I think it would be a good idea to have a subcategory of associative
> algebras
> (and inheritance of classes from an associative class). Morphisms need to
> know to check associativity.
>
The current heirarchy is to start Magmatic algebras (no assumptions), then
you add the axioms "associative" and "unital", giving you the categories
(with the current naming scheme):
MagmaticAlgebras
/ \
UnitalAlgebras AssociativeAlgebras
\ /
Algebras
> On the other hand, I was convinced years ago (by an argument of Bergman
> at Berkeley) that algebras should be unital. A non-unital "algebra" can
> be
> embedded in a universal unital algebra, such that the non-unital object is
> a
> two-sided ideal.
>
> In Magma, I find it awkward that the algebras include the ideals, and
> there is
> no class (type in Magma) to know whether an "algebra" admits a morphism
> from its base ring or is just a module. Consequently Ideals as objects of
> study
> is essentially absent in Magma. E.g. ideals over a number ring are hacked
> as elements of a (fractional) ideal group or ideal monoid, but do not
> support
> elements.
>
> Peter argues that certain Hecke algebras are non-unital, but should these
> not be viewed as ideals? I would be tempted to have associative algebras
> inherit from unital algebras, and to view non-unital "algebras" as
> ideals.
>
> The (important) case of Lie algebras is worth considering, whether it is
> too
> restrictive to assume that a general Lie algebra is an ideal over a
> universal
> algebra. This would conflict with the usual definition (a Lie algebra
> "over R"
> has no embedding of R). Moreover, should the * operation be the Lie
> bracket,
> or be reserved for its envelopping algebra? (Using [x,y] would be nice
> but
> would conflict with lists.) With * as the algebra operation, do the
> special
> properties of Lie algebras (e.g. the set-theoretic inclusion in an
> envelopping
> algebra is not a homomorphism) suggst they should be implemented outside
> the class and categorical hierarchy of algebras. To me, the main question
> is whether one needs morphisms between Lie algebras and associative or
> non-associative algebras, and (in practice) what code can be shared
> between
> the associated classes. It seems clear that Lie algebras need to inherit
> from
> general (non-unital non-associative) algebras, but the relation with other
> algebras needs some consideration.
>
> What I'm currently doing with #14901 (Lie algebras) is to have Lie
algebras inherit from Modules (to avoid the situation mentioned above),
implement some syntactic sugar for [x, y] notation but mostly to use a call
to a method bracket(), and let * automatically lift to the universal
enveloping algebra.
> The unital condition is important because it determines whether it is
> reasonable
> to coerce an element of the base ring, or dismiss such a request without
> solving a potentially hard question whether there exists a canonical
> embedding.
> For a unital algebra, we want to require an efficient canonical coercion
> from
> the base ring, whereas for ideals, either a not implemented error or a
> potentially
> expensive but correct algorithm would be acceptable and expected.
>
> In short, to the extent possible I would impose the unital condition as
> widely as
> possible, and develop the ideal theory of algebras in its own right.
>
>
>
We currently don't have any (non-associative) unital algebras AFAIK, so
there currently is no precedence. For Jordan algebras a (magmatic) algebra,
IIRC I just disabled the coercion from the base ring and just made it an
action, similar to what we do for modules. +1 to further developing ideals
in Sage.
I wouldn't like so much to denote something as "non-bla" (where "bla"
> can be associative, commutative, unital, finite, ...), when "non-bla"
> just means "not necessarily bla".
>
> So, please don't name them "category of non-associative algebras". I'd
> prefer to name them "category of magmatic algebras".
>
I agree, although we currently say an associative magmatic algebra simply
as an associative algebra (and in the doc say it is not necessarily unital).
>
> Sure, I once learnt that an "algebra" is not necessarily associative.
> Just think of Lie algebras. On the other hand, I wouldn't go as far as
> saying that our notion of Algebras() should change, because Algebras()
> in Sage has always included "associative unital". Changing that would be
> backwards incompatible.
>
Yes, it would be. However, there are many ways I can think of that could
possibly be used to transition between the conventions. Although that's a
bit off the topic of this thread, but is definitely something we should
discuss.
I am looking forward to the discussion about "skew fields". Which
> are not fields. Except perhaps in France (but there zero is positive,
> so people clearly cannot be trusted).
>
Division rings FTW!
--
You received this message because you are subscribed to the Google Groups
"sage-devel" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to sage-devel+unsubscr...@googlegroups.com.
To post to this group, send email to sage-devel@googlegroups.com.
Visit this group at http://groups.google.com/group/sage-devel.
For more options, visit https://groups.google.com/d/optout.
