Talk:Supercommutative algebra

Definition with 2-torsion

I notice that R.e.b. made a correction to this article sometime ago regarding the square of odd elements. With the present definition we have

$2x^{2}=0$

for all odd x. If there is no 2-torsion this implies that

$x^{2}=0$.

However, it seems to be an crucial fact when dealing with commutative superalgebras that the odd elements square to zero. In all of the references I can find, authors either (implicitly or explicitly) assume that 2 is invertible or that there is no 2-torsion, or they explicitly require that $x^{2}=0$ for all odd x. I may edit the article to this effect, but I'm interested in what others have to say. -- Fropuff (talk) 22:03, 7 February 2008 (UTC)

Actually, the more I think about, modifying the definition of supercommutative to insist that $x^{2}=0$ for all odd x seems like the wrong thing to do. It would no longer be true that a superalgebra is commutative iff it is equal to its opposite, or to its supercenter, or iff the supercommutator vanished identically. I guess the correct thing to do is simply assume that 2 is invertible whenever necessary (or at least assume there is no 2-torsion). -- Fropuff (talk) 08:26, 8 February 2008 (UTC)

Yes; Bourbaki distinguishes between anticommutative and alternating algebras, and it seems to me (on the surface) to be exactly this distinction. —Quondum 06:31, 18 December 2016 (UTC)

Supercommutative vs. anticommutative

This article describes essentially what Bourbaki defines as an anticommutive algebra. There is a subtle distinction between between an anticommutative algebra (defined as Z-graded with the sign-changing property as given in the article with |x| interpreted as the Z-grading) and a supercommutative algebra (defined as Z₂-graded): an anticommutative algebra is necessarily also Z₂-graded and hence also a supercommutative algebra, but the converse seems to be false, because a Z₂-graded algebra need not be amenable to Z-grading. Thus, I would expect supercommutative algebras to exist that are not classifiable as anticommutative algebras. —Quondum 06:50, 18 December 2016 (UTC)

Currently anticommutative algebra is a redirect back to this article, and no explicit definition is given, for what that is, or how a Z-grading might arise. 67.198.37.16 (talk) 19:10, 24 May 2024 (UTC)

"Commutative superalgeba" listed at Redirects for discussion

The redirect Commutative superalgeba has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2025 September 28 § Commutative superalgeba until a consensus is reached. 1234qwer1234qwer4 05:28, 28 September 2025 (UTC)