[eigenspaces]] of $J \otimes_{{}_\mathbb{R}} \mathbb{C}$:

\linebreak

\linebreak

\[\label{ComplexifiedComplexStructure}\]
\begin{imagefromfile}
    "file_name": "ComplexifiedComplexStructure-231110.jpg",
    "width": 320,
    "unit": "px",
    "margin": {
        "top": -75,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

With this, the isometry condition $g \circ (J \otimes J) = g$  --- which means that $g$ factors through the $(+1)$-eigenspace of $J \!\otimes_{{}_\mathbb{R}}\! J$ --- implies by functoriality of the equivalence (eq:RModulesAsRealBundlesViaComplexification) that the pairing on the right of (eq:RealInnerProductsAsHermitianForms) factors through the $(+1)$-eigenspace of $\mathrm{I} \otimes \mathrm{I}$ (eq:ComplexifiedComplexStructure), which already makes it a Hermitian form on $\mathscr{V}_{+J}$, as shown on the right of (eq:RealInnerProductsAsHermitianForms).
Explicit computation shows that this is indeed the one given by the traditional formula:

\begin{imagefromfile}
    "file_name": "RealInnerProductProducesTradFormula-231110.jpg",
    "width": 880,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

\end{proof}

\begin{example}\label{HermtianSpacesAsRealComplexModules}
**(Complex Hermitian spaces as Euclidean Real complex modules)**
\linebreak

For $H, \langle-\vert-\rangle_H$ a [[finite-dimensional vector space|finite-dimensional]] [[complex vector space|complex]] [[Hermitian form|Hermitian]] [[inner product space]] (assumed non-degenerate), its [[direct sum]] with its [[linear dual space]] carries the [[anti-linear map|anti-linear]] [[involution]] induced
(eq:AntiLinearIsomorphismWithDualVectorSpace) by the Hermitian form:

\begin{imagefromfile}
    "file_name": "HermitianSpaceAsRealComplexModule-230905.jpg",
    "width": 270,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

Therefore the Hermitian space $H$ induces a Real complex module by Lem. \ref{AntilinearInvolutionFromRealComplexModules}, which we denote by the corresponding script symbol:
$$
  \mathscr{H}
  \;\coloneqq\;
  \mathbb{Z}_2 \curvearrowright
  \big(
    H  \oplus H^\ast
  \big)
  \;\;
  \in
  \;\;
  Mod_{
   \mathbb{Z}_2 \curvearrowright \mathbb{C}
   \,.
  }
$$

Notice that the Real complex modules $\mathscr{H}$ arising this way have special properties:

1. $\mathscr{H}$ is a [[self-dual object]], via the following ([[coevaluation map|co]]-)[[evaluation maps]]

\begin{imagefromfile}
    "file_name": "SelfDualStructureOnRealComplexModule-230905.jpg",
    "width": 700,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

1. $\mathscr{H}$ carries an [[internalization|internal]] [[complex structure]], namely an [[automorphism]] (now of Real complex modules!) which squares to minus the [[identity morphism]]:

\begin{imagefromfile}
    "file_name": "ComplexStructureOnRealComplexModule-230905.jpg",
    "width": 300,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

Due to this complex and [[self-dual object|self-dual]] structure we may think of $\mathscr{H}$ as being a *[[Euclidean space|Euclidean]]* (instead of Hermitian) complex linear space but now *[[internalization|internal to]]* Real complex modules.
\end{example}

\begin{proposition}
  Given a [[pair]] of ([[finite-dimensional vector space|finite-dimensional]], non-degenerate) complex Hermitian spaces $H, \langle-\vert-\rangle_H$ and $K, \langle-\vert-\rangle_K$, there the complex [[linear maps]] between them are in bijection to the [[homomorphisms]] between the corresponding Real complex modules according to Exp. \ref{HermtianSpacesAsRealComplexModules}, as follows:

1. the ordinary $\mathrm{i}$-complex linear maps $H \to K$ correspond to the $\mathrm{i}\beta$-complex homomorphism $\mathscr{H} \to \mathscr{K}$

1. the ordinary [[unitary maps]] $H \to K$ correspond to the [[orthogonal maps]] $\mathscr{H} \to \mathscr{K}$ (namely those which respect the [[evaluation maps]] on these [[self-dual objects]]).

\end{proposition}
\begin{proof}
  This follows by straightforward unwinding of the definitions:

First, for a homomorphism $\mathscr{H} \to \mathscr{K}$ to commute with $\mathrm{i}\beta$ its [[underlying]] complex linear map clearly needs to respect the [[direct sum]], structure $H \oplus H^\ast \to K \oplus K^\ast$, hence it needs to come from complex linear map $g \,\colon\,H \to K$. But then the respect for the complex involution uniquely fixes the action $H^\ast \to K^\ast$. Interestingly, it fixes them to be given by the linear dual of the [[adjoint operator|operator adjoint]] $g^\dagger$:

\begin{imagefromfile}
    "file_name": "ComplexLinMapsAsRealComplexHoms-230905.jpg",
    "width": 700,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

Second: A unitary map is a complex linear map that preserves the Hermitian form. But with the first point above one sees that this is equivalent to preserving the evaluation map on the corresponding Real complex modules:

\begin{imagefromfile}
    "file_name": "UnitaryCLinMapIsOrthRealComplexMap-230905.jpg",
    "width": 700,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

\end{proof}

In **summary**: After [[internalization]] as Real complex modules, complex Hermitian/unitary space look like complex Euclidean/orthogonal spaces:

\begin{imagefromfile}
    "file_name": "HermitianSpAsEuclidRealComplexMods-230906.jpg",
    "width": 650,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

\linebreak

<center>
<img src="/nlab/files/HermitianSpacesAsEquivariantModules-230901.jpg" width="780">
</center>

\linebreak

(...)

