Talk:Lie superalgebra

Error in the article

The picture representing the graded Jacobi identity (in the last section of the article) is wrong. More precisely $\sigma ^{2}$ hasn't been drown correctly. 83.87.165.29 (talk) 01:57, 6 November 2008 (UTC)

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I cannot understand the classification part

The sections about classification need (I believe) to be seriously edited. Unfortunately I do not feel knowledgeable enough to do it myself, but here are the places which I find very unclear:

SU(m/n) These are the superunitary Lie algebras which have invariants:

$z.{\overline {z}}+iw.{\overline {w}}$

There is no mention of invariants anywhere before this place and I do not understand what kind of invariants are meant, how the indicated expression determines them and what do they have to do with classification of Lie superalgebras.

The objects are interchangeably referred to as algebras and groups. It confuses me a lot. Besides, supergroups are much more esoteric objects than superalgebras - the latter are just plain algebraic structures while the former require the notion of supermanifold, etc.

This happens all along the whole section. In the end, I cannot figure out what exactly are those superalgebras listed. According to the previous material, they should be described as pairs of vector spaces with appropriate bilinear operations between them. None of such structures are ever mentioned in the classification section. — Preceding unsigned comment added by 109.172.129.12 (talk) 17:59, 28 August 2018 (UTC)

Grading of the product

In the current form of this article, the second sentence of the lede is the perfectly cryptic:

In most of these theories, the even elements of the superalgebra correspond to bosons and odd elements to fermions (but this is not always true; for example, the BRST supersymmetry is the other way around).

What this sentence is trying to say is that there are two possible gradings for the superbracket: For ordinary bosons and fermions, it is

$|[a,b]|=|a|+|b|$

while for BRST and ghosts, it is

$|[a,b]|=|a|+|b|-1$

I claim it is necessary to update this article to mention that different gradings are possible. I raised this issue on Talk:Graded ring so maybe continue conversation there?

BTW, I claim that it would also be useful to mention that there are sometimes some algebras which have not one, but two products: a "normal" associative product ab which may or may not be super-graded, and also a second product, the Lie bracket [a,b] which also may or may not be super-graded. Stating up front, in a prominent location, that there might be two products, would help with drawing distinctions between this article, and the Poisson superalgebra and Gerstenhaber algebra and other super-or-not algebras one hop away from here. I'm gonna meatball surgery this now. 67.198.37.16 (talk) 23:16, 24 May 2024 (UTC)