Dear John (et
al.),
I think the inner product should be the same irrespective of the
field.
The inner product as dot product is relevant to the study of
quadratic
forms, conics, and orthogonal groups. For instance finding a
rational
point on the conic x^2 + y^2 + z^2 = 0 over CC is equivalent to
a
representation of zero of the quadratic form, but not if the form
is
replaced by x\bar{x} + y\bar{y} + z\bar{z}, and I don't think we
want
to break this correspondence between points on conics and
isotropic
vectors.
What is missing is a class for Hermitian modules, which would
have
a ring with involution as base ring. Such a class would be useful
for
studying Riemannian lattices such as the period lattice of an
abelian
variety, and unimodular groups. This class would allow one to
define
the Hermitian inner product x\bar{x} + y\bar{y} + z\bar{z} on a
ring
with inner product x |->
\bar{x}.
One choice of inner product is not sufficient to represent all such
forms,
and defining the default inner product to be the Hermitian product
for
complex fields would require making similar definition for
imaginary
quadratic fields, cyclotomic fields, and other CM fields for
consistency.
Even recognition of a CM field is nontrivial and this would lead
to
arbitrary choices. Worse, for other number fields, a Hermitian
product
would depend on a particular choice of
embedding.
So if this is entered into a trac, I think it should be for creation
of a class
of Hermitian forms together with a class of rings with involution
(from
which CC, quadratic rings, cyclotomic rings, and a class of CM
fields
would
inherit).
--
David
--~--~---------~--~----~------------~-------~--~----~
To post to this group, send email to sage-devel@googlegroups.com
To unsubscribe from this group, send email to [EMAIL PROTECTED]
For more options, visit this group at http://groups.google.com/group/sage-devel
URLs: http://sage.scipy.org/sage/ and http://modular.math.washington.edu/sage/
-~----------~----~----~----~------~----~------~--~---