\linebreak

## Related concepts

* [[Hilbert space]]

* [[bilinear form]], [[quadratic form]], [[sesquilinear form]]

* [[symplectic form]], [[Kähler form]]

* [[hermitian matrix]]

* [[Hermitian manifold]]

* [[quadratic form]]

* [[bra-ket]]

* [[Hermitian K-theory]]

## References

Hermitian forms are named in honor of the discussion of [[quadratic forms]] due to:

* [[Charles Hermite]], *Sur la théorie des formes quadratiques. Premier mémoire*,  Journal für die reine und angewandte Mathematik **47** (1854) 313-342 [[doi:10.1515/crll.1854.47.343](https://doi.org/10.1515/crll.1854.47.343)]

maybe starting with

* [[Luigi Bianchi]], *Forme definite di Hermite*, §24 in:  *Sui gruppi di sostituzioni lineari con coefficienti appartenenti a corpi quadratici immaginarî*, Mathematische Annalen **40** (1892) 332–412 [[doi:10.1007/BF01443558](https://doi.org/10.1007/BF01443558)]

see:

* Jürgen Elstrodt, Fritz Grunewald, Jens Mennicke: *Integral Binary Hermitian Forms*, Ch. 9 in *Groups Acting on Hyperbolic Space*, Springer (1998) [[doi:10.1007/978-3-662-03626-6_9]( https://doi.org/10.1007/978-3-662-03626-6_9)]

Discussion of Hermitian forms over the [[complex numbers]], as understood today, originates in the definition of [[Hilbert spaces]] (in laying of mathematical foundations of [[quantum mechanics]]):

* {#vonNeumann30} [[John von Neumann]], p. 64 in: *Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren*, Math. Ann. **102** (1930) 49–131 [[doi:10.1007/BF01782338](https://doi.org/10.1007/BF01782338)]

* [[John von Neumann]]:

  p. 21 in: *Mathematische Grundlagen der Quantenmechanik*, Springer (1932, 1971) [[doi:10.1007/978-3-642-96048-2](https://link.springer.com/book/10.1007/978-3-642-96048-2)]

  pp. 38 in: *Mathematical Foundations of Quantum Mechanics* Princeton University Press (1955) [[doi:10.1515/9781400889921](https://doi.org/10.1515/9781400889921), [Wikipedia entry](https://en.wikipedia.org/wiki/Mathematical_Foundations_of_Quantum_Mechanics)]

Textbook account:

* {#Bourbaki81} [[Nicolas Bourbaki]], §V.1.1 in: *Topological Vector Spaces*, Chapters 1–5, Masson (1981), Springer (2003) [[doi:10.1007/978-3-642-61715-7](https://doi.org/10.1007/978-3-642-61715-7)]

* {#Huybrechts04} [[Daniel Huybrechts]],  §1.2 in: _Complex geometry: An introduction_, Universitext, Springer (2004) [[doi:10.1007/b137952](https://doi.org/10.1007/b137952), [pdf](http://www.im.ufrj.br/~andrew/ensino/VC20-1/Huybrechts-Complex_Geometry.pdf)]

Textbook accounts in the context of [[operator algebras]]:

* [[Richard V. Kadison]], [[John R. Ringrose]], §2.1 in: *Fundamentals of the theory of operator algebras* Vol I *Elementary Theory*, Graduate Studies in Mathematics **15**, AMS (1997) [[ISBN:978-0-8218-0819-1](https://bookstore.ams.org/gsm-15)]

* [[Bruce Blackadar]], §I.1.1 in: *Operator Algebras -- Theory of $C^\ast$-Algebras and von Neumann Algebras*, Encyclopaedia of Mathematical Sciences **122**, Springer (2006) [[doi:10.1007/3-540-28517-2](https://doi.org/10.1007/3-540-28517-2)]

* {#Landsman17} [[Klaas Landsman]], §A.1 in: *Foundations of quantum theory -- From classical concepts to Operator algebras*, Springer Open (2017) [[doi:10.1007/978-3-319-51777-3](https://link.springer.com/book/10.1007/978-3-319-51777-3), [pdf](https://link.springer.com/content/pdf/10.1007%2F978-3-319-51777-3.pdf)]

See also:

* Wikipedia, _[Hermitian form](https://en.wikipedia.org/wiki/Sesquilinear_form#Hermitian_form)_

and see the references at *[[Hilbert space]]*.

