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geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
(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
such that this is symmetric sesquilinear, in that:
$h$ is complex-linear in the first argument;
$h(w,v) = \left(h(v,w) \right)^\ast$ for all $v,w \in V$
where $(-)^\ast$ denotes complex conjugation.
A Hermitian form is positive definite (often assumed by default) if for all $v \in V$
$h(v,v) \geq 0$
$h(v,v) = 0 \phantom{AA} \Leftrightarrow \phantom{AA} v = 0$.
A complex vector space $(V,J)$ equipped with a (positive definite) Hermitian form $h$ is called a (positive definite) Hermitian space.
A positive-definite and complete Hermitian vector space is called a Hilbert space.
(basic properties of Hermitian forms)
Let $((V,J),h)$ be a positive definite Hermitian space (def. ). Then
the real part of the Hermitian form
is a Riemannian metric, hence a symmetric positive-definite real-bilinear form
the imaginary part of the Hermitian form
is a symplectic form, hence a non-degenerate skew-symmetric real-bilinear form
hence
The two components are related by
Finally
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
These are called the Hermitian metrics.
The positive-definiteness of $g$ is immediate from that of $h$. The symmetry of $g$ follows from the symmetric sesquilinearity of $h$:
That $h$ is invariant under $J$ follows from its sesquilinarity
and this immediately implies the corresponding invariance of $g$ and $\omega$.
Analogously it follows that $\omega$ is skew symmetric:
and the relation between the two components:
as well as
(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:
$\omega \in \wedge^2 V^\ast$ is a linear Kähler structure (def. );
$g \in V \otimes V \to \mathbb{R}$ is a Hermitian metric (2)
where $\omega$ and $g$ are related by (1)
While a non-degenerate inner product $(-\vert-)$ on a finite-dimensional real vector space $V$ is equivalently a linear isomorphism to its dual vector space
the analogous statement for Hermitian complex inner products $\langle - \vert - \rangle$ fails, since the corresponding maps
are now complex 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 space?.
Another way to regard this situation is to observe that complex anti-linear involutions $\mathscr{H} \leftrightarrow \mathscr{H}^\ast$ on non-degenerate Hermitian spaces $\mathscr{H}$ are equivalently $(\mathbb{Z}/2 \curvearrowright \mathbb{C})$-module structures on the direct sum $\mathscr{H} \oplus \mathscr{H}^\ast$, regarded in the topos of $\mathbb{Z}/2$-sets, for $(\mathbb{Z}/2 \curvearrowright \mathbb{C})$ the ring object given by the complex numbers equipped with their involution by complex conjugation:
Notice that these $(\mathbb{Z}/2 \curvearrowright \mathbb{C})$-modules arising from (non-degenerate, finite-dimensional) Hermitian vector spaces this way happen to carry also a complex structure, hence a compatible module-structure by the actual complex numbers (i.e. equipped with the trivial involution), given by $\underline{\mathrm{i}}$.
Using this, one may identify:
the space of linear operators on a Hermitian vector space as the equalizer of this imaginary rotation on the tensor square of the $(\mathbb{Z}/2 \curvearrowright \mathbb{C})$-module with the braiding and the identity,
among these that of hermitian operators as the further fixed locus of the involution action:
Among original articles:
See also:
and see the references at Hilbert space.
Last revised on October 21, 2022 at 07:32:08. See the history of this page for a list of all contributions to it.