#16820: Implement ABCs for Lie algebras and finite dimensional given by
structure
cofficients
-------------------------------------+-------------------------------------
Reporter: tscrim | Owner: tscrim
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-6.7
Component: algebra | Resolution:
Keywords: lie algebras, | Merged in:
days64, sd67 | Reviewers:
Authors: Travis Scrimshaw | Work issues:
Report Upstream: N/A | Commit:
Branch: | 09281457c8cf34a075be3b72ae88296a75c54d93
public/lie_algebras/fd_structure_coeff-16820| Stopgaps:
Dependencies: |
-------------------------------------+-------------------------------------
Comment (by tscrim):
Replying to [comment:38 darij]:
> Replying to [comment:37 tscrim]:
> > comment:31
> > I would say the set of 0x0 matrices is the empty set, not the set
containing a unique element, the empty matrix.
>
> There are very good reasons for considering it a one-element set.
Matrices of size m \times n correspond to morphisms `R^n \to R^m`. How
many morphisms are there from `R^0` to `R^0` (that is, from 0 to 0) ? One
-- the zero morphism.
I would say that this is a degenerate case of the general equality, but
fair enough. More fun with corner cases that no one will likely construct
except for corner case testing... Will fix.
> > As for the oddity, I think we need a better/more uniform system for
equality for things that behave like 0. At the very least, this is an
issue with `CombinatorialFreeModuleElement` that deserves a separate
ticket:
> > {{{
> > sage: C = CombinatorialFreeModule(ZZ, ['a','b'])
> > sage: C.zero() == 0
> > True
> > sage: C.zero() == QQ(0)
> > False
> > }}}
>
> +1.
This is #18251.
> > comment:34
> > Yes, that is definitely bad. I'm thinking we should have `lift` always
return an element of the UEA and state that explicitly. For the lifting to
the defining associative algebra, perhaps call that `lift_associative`?
>
> That's a good idea.
>
> The Lie subalgebra of an associative algebra generated by a subset will,
so far, have no UEA, since we don't compute its full ground set. But now
you made me wonder if we really need the Lie subalgebra of an associative
algebra generated by a subset... If we don't compute its ground set, and
only let the user compute "inside" it, then why don't we just use the
whole associative algebra as a Lie algebra? The generators seem to be
useless...
Then you'd have gl,,n,, = sl,,n,, and there will be Lie subalgebras
eventually implemented. More generally, the UEA of the (Lie) subalgebra
will be a (assoc) subalgebra of the UEA of the ambient Lie algebra.
However, we will need general support for subalgebras, which I don't think
#11111 provides...
--
Ticket URL: <http://trac.sagemath.org/ticket/16820#comment:39>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
--
You received this message because you are subscribed to the Google Groups
"sage-trac" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to [email protected].
To post to this group, send email to [email protected].
Visit this group at http://groups.google.com/group/sage-trac.
For more options, visit https://groups.google.com/d/optout.