Hermitian forms as isomorphisms to the [[anti-dual linear space]]:

* {#Karoubi10} [[Max Karoubi]], §1 in: *Le théorème de périodicité en K-théorie hermitienne*, Quanta of Maths **1**, AMS and Clay Math Institute Publications (2010) [[arXiv:0810.4707](https://arxiv.org/abs/0810.4707)]

  > (in the context of [[Hermitian K-theory]])

Hermitian forms in the generality over [[noncommutative ring|noncommutative]] [[ground rings]] (and discusssed in the context of [[Hermitian K-theory]]):

* {#KaroubiVillamayor73} [[Max Karoubi]], [[Orlando Villamayor]], Ex. 1 in: *K-théorie algébrique et K-théorie topologique II*, Math. Scand. **32** (1973) 57-86 [[jstor:24490565](https://www.jstor.org/stable/24490565)]

* [[C. T. C. Wall]], _On the axiomatic foundations of the theory of Hermitian forms_, Proc. Camb. Phil. Soc. **67** (1970) 243-250 [[doi:10.1017/S0305004100045515](https://doi.org/10.1017/S0305004100045515), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/wall9.pdf)]

* [[John Milnor]], [[Dale Husemöller]], *Hermitian forms*, Appendix 2 of: *Symmetric Bilinear Forms*,  Ergebnisse der Mathematik und ihrer Grenzgebiete **73**, Springer (1973) [[doi:10.1007/978-3-642-88330-9](https://doi.org/10.1007/978-3-642-88330-9)]

{#ReferencesViaDaggerCompactStructure} Hermitian forms expressed through [[dagger-compact category]]-[[structure]] (notably for [[quantum information theory via dagger-compact categories]]):

* {#AbramskyCoecke04} [[Samson Abramsky]], [[Bob Coecke]], Def. 7.5 in: *A categorical semantics of quantum protocols*, Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS'04). IEEE Computer Science Press (2004) $[$[arXiv:quant-ph/0402130](http://arxiv.org/abs/quant-ph/0402130), [doi:10.1109/LICS.2004.1319636](https://doi.org/10.1109/LICS.2004.1319636)$]$

* {#AbramskyCoecke05} [[Samson Abramsky]], [[Bob Coecke]], p. 6 in: *Abstract Physical Traces*, Theory and Applications of Categories, **14** 6 (2005) 111-124. \[<a href="http://www.tac.mta.ca/tac/volumes/14/6/14-06abs.html">tac:14-06</a>, [arXiv:0910.3144](https://arxiv.org/abs/0910.3144)\]

* [[Bob Coecke]], Def. 2.5 in: *De-linearizing Linearity: Projective Quantum Axiomatics from Strong Compact Closure*, Proceedings of the [3rd International Workshop on Quantum Programming Languages (2005)](https://www.mathstat.dal.ca/~selinger/qpl2005/), Electronic Notes in Theoretical Computer Science **170** (2007) 49-72 \[<a href="https://doi.org/10.1016/j.entcs.2006.12.011">doi:10.1016/j.entcs.2006.12.011</a>, <a href="https://arxiv.org/abs/quant-ph/0506134">arXiv:quant-ph/0506134</a>\]

* [Duncan 2006](quantum+information+theory+via+dagger-compact+categories#Duncan06), [p. 31](http://personal.strath.ac.uk/ross.duncan/papers/rduncan-thesis.pdf#page=39)

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Hermitian K-theory

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## Idea

Where [[algebraic K-theory]] concerns (in degree zero) the [[Grothendieck groups]] of ([[finitely generated module|finitely generated]] [[projective module|projective]]) [[modules]]/[[vector spaces]], Hermitian K-theory is concerned with Grothendieck groups of modules equipped with [[Hermitian form|Hermitian]] [[quadratic forms]].

Since the resulting groups were originally denoted (in [Wall 1969](#Wall69)) by the letter "L" (just the next letter in the alphabet after "K" for "class"), this is also known as "[[L-theory]]" (cf. [Karoubi 1973, p. 306](#Karoubi73)).

## Related concepts

* [[hyperbolic functor]]

* [[KR-theory]]

## References

### General

Precursor discussion:

* [[Hyman Bass]] (notes by Amit Roy), §5 in: *Lectures on Topics in Algebraic K-Theory*, Tata Institute of Fundamental Research (1965) [[[Bass-HyperbolicFunctor.pdf:file]]]

* {#Wall69} [[C. T. C. Wall]] (ed. [[Andrew Ranicki]]), *Surgery on compact Manifolds*, Math. Surveys and Monographs **69** (1969) [[pdf](https://www.maths.ed.ac.uk/~v1ranick/books/scm.pdf)]

Introduction of Hermtian K-theory as such:

* [[Max Karoubi]], [[Orlando Villamayor]], *K-théorie algébrique et K-théorie topologique II*, Math. Scand. **32** (1973) 57-86 [[jstor:24490565](https://www.jstor.org/stable/24490565)]

Further early discussion:

* [[Hyman Bass]] (ed.), *Algebraic K-Theory III -- Hermitian K-Theory and Geometric Applications*, Lecture Notes in Mathematics **343** (1973) [[doi:10.1007/BFb0061366](https://doi.org/10.1007/BFb0061366)]

* {#Mishchenko76} [[Alexandr S. Mishchenko]], *Hermitian K-Theory. The Theory of characteristic classes and methods of functional analysis*, Uspeki Mat. Nauk **31** 2 (1976) 69-134,  Russ. Math. Surv. **31** 71 (1976) [[doi:10.1070/RM1976v031n02ABEH001478](https://iopscience.iop.org/article/10.1070/RM1976v031n02ABEH001478), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/mishch5.pdf)]

