It's clear what your algebra is: over the base field K=Q(i) it's the quaternion algebra with parameters 2,5.
I think that sage-nt would be a better forum for this than sage-devel. Ask to join (at http://groups.google.co.uk/group/sage-nt). John 2009/7/9 Leonard Foret <a314s...@gmail.com>: > > The problem is about finding co-compact lattices in SL(2, C) by using > quaternion algebras. > > The example we are working out now is based on the Quaternion algebra > over Q(i) defined by the quadratic extension Q(i)[X]/(X^2 - 2) and > additional (non-commutative) relation s^2 = 5. > > We need the algebra to be over Q(i) - that's important for the > construction of the lattice. > > The lattice is constructed by collecting the elements of the algebra > of reduced norm 1 with and with coefficients in the ring of Gaussian > integers. The reduced norm is defined by using the left multiplication > by an element on the algebra. > > It is possible to figure out how long I should compute before getting > all the generators of the lattice, but the idea is to avoid the use of > the corresponding theory for this example, and try to work out an > approach for any particular case. > > Finding the elements of norm one is related to solving Diophantine > equations over Z. Do you know if there is a software for finding > generators of groups (like the one we are dealing with in our > example)? > > On Jul 7, 9:41 pm, William Stein <wst...@gmail.com> wrote: >> On Sat, Jul 4, 2009 at 8:39 PM, Leonard Foret<a314s...@gmail.com> wrote: >> >> > Hello all, >> >> > This is my first time in sage-devel. I have a project with a >> > professor til the end of August to construct cocompact/uniform >> > lattices on SL2(Z[i]) basically by quaternion algebras. >> >> What is a "lattice on SL2(Z[i])"? >> >> > I figure that >> > since I'm writing code using python and sage, I might as well do it >> > right and incorporate it into sage. >> >> Yes, definitely. >> >> > The two contributions I could >> > make to sage would be 1) to redo a polished version of the code which >> > computes the generators of elements of reduced norm one within a >> > certain radius for an explicit example (hopefully extend it to a >> > general skew field/quaternion algebra) >> >> Good. >> >> > and 2) functionality for >> > quaternion algebras over the field Q(i) rather than Q. >> >> What functionality do you want to add? >> >> > What I have right now are some python/sage code which looks at the >> > quaternion algebra over Q(i) given by the field extension Q(i)(sqrt >> > (2)) over Q(i) and the added relation j^2 = 5, (similar to the >> > construction that's already implemented over Q). >> >> As a non-commutative ring, isn't that precisely exactly the same thing as >> the quaternion algebra >> >> sage: R.<i,j,k> = QuaternionAlgebra(-1,5) >> sage: R >> Quaternion Algebra (-1, 5) with base ring Rational Field >> >> already in Sage? Is the point just that you're viewing it differently >> as a quadratic >> extension of Q(i)? >> >> > The resulting algebra >> > produces a lattice which will be cocompact/uniform and I've >> > implemented the following algorithms: >> >> Which lattice? In what space? >> >> > 1) compute the elements of reduced norm one within a ball. >> >> elements in what? >> >> > 2) compute left multiplication by an element >> >> left multiplication on what? >> >> > 3) compute a norm for these elements (that is, by a norm for the >> > matrix computed in 2) >> >> > When the elements of reduced norm one are considered with >> > multiplication, they form a group and the following algorithm is >> > applicable: >> > 4) compute generators for the elements of reduced norm one. >> >> > The trouble with algorithm 4 and 1, is that it's by complete brute >> > force to the point where the algorithm works but I don't know how long >> > it would take to find all of them (for 1 there's infinitely many). >> >> Since you seem to be doing this for exactly the 1 single ring >> Q(i)(sqrt(2)), shouldn't you know? >> >> >> >> > As for adding functionality to Quaternion Algebra I would like to work >> > on the following: >> > 1) extend .is_division_algebra() to the base field Q(i). >> > 2) .is_anisotropic() >> > 3) and any others. >> >> > I'm a beginning graduate student at the Florida International >> > University and am working closely with a professor there. If anyone >> > is interested or can offer any advice (books, articles to read, ideas >> > for the algorithms, etc), it would be well received and I'll implement >> > them immediately. >> >> > Thanks in advance! >> > Leonard Foret >> >> -- >> William Stein >> Associate Professor of Mathematics >> University of Washingtonhttp://wstein.org > > > --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-devel@googlegroups.com To unsubscribe from this group, send email to sage-devel-unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
