#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: |
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Comment (by darij):
> Sorry for the delay in getting back to you; traveling plus clearing off
my built up work.
Don't worry. I'm again swamped in work, so the sorriness is mutual.
Replying to [comment:37 tscrim]:
> comment:29
> I think the issue is that there isn't enough checking of coefficients
and maybe a default assumption somewhere that algebras are unital? Perhaps
it's also lifting up in some fashion? I'll take a look at that this week.
Forget about this part -- I've since noticed that you define sl(2) as a
Lie subalgebra of an associative algebra generated by a subset, and since
you don't have methods which compute such a thing exactly, there is no
surprise that it accepts input too liberally.
> 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.
> 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.
> 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...
--
Ticket URL: <http://trac.sagemath.org/ticket/16820#comment:38>
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