* [[Anthony Bak]], *Grothendieck Groups of Modules and Forms Over Commutative Orders*, American Journal of Mathematics, **99** 1 (1977) 107-120 [[jstor:2374010](https://www.jstor.org/stable/2374010), [doi:10.2307/2374010](https://doi.org/10.2307/2374010)]

For more see:

* [[Max Karoubi]], *K-théorie hermitienne* [[webpage](https://webusers.imj-prg.fr/~max.karoubi/K-theorie.hermitienne.html)]

Further developments:

* [[Damjan Kobal]], *K-Theory, Hermitian K-Theory and the Karoubi Tower*, K-Theory **17** 113–140 (1999) [[doi:10.1023/A:1007799508729](https://sci-hub.ru/10.1023/A:1007799508729)]

* [[Max Karoubi]], *Le théorème de périodicité en K-théorie hermitienne*, in *Quanta of Maths*, Clay Mathematics Proceesings **11**, AMS and Clay Math Institute Publications (2010) 257-282 [[arXiv:0810.4707](https://arxiv.org/abs/0810.4707), [pdf](https://www.claymath.org/library/proceedings/cmip011c.pdf)]

* [[Max Karoubi]], [[Mariusz Wodzicki]], *Algebraic and Hermitian K-theory of $\mathcal{K}$-rings*, The Quarterly Journal of Mathematics **64** 3 (2013) 903–940 [[arXiv:1310.4123](https://arxiv.org/abs/1310.4123), [doi:10.1093/qmath/hat030](https://doi.org/10.1093/qmath/hat030)]

* [[Marco Schlichting]], *Higher $K$-Theory of Forms I. From Rings to Exact Categories*, Journal of the Institute of Mathematics of Jussieu **20** 4 (2021) 1205-1273 [[doi:10.1017/S1474748019000410](https://doi.org/10.1017/S1474748019000410)]

### Topological Hermitian K-theory
 {#ReferencesTopologicalHermitianKTheory}

The case of [[topological K-theory|topological]] Hermitian K-theory:

* {#Karoubi73} [[Max Karoubi]], §III in: *Périodicité de la K-théorie hermitienne*, in: [[Hyman Bass]] (ed.), *Algebraic K-Theory III -- Hermitian K-Theory and Geometric Applications*, Lecture Notes in Mathematics **343** (1973) 301-411 [[doi:10.1007/BFb0061366](https://doi.org/10.1007/BFb0061366)]

and its relation to [[KR-theory]] via the [[hyperbolic functor]]:

* {#Crabb80} [[Michael C. Crabb]], §9 of: *$\mathbb{Z}/2$-Homotopy Theory*, Lond. Math. Soc. Lecture Notes **44**, Cambridge University Press (1980) [[ark:/13960/t3jx7bg4w](https://archive.org/details/zz2homotopytheor0000crab), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/crabb.pdf)]

* [[Po Hu]], [[Igor Kriz]], Appendix of: *Topological Hermitian cobordism*, Journal of Homotopy and Related Structures **11** (2016) 173–197 [[doi:10.1007/s40062-014-0100-9](https://doi.org/10.1007/s40062-014-0100-9)]

* [[Max Karoubi]], [[Charles Weibel]], Thm. 3.5 in: *Twisted $K$-theory, Real $A$-bundles and Grothendieck–Witt groups*, Journal of Pure and Applied Algebra **221** 7 (2017) 1629-1640 [[doi:10.1016/j.jpaa.2016.12.020](https://doi.org/10.1016/j.jpaa.2016.12.020)]

See also:

* [[Max Karoubi]], [[Marco Schlichting]], [[Charles Weibel]], *The Witt group of real algebraic varieties*, Journal of Topology **9** 4 (2016) 1257-1302 [[arXiv:1506.03862](https://arxiv.org/abs/1506.03862), [doi:10.1112/jtopol/jtw024]( https://doi.org/10.1112/jtopol/jtw024)]

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Hermitian manifold

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## Idea

An _(almost) Hermitian manifold_ is a [[smooth manifold]] compatibly equipped with

* ([[almost complex structure|almost]]) [[complex structure]]

* [[Riemannian structure]].

Equivalently an _[[almost Hermitian structure]]_, namely a [[G-structure]] for $G = U(n) \hookrightarrow GL(2n,\mathbb{R})$ the [[unitary group]].

If in addition it carries compatibly [[symplectic structure]] it is called a _[[Kähler manifold]]_.

| [[complex structure]] | + [[Riemannian structure]] | + [[symplectic structure]] |
|--|--|--|
[[complex structure]] | [[Hermitian structure]] | [[Kähler structure]] |

## Definition

+-- {: .num_defn}
###### Defnition

Given an [[orientation|oriented]] [[smooth manifold]] $X$, an **(almost) Hermitian structure** on $X$ is

1. an ([[almost complex structure|almost]]) [[complex structure]] $J$;

1. a [[Riemannian structure]] $g$

such that the rank $(0,2)$ [[tensor field]]

$$
  \omega(-,-) \coloneqq g(-,J(-))
$$

is a non-degenerate [[differential 2-form]], then called the **Hermitian form** or similar.

