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A real structure on a complex vector space is an antilinear map which is an involution.
Equivalently this is a real vector space and an isomorphism of with its complexification.
Here is the eigenspace of for eigenvalue 1 and is the eigenspace for eigenvalue -1.
(real structures on the complex line)
The standard real structure on the complex line is complex conjugation :
The fixed locus of this real structure are the real numbers under their defining embedding into the complex numbers:
with
Restricting the canonical complex-bilinear form
along this inclusion yields the canonical -bilinear form on :
Another real structure on is:
Now the fixed locus is the imaginary numbers
with
Restricting the canonical complex bilinear form (1) along this inclusion now yields minus the canonical real bilinear form
(real structures on the complex plane)
On the complex plane we have first of all the real structures inherited from (Ex. ).
and
In addition, there is for instance
whose fixed locus is
the restriction to which of the canonical complex bilinear form on the complex plane
is the hyperbolic form on the real plane:
Yet another real structure is
whose fixed locus is
the restriction to which of the canonical complex bilinear form (2) is twice the hyperbolic form on
(real structure as dagger-self-dual Hermitian structure)
Let be a finite-dimensional complex vector space equipped with both
a non-degenerate sesquilinear (Hermitian) inner product ,
a non-degenerate symmetric bilinear inner product
such that
then this induces a real structure on .
Moreover, for another finite-dimensional complex vector space equipped with a compatible pair of such structures, then the linear maps
which preserve both structures also preserve that real structure, hence come from -linear maps of underlying real vector spaces.
To start with, we note some generalities:
the non-degenerate sesquilinear form is equivalently given by an antilinear map
to the dual linear space, via
the non-degenerate bilinear form is equivalently given by a complex linear map , via
(where the second argument means evaluation).
In bra-ket notation this is suggestively written as:
For example, the Hermitian adjoint of any linear map
is
Moreover, if is any orthonormal basis of with respect to , then the coevaluation map which exhibits as the linear dual space to may be written as
and hence equivalently as
which we use at the very end below.
Now regarding the concrete proof:
The composite
is an antilinear endomorphism of , 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
because then
We now show that this condition (3) is equivalent to the assumption that is the coevaluation map.
Namely, with
we have
which by the zig-zag identity for & equals
On the other hand, the coevaluation which exhibits as a self-duality is clearly
which by the above discussion equals
This being equal to above is clearly equivalent to (3).
More generally in spectral geometry (via spectral triples) and KR-theory:
Last revised on November 9, 2023 at 06:06:48. See the history of this page for a list of all contributions to it.