David, I don't think you understood my suggestion. We are talking
about groups A which are finitely-generated and torsion-free, so
abstractly isomorphic to Z^n, together with a suitable blinear
function on AxA taking values in Z or Q, and I wish to include R
-valued forms.
John
2008/4/28 David Joyner <[EMAIL PROTECTED]>:
>
>
> On Mon, Apr 28, 2008 at 2:02 PM, Robert Miller <[EMAIL PROTECTED]> wrote:
> >
> > SEP
> >
> > Implement Lattices over ZZ, with pairings into QQ or ZZ
> >
> > 0. (Maybe) Implement a FreeModule_ZZ_quotient class. This would also
> > allow for constructing abelian groups in the sort of canonical way
> > (something people have been asking for...)
> >
> > 1. Implement a LatticeModule class, which will inherit from
> > FreeModule_generic_pid: instances of LatticeModule will inherit an
> > underlying free ZZ module and make use of the optional
> > inner_product_matrix property.
> >
> > This shouldn't just be a free ZZ module with inner product matrix,
> > since we want specific functions for computing the dual lattice, etc.
> > which are more appropriate in a Lattice class.
> > a. Attributes will include
> > - is_euclidean (whether the inner product matrix is symmetric,
> > rather than skew-symmetric)
> > - is_integral (whether the image of the pairing is in ZZ or QQ)
> > - discriminant (the determinant of the matrix [<a_i,a_j>], where
> > {a_i} is a basis for the module). A lattice is nondegenerate if its
> > discriminant is nonvanishing.
> > b. Euclidean lattices also have the attributes:
> > - signature
> > - even/odd (whether <a,a> \in 2 ZZ for all a)
> > c. Use L.<a,b> for the pairing induced on module elements by the
> > inner product matrix.
> >
> > 2. Implement a SubLatticeModule class, which will inherit from
> > FreeModule_submodule_with_basis_pid and from Lattice, but override
> > L.<a,b> for the inner product.
> > a. Function is_primitive (a sublattice M of a lattice L is
> > primitive if L/M is a free ZZ-module)
> > b. Functions to get parent lattice and sublattice as LatticeModule
> > objects.
> >
> > 3. Implement a LatticeQuotient class (for now, just full sublattices,
> > i.e., finite quotients).
> > -- Inherit from FreeModule_ZZ_quotient?
> > -- Inherit from AbelianGroup?
>
> -1 is my vote on this. Infinite AbelianGroup instances are not
> completely implemented.
>
>
>
> > -- Inherit from nothing?
> > ( The question here is what the underlying structure for a
> > LatticeQuotient should actually be. The important thing is how will
> > someone want to access elements of a LatticeQuotient? )
> > a. Attributes will include a quadratic_form_matrix with entries
> > defined over QQ/ZZ or QQ/2ZZ
> >
> > 4. Create a dual_lattice function for integral euclidean lattices,
> > with optional "embedding" argument
> >
> > 5. Implement a dual_quotient function for integral euclidean lattices
> > which returns a LatticeQuotient.
> >
> > 6. Implement isomorphism tests for indefinite integral euclidean
> > lattices.
> >
> > -- Robert Miller, Andrey Novoseltsev, Ursula Whitcher
> >
> > >
> >
>
> >
>
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