=--

+-- {: .num_remark}
###### Remark

If in addition the Hermitian form $\omega(-,-) = g(-,J-)$ is closed, hence a [[symplectic structure]], then $(X,J,g)$ is an ([[almost Kähler manifold|almost]]) [[Kähler manifold]].

=--

## Properties

### Relation to Spin-structure

+-- {: .num_prop}
###### Proposition

A spin structure on a [[compact topological space|compact]] [[Hermitian manifold]] ([[Kähler manifold]]) $X$ of complex [[dimension]] $n$ exists precisely if, equivalently

* there is a choice of [[square root]] $\sqrt{\Omega^{n,0}}$ of the [[canonical line bundle]] $\Omega^{n,0}$ (a "[[Theta characteristic]]");

* there is a trivialization of the [[first Chern class]] $c_1(T X)$ of the [[tangent bundle]].

=--

In this case one has:

+-- {: .num_prop}
###### Proposition

There is a [[natural isomorphism]]

$$
  S_X \simeq \Omega^{0,\bullet}_X \otimes \sqrt{\Omega^{n,0}}_X
$$

of the [[sheaf]] of [[sections]] of the [[spinor bundle]] $S_X$ on $X$ with the [[tensor product]] of the  [[Dolbeault complex]] with the corresponding [[Theta characteristic]];

Moreover, the corresponding [[Dirac operator]] is the [[Dolbeault-Dirac operator]] $\overline{\partial} + \overline{\partial}^\ast$.

=--

This is due to ([Hitchin 74](#Hitchin74)). A textbook account is for instance in ([Friedrich 74, around p. 79 and p. 82](#Friedrich74)).

## Related concepts

* [[Hermitian space]]

* [[para-Hermitian manifold]]

* [[Kähler manifold]]

* [[p-convex polarization]]

## References

[[spin structure]] on Hermitian manifolds is discussed for instance in

* {#Friedrich97} [[Thomas Friedrich]], _Dirac operators in Riemannian geometry_, Graduate studies in mathematics 25, AMS (1997)

Discussion of the relation between Hermitian metrics and almost Kähler metrics is in

* Vestislav Apostolov, Tedi Draghici, _Hermitian conformal classes and almost Kähler structures on 4-manifolds_ ([pdf](http://www2.fiu.edu/~draghici/research/paperfiles/dga.pdf))

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hermitian matrix

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## Hermitian adjoints

Suppose $\mathcal{H}$ is a [[Hilbert space]] with an inner product $\langle\cdot,\cdot\rangle$.  Consider a continuous [[linear operator]] $A: \mathcal{H} \to \mathcal{H}$.  One can show that there exists a unique continuous linear operator $A^{*}: \mathcal{H} \to \mathcal{H}$ with the following property:

$\langle A x,y\rangle=\langle x,A^{*}y\rangle$ for all $x,y \in \mathcal{H}$.

This is a generalization of the concept of an [[adjoint matrix]] (also known as a [[conjugate transpose matrix|conjugate transpose]], Hermitian conjugate, or Hermitian adjoint).  The adjoint of an $m \times n$ matrix $A$ with complex entries is the $n \times m$ matrix whose entries are defined by

$(A^{*})_{ij}=\bar{A_{ji}}$.

As such,

$A^{*}=(\bar{A})^{T}=\bar{A^{T}}$

where $A^{T}$ is the [[transpose matrix]] of $A$ and $\bar{A}$ is the matrix with complex conjugate entries of $A$.

## Hermitian matrices

A matrix, $A$, is said to be Hermitian if

$A^{*}=A$

where $A^{*}$ is the Hermitian adjoint of $A$.

## Notation

The notation used here for the adjoint, $A^{*}$, is commonly used in linear algebraic circles (as is $A^{H}$).  In [[quantum mechanics]], $A^{\dagger}$ is exclusively used for the adjoint while $A^{*}$ is interpreted as the same thing as $\bar{A}$.

## Related concepts

* [[symmetric matrix]]

* [[celestial sphere]]

* [[Hermitian form]]

* [[Hermitian operator]]

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Hessel Posthuma

* [personal page](https://staff.fnwi.uva.nl/h.b.posthuma/)

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=164307)

## Selected writings

On [[equivariant differential topology]] via [[Lie groupoids]]:

* {#PflaumPosthumaTang11} [[Markus Pflaum]], [[Hessel Posthuma]], X. Tang, *Geometry of orbit spaces of proper Lie groupoids*, Journal für die reine und angewandte Mathematik (Crelles Journal) 2014.694 [[arXiv:1101.0180](https://arxiv.org/abs/1101.0180), [doi:10.1515/crelle-2012-0092](https://doi.org/10.1515/crelle-2012-0092)]

On the [[K-theory classification of topological phases of matter]] for the case of [[topological crystalline insulators]] classified by [[equivariant K-theory]]:

* [[Luuk Stehouwer]], [[Jan de Boer]], [[Jorrit Kruthoff]], [[Hessel Posthuma]], *Classification of crystalline topological insulators through K-theory*, Adv. Theor. Math. Phys, **25** 3 (2021) 723–775 [[arXiv:1811.02592](https://arxiv.org/abs/1811.02592), [doi:10.4310/ATMP.2021.v25.n3.a3](https://doi.org/10.4310/ATMP.2021.v25.n3.a3)]

category: people


Hessian

# Hessians
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{: toc}

##Idea

In all cases, a Hessian is a symmetric [[bilinear form]] on a [[tangent space]], encoding second-order information about a twice-[[differentiable function]].  (Compare the [[differential]] of a once-differentiable function, which is a [[1-form]] on the tangent space.)

