Talk:Alternating algebra

Anticommutative

Is "anticommutative ring" = "graded-commutative ring"? The answer is yes if "anticommutative" = "skew-commutative". -- Taku (talk) 22:57, 21 December 2016 (UTC)

Can you give more context to why you're posting this here?

I think it is worth pointing out that anticommutative/skew-commutative seems to assume a Z₂-grading, whereas graded-commutative seems to assume a Z-grading, the latter being a more stringent criterion. (Every Z-grading induces/implies a Z₂-grading, but not the other way around.) Under this characterization, "anticommutative ring" ≠ "graded-commutative ring", because the latter will be a proper subset of the former. See my comment at Talk:Supercommutative_algebra#Supercommutative_vs._anticommutative. However, if "graded-commutative" assumes only that the grading monoid has Z₂ as a quotient monoid (and hence implies a Z₂-grading), then "anticommutative ring" = "graded-commutative ring". That article is not clear about it, however. —Quondum 23:44, 21 December 2016 (UTC)

I asked it because I think "anticommuattive ring" here should really be "graded-commutative algebra". I do agree we need to distinguish Z-graded algebra and Z/2-graded ones, basically because Z is not Z/2. Strictly speakingly, graded rings with different indexing sets need to be distinguished, especially the questions about the definitions. In practice, context makes it clear what is meant. I think, at least in mathematics, unqualified "graded ring" means either N or Z-graded rings and not Z/2-graded. While anticommutative algebra considers Z/2-algebra, graded-commutative ring considers a Z-graded algebra. So the latter seems like a better target. -- Taku (talk) 02:52, 22 December 2016 (UTC)

I agree with you (i.e. the suggested change to the link target), though there are so many subtleties. Notice though that a ring is not necessarily an algebra, and an alternating algebra is not necessarily a ring (not necessarily associative, but then, neither is a graded ring). I wish terminology was more consistent ... —Quondum 03:07, 22 December 2016 (UTC)

Simpler definition?

If someone has a reference that gives an intuitively simpler definition of an alternating algebra, this would be nice to add to the article. For example, something like "An alternating algebra is a Z-graded algebra in which the homogeneous elements of degree 1 square to zero and which generate the entire algebra." I'm not sure that this is even correct (e.g. it may fall apart for non-associative algebras), but it should give the idea. —Quondum 22:03, 15 July 2017 (UTC)

This characterization should simplify by removing the reference to grading:

• An alternating algebra A is an algebra for which there exists a generating set S ⊆ A such that x ∈ S implies x² = 0.
• A skew-commutative algebra A is an algebra for which there exists a generating set S ⊆ A such that x,y ∈ S implies xy + yx = 0.

This should work generally, i.e. with nonunital nonassociative algebras. (Note to self: Bourbaki gives Proposition 13, which is very close to, but not identical my first characterization above; I'll add this at some point.) —Quondum 15:17, 30 July 2017 (UTC)

Note on possible merge

Serge Lang uses the name "alternating algebra" as a direct synonym for "exterior algebra". I have not seen anything to suggest that Bourbaki's definition of an alternating algebra is not equivalent to that of an exterior algebra. Unless there is a difference, it would seem appropriate to merge this article into Exterior algebra. —Quondum 15:32, 12 February 2021 (UTC)