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)