For the purpose of this page, a tangent vector "is" a local [[derivation]] on [[germ]]s of functions.  If you don't like this model, take it that the natural isomorphism is unwritten, but trivial to insert; we do not actually name any derivations, but only germs.

## Trivial preliminary lemma

If an $\mathbb{R}$-bilinear form $G$ on germs $X,Y$ of $T M$ at a point $x$ of a twice-[[differentiable manifold]] $M$ is $C^2$-linear in one argument and symmetric, then it descends to a bilinear form on the tangent space $T_x M$.

## Definitions

Given a $C^2$ manifold $M$ and $C^2$ function $\varphi:M\to \mathbb{R}$, the (ordered) second [[derivative]] of $\varphi$, for germs $X$ and $Y$ of tangent fields, is simply
$$ X (Y \varphi).$$
At a [[critical point]] $x$, (i.e., $(d\varphi)_x = 0$) [[exercise|one checks]] that
$$ X (Y \varphi) = Y (X \varphi) $$
so that for a real function germ $f$,
$$ X ((f\cdot Y)\varphi) = f \cdot (X (Y \varphi)) $$
and hence the symmetric bilinear form
$$ Hesse_{\varphi,x} (X,Y) = X(Y \varphi)$$
descends to the tangent space at $x$, and is pronounced as the __Hessian__ of $\varphi$ at the critical point $x$.

When $M$ carries a torsion-free [[connection]] $\nabla$, (e.g., the [[Levi-Civita connection]] of a [[Riemannian structure]]), one may define a global Hessian, defined for germs by
$$ Hesse_{\varphi,\nabla} (X,Y) = X(Y \varphi) - (\nabla_X Y) \varphi $$
which again is symmetric, and hence descends to the tangent space.  Note, however, this obviously depends on the particular connection used.

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Hessian manifold

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## Definition

\begin{definition}
*In coordinates*, a *Hessian manifold* is a [[pseudo-Riemannian manifold]] $(M,g)$ that admits local coordinates $\{q^a\}$, called *special coordinates*, that realize the metric as the [[Hessian]] of a function $H$, the *Hessian potential*:
$$
g_{a,b} = \partial_{a} \partial_{b} H = H_{ab}
$$
such a pseudo-Riemannian metric $g$ is known as a *Hessian metric*.

The defining relation is only invariant under [[affine space|affine transformations]].

Equivalently, in a *coordinate-free expression* (see ([Shima (2013)](#Shima13))), a Hessian manifold is a pseudo-Riemannian manifold $(M,g)$ endowed with a torsion-free [[flat connection]] $\nabla$, such that the rank-3 tensor
$$
S=\nabla g
$$
is totally anti-symmetric.

\end{definition}

## Related concepts

* [[special geometry]]

## References

General:

* {#Shima13} Hirohiko Shima. (2013). "Geometry of Hessian Structures". In: Nielsen, F., Barbaresco, F. (eds) *Geometric Science of Information. GSI 2013. Lecture Notes in Computer Science, vol 8085*. Springer, Berlin, Heidelberg. ([doi](https://doi.org/10.1007/978-3-642-40020-9_4))

* Gabriel Lopes Cardoso, [[Thomas Mohaupt]]. *Special geometry, Hessian structures and applications*. Physics Reports 855, 25 April 2020, Pages 1-141. ([doi](https://doi.org/10.1016/j.physrep.2020.02.002))


heterogeneous equality

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## Idea

A form of [[equality]] which allows for elements of [[families]] of [[types]] to be compared for equality.

## Definition

### Typal heterogeneous equality

In the [[type theory]] literature, there are different kinds of [[types]] which are called *heterogeneous equality* types:

* *[[heterogeneous identity types]]* are a variant of [[identity types]] but parameterized over pairs of possibly different types,

* *[[fibered heterogeneous identity types]]* are the [[dependent sum]] of heterogeneous identity types over all relevant [[identifications]].

### Propositional heterogeneous equality

In any two-layer type theory with a layer of [[types]] and a layer of [[propositions]], or equivalently a [[first order logic]] over [[type theory]] or a [[first-order theory]], every type $A$ has a binary [[relation]] called [[propositional equality]], according to which two elements $x$ and $y$ of $A$ are related if and only if they are equal; in this case we write $x =_A y$. Similarly, every type family $B(x)$ indexed by $x:A$ has a family of binary relations, according to which, given two elements $x$ and $y$ of $A$ where $x =_A y$, and two elements $a$ of $B(x)$ and $b$ of $B(y)$, $a$ and $b$ are related if and only if they are equal across the equality $x =_A y$. Since relations are propositions in the context of a term variable judgment in the type layer, this is called **[[propositional heterogeneous equality]]** or **[[heterogeneous propositional equality]]**. The formation and introduction rules for propositional heterogeneous equality is as follows

