Hi All, 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.
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. 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. Cheers, David On Wednesday, May 6, 2015 at 10:21:58 PM UTC+2, Peter Bruin wrote: > > Hello, > > Travis Scrimshaw wrote: > > > On #15635, we are trying to decide whether we want non-associative > > algebras to be included in the catalog of algebras. > > For a general mathematical software system such as Sage, I think it is > overly restrictive to impose the rule that algebras are associative. > There are too many interesting non-associative algebras (such as Lie > algebras), or non-unital algebras (such as certain Hecke algebras) to > make associativity part of the definition of an algebra. > > Moreover, it is in my opinion an unfortunate choice of terminology if > non-associative algebras are not in general algebras. > > > The argument against including them is "most" people think of algebras > > as being associative (and maybe even unital), and as such, might > > surprise people when they come across the non-associativity in their > > computations. > > Anyone has the right to think of algebras as being associative, just > like many people think of vector spaces as being finite-dimensional, > say. This is a bit like books or papers using the convention that all > the algebras that one considers are assumed to be associative (or that > vector spaces are assumed to be finite-dimensional, etc.) However, it > feels wrong to elevate such conventions to definitions. > > > However, the community was at one point considering renaming magmatic > > algebras into algebras and having to specify the associative axiom > > explicitly. > > I would be in favour of this. > > When finite-dimensional algebras over a field were implemented in > #12141, we explicitly included non-associative algebras. In case the > user already knows that his algebra is associative, he can pass a flag > "assume_associative" (default: False) to avoid a lengthy computation to > check this. > > Peter > > -- 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.