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma, x:A, y:A, x =_A y \; \mathrm{true}, a:B(x), b:B(y) \vdash a =_{\underline{ }.B}^{x =_A y} b \; \mathrm{prop}} \quad \frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma, x:A, a:B(x) \vdash a =_{\underline{ }.B}^{x =_A x} a \; \mathrm{true}}$$

Then we have the elimination rules for propositional heterogeneous equality:

$$\frac{
\begin{array}{c}
\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \\
\Gamma, x:A, y:A, x =_A y \; \mathrm{true}, a:B(x), b:B(y), a =_{\underline{ }.B}^{x =_A y} b \; \mathrm{true} \vdash P(a, b, x, y) \; \mathrm{prop} \\
\Gamma, x:A, a:B(x), \vdash P(x, x, a, a) \; \mathrm{true}
\end{array}
}{\Gamma, x:A, y:A, x =_A y \; \mathrm{true}, a:B(x), b:B(y), a =_{\underline{ }.B}^{x =_A y} b \; \mathrm{true} \vdash P(a, b, x, y) \; \mathrm{true}}$$

## Properties

### Heterogeneous equalities are equivalence relations

The introduction rule of heterogeneous equality says that heterogeneous equality is reflexive.

We can show that heterogeneous equality is symmetric, that for all $x:A$ and $y:A$ such that $x =_A y$, and for all $a:B(x)$ and $b:B(y)$ such that $a =_{\underline{ }.B}^{x =_A y} b$, we have $b =_{\underline{ }.B}^{y =_A x} a$. Non-heterogeneous equality is symmetric, which means that for all $x:A$ and $y:A$ such that $x =_A y$, $y =_A x$. By the introduction rule, we have that for all $x:A$ and $a:B(x)$, we have that $a =_{\underline{ }.B}^{x =_A x} a$, and because all propositions imply themselves, we have that $a =_{\underline{ }.B}^{x =_A x} a$ implies $a =_{\underline{ }.B}^{x =_A x} a$. Thus, by the elimination rules for heterogeneous equality, for all $x:A$ and $y:A$ such that $x =_A y$, and for all $a:B(x)$ and $b:B(y)$ such that $a =_{\underline{ }.B}^{x =_A y} b$, we have $b =_{\underline{ }.B}^{y =_A x} a$.

We can also show that heterogeneous equality is transitive, that for all $x:A$, $y:A$, and $z:A$ such that $x =_A y$ and $y =_A z$, and for all $a:B(x)$, $b:B(y)$, and $c:B(z)$ such that $a =_{\underline{ }.B}^{x =_A y} b$, $b =_{\underline{ }.B}^{y =_A y} c$ implies $a =_{\underline{ }.B}^{x =_A z} c$. Non-heterogeneous equality is transitive, that for all $x:A$, $y:A$, and $z:A$ such that $x =_A y$ and $y =_A z$, $x =_A z$. By the introduction rule, we have that for all $x:A$ and $a:B(x)$, we have that $a =_{\underline{ }.B}^{x =_A x} a$, and because all propositions imply themselves, we have that $a =_{\underline{ }.B}^{x =_A z} c$ implies $a =_{\underline{ }.B}^{x =_A z} c$, and because true propositions imply true propositions, we have that $a =_{\underline{ }.B}^{x =_A x} a$ implies that $a =_{\underline{ }.B}^{x =_A z} c$ implies $a =_{\underline{ }.B}^{x =_A z} c$. Thus, by the elimination rules for heterogeneous equality, or all $x:A$, $y:A$, and $z:A$ such that $x =_A y$ and $y =_A z$, and for all $a:B(x)$, $b:B(y)$, and $c:B(z)$ such that $a =_{\underline{ }.B}^{x =_A y} b$, $b =_{\underline{ }.B}^{y =_A y} c$ implies $a =_{\underline{ }.B}^{x =_A z} c$.

Thus, heterogeneous equality is an equivalence relation.

### Dependent functions are extensional

For all dependent function $f:\prod_{x:A} B(x)$ and elements $x:A$ and $y:A$ such that $x =_A y$, $f(x) =_{\underline{ }.B}^{x =_A y} f(y)$:

$$\forall f:\prod_{x:A} B(x).\forall x:A.\forall y:A.x =_A y \implies f(x) =_{\underline{ }.B}^{x =_A y} f(y)$$

This is because for all dependent function $f:\prod_{x:A} B(x)$, by the introduction rule for heterogeneous equality, for all elements $x:A$, $f(x) =_{\underline{ }.B}^{x =_A x} f(x)$, and the elimination rule for heterogeneous equality states that if for all elements $x:A$, $f(x) =_{\underline{ }.B}^{x =_A x} f(x)$, then for all elements $x:A$ and $y:A$ such that $x =_A y$, $f(x) =_{\underline{ }.B}^{x =_A y} f(y)$.

### Dependent function extensionality

The extensionality principle for [[dependent product types]] (dependent [[function extensionality]]) states that for all dependent functions $f:\prod_{x:A} B(x)$ and $g:\prod_{x:A} B(x)$, $f =_{\prod_{x:A} B(x)} g$ if and only if for all $a:A$ and $b:A$ such that $a =_A b$, $f(a) =_{\underline{ }.B}^{a =_A b} g(b)$

$$\forall f:\prod_{x:A} B(x).\forall g:\prod_{x:A} B(x).f =_{\prod_{x:A} B(x)} g \iff \forall a:A. \forall b:A.a =_A b \implies (f(a) =_{\underline{ }.B}^{a =_A b} g(b))$$

### Relation to non-heterogeneous propositional equality

Given elements $x:A$, and for all $a:B(x)$ and $b:B(x)$, $a =_{\underline{ }.B}^{x =_A x} b$ if and only if $a =_{B(x)} b$

$$\forall a:B(x).\forall b:B(x). a =_{\underline{ }.B}^{x =_A x} b \iff a =_{B(x)} b$$

This is because by the introduction rules of propositional equality and heterogeneous propositional equality, we have that for all $a:B(x)$, $a =_{B(x)} a$ and $a =_{\underline{ }.B}^{x =_A x} a$, and since true propositions imply true propositions, we have $a =_{B(x)} a$ implies $a =_{\underline{ }.B}^{x =_A x} a$ and $a =_{\underline{ }.B}^{x =_A x} a$ implies $a =_{B(x)} a$. By the elimination rules for propositional equality, for all $a:B(x)$ and $b:B(x)$, we have $a =_{B(x)} b$ implies $a =_{\underline{ }.B}^{x =_A x} b$, and by the elimination rules for heterogeneous propositional equality, we have $a =_{\underline{ }.B}^{x =_A x} b$ implies $a =_{B(x)} b$. Thus, $a =_{\underline{ }.B}^{x =_A x} b$ if and only if $a =_{B(x)} b$.

### Transport functions

The analogue of [[identity functions]] on a type $A$, a function $f:A \to A$ such that $x =_A f(x)$ for all elements $x:A$, is the notion of [[transport functions]] between a family of types $B(x)$ indexed by elements $x:A$.

Given elements $x:A$ and $y:A$ such that $x =_A y$, a [[transport function]] is a function $f:B(x) \to B(y)$ such that for all $a:B(x)$, $a =_{\underline{ }.B}^{x =_A y} f(a)$

$$\forall a:B(x).a =_{\underline{ }.B}^{x =_A y} f(a)$$

These transport functions, if they exist, are [[uniqueness quantifier|unique up to equality]]. Suppose we have a second function $g:B(x) \to B(y)$ such that $x =_A y$ implies that for all $a:B(x)$, $a =_{\underline{ }.B}^{x =_A y} g(a)$. Then by symmetry and transitivity of heterogeneous equality, we have $f(a) =_{\underline{ }.B}^{y =_A y} g(a)$, which is logically equivalent to $f(a) =_{B(y)} g(a)$. Then by function extensionality, we have $f =_{B(x) \to B(y)} g$, which implies that transport functions, if they exist, are unique up to equality.

Suppose that for all elements $x:A$ and $y:A$ such that $x =_A y$, there exists a transport function $\mathrm{tr}_{\underline{ }.B}^{x =_A y}:B(x) \to B(y)$. Then for all $a:B(x)$ and $b:B(y)$, $a =_{\underline{ }.B}^{x =_A y} b$ if and only if $\mathrm{tr}_{\underline{ }.B}^{x =_A y}(a) =_{B(y)} b$

$$\forall a:B(x).\forall b:B(y). a =_{\underline{ }.B}^{x =_A y} b \iff \mathrm{tr}_{\underline{ }.B}^{x =_A y}(a) =_{B(y)} b$$

In particular, this means that we have $\mathrm{tr}_{\underline{ }.B}^{y =_A x}(b) =_{B(x)} a$, and by the fact that functions preserve equality, $\mathrm{tr}_{\underline{ }.B}^{y =_A x}(\mathrm{tr}_{\underline{ }.B}^{x =_A y}(a)) =_{B(x)} \mathrm{tr}_{\underline{ }.B}^{y =_A x}(b)$, which by transitivity of propositional equality implies that $\mathrm{tr}_{\underline{ }.B}^{y =_A x}(\mathrm{tr}_{\underline{ }.B}^{x =_A y}(a)) =_{B(x)} a$. Similarly, we have $\mathrm{tr}_{\underline{ }.B}^{x =_A y}(\mathrm{tr}_{\underline{ }.B}^{y =_A x}(b)) =_{B(y)} \mathrm{tr}_{\underline{ }.B}^{x =_A y}(a)$, which by transitivity of propositional equality implies that $\mathrm{tr}_{\underline{ }.B}^{x =_A y}(\mathrm{tr}_{\underline{ }.B}^{y =_A x}(b)) =_{B(y)} b$. Thus, the transport functions $\mathrm{tr}_{\underline{ }.B}^{x =_A y}:B(x) \to B(y)$ and $\mathrm{tr}_{\underline{ }.B}^{y =_A x}:B(y) \to B(x)$ are [[inverse functions]] of each other, and thus both transport functions are [[bijections]], and could be written as $\mathrm{tr}_{\underline{ }.B}^{x =_A y}:B(x) \cong B(y)$ and $\mathrm{tr}_{\underline{ }.B}^{y =_A x}:B(y) \cong B(x)$ respectively.

## Related concepts

* [[equality]]

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