Talk:Graded Lie algebra

Todo

• Outline a few applications of graded Lie algebras in the Other uses sense.
• Maybe motivate graded Lie algebras (in the "usual" sense -- that of the article) by discussing (linking to?) the deformation equation.

Silly rabbit 22:07, 18 June 2006 (UTC)

How to handle the two conflicting uses

Really there should be a better way to disambiguate between the two uses of the term "graded Lie algebra". Both usages are of comparable importance, even within the same fields of research. It's tempting to start inventing new words for things, but that seems a bit un-Wikipedian. Silly rabbit 22:07, 18 June 2006 (UTC)

The graded Lie algebras of this article are essentially the same as Lie superalgebras; you might want to merge much of your work on this article into the Lie superalgebra article. This would solve the problem you mention of having two uses for the term. R.e.b. 03:02, 19 June 2006 (UTC)

Although the two notions are similar, they are not the same. For example, from geometry the tensor product of the ring of differential forms with the tangent bundle forms a graded Lie algebra. But it would be disingenuous to call it a Lie superalgebra because it actually carries a pair of gradings, only one of which is over Z/2Z. Thus the underlying Lie superalgebra contains strictly less information about the algebraic structure.

A more basic example is the exterior algebra over a vector space. (Well, this isn't a Lie *anything* but it still helps to illustrate how supersymmetry isn't appropriate for a wide variety of structures.) Yes, this is a superalgebra -- technically -- if all you care about is the parity of the form degree. But most uses of the Grassmann algebra require you to know more than just the parity.

That said, I do see some merits of merging this in with Lie superalgebra. But I *would* have to change the definition of Lie superalgebra, and that seems unjustifiably revisionist. Silly rabbit 04:01, 19 June 2006 (UTC)

My recommendation is to merge (on the grounds that the difference between these two sorts of grading is not really enough for a separate article, and there is a danger of ending up with two parallel articles on essentially the same topic), without changing the definition of Lie superalgebra, and have a section explaining that the Z/2Z grading can often be lifted to a Z grading. If you decide to keep the articles separate then they could do with some dablinks at the top of each pointing to the other, together with an expanation at the top of both explaining exactly what the difference between them is. R.e.b. 05:15, 19 June 2006 (UTC)

I think I have a mutually satisfactory solution. I'll expand the Lie superalgebra article to include a section on graded Lie superalgebras. This may not be a widely accepted terminology, since most of the work I know of on graded Lie algebras (in my sense) predates 1970. At any rate, it's better this way since the "super" prefix unambiguously suggests the Z/2Z gradation. This article can then focus exclusively on the other uses of the term, with a dablink over to super. Thanks, Silly rabbit 13:33, 19 June 2006 (UTC)

Or, on second thought, it may be more uniform to define both graded Lie algebra (in the "Other sense") and graded Lie superalgebra in this article (but using the "super" prefix as per my last remark). I think I'll set it up that way first, and then deal with a merger if it still seems to be warranted. Silly rabbit 13:49, 19 June 2006 (UTC)

[1]has a discussion by Deligne of two of the different meanings of "graded superalgebra" R.e.b. 14:27, 19 June 2006 (UTC)

Thanks for the link. This illustrates a very good reason for keeping the articles separate. Specifically, what if we aren't working on a super category, but happen to come across something which a supersymmetrist would call a super object? If we follow Deligne's convention, then we're essentially forced to revise our point of view and so say: "Clearly we were mistaken and the category was super all along. Here are some auxiliary Grassmann variables which we'll never use." Bernstein's convention seems to be nearer the mark since, in the classical picture, the cohomological degree and super degree are going to be coupled ab initio. (See my example above of form-valued derivations.) Silly rabbit 15:24, 19 June 2006 (UTC)

I would be very wary of using a convention different from Deligne's; I have made this mistake before, and found that he always turns out to be right in the end. He spent a lot of time trying to figure out the best conventions. R.e.b. 17:18, 19 June 2006 (UTC)

Good, so we are in agreement. Keep graded Lie algebras separate from Lie superalgebras. ;-) Silly rabbit 23:46, 19 June 2006 (UTC)

Allow me to explain a little more thoroughly. Deligne says that there are some super-like objects living in the classical world -- let's call these objects "dupers" for reasons to become clear in a moment. For example, graded Lie superalgebras arise in purely classical situations (derivations on graded algebras, deformations of complex manifolds, homotopy theory, Lie algebra cohomology, etc.). So graded Lie superalgebras are duper (within this classical setting). The purpose of his appendix is then to describe how to generalize these to the super domain, to obtain (if you'll pardon the terminology) super duper structures. There are two common ways to do this. One is Deligne's own approach: keep the super and the duper separate (possibly until the very end). The only problem with this is that one needs a well-defined theory of duper structures in order for this to even get started. This article should fill that gap (at least for the graded Lie algebra case). Silly rabbit 00:10, 20 June 2006 (UTC)

The other approach is that of Bernstein, which applies equally well to duper objects and super duper objects. Hence, it would seem that a sort of Bernsteinian point of view is needed for this article. Silly rabbit

Perhaps the graded Lie algebra article could discuss ALL the possible meanings of "graded Lie algebra" and the relations between them, instead of trying to select the "correct" one. R.e.b. 01:26, 20 June 2006 (UTC)

To hell with it. Someone else can write the damn article. Let the supersymmetrists appropriate whatever they want to. Silly rabbit 03:50, 20 June 2006 (UTC)

Oh, just forget I said that. I'm feeling a bit frustrated at the moment. :-( Silly rabbit 06:21, 20 June 2006 (UTC)

blank page

blanking doesn't seem like a good idea, folks. i am going to revert to last version, by Silly rabbit, before the blank. Mct mht 05:21, 20 June 2006 (UTC)

the article seems far from being in sufficiently poor state as to provoke one to take such drastic action anyway. :) Mct mht 05:39, 20 June 2006 (UTC)

Graded Lie Algebra Definition

I've never contributed to one of these discussions before, but would like to comment on this one.

Published definitions of graded Lie algebra are hard to find. It's very common to assume the grading group is the integers (or nonnegative integers) and the bracket is graded and satisfies super-skew-symmetry and Jacobi identities.

Like you, most of the articles I've seen on graded Lie algebras were published before 1970. It would be great if you could include a book reference for the definition, since I've never seen one. Perhaps topology books cover it.

The postings on graded Lie algebras and Lie superalgebras could be collected in a single one on Lie color algebras, which are also called color Lie superalgebras, generalized Lie algebras, or epsilon-Lie algebras. These are graded over an abelian group G and equipped with a bracket satisfying an ε-skew symmetry and Jacobi identity. Lie superalgebras and graded Lie algebras are just specializations. This is made precise following the definition included at the end of these comments.

An advantage of the Lie color algebra terminology is that you don't have to specify the grading group or bicharacter when setting conventions and making definitions. One works within the category of G-graded vector spaces with ε-commutative tensor products defined on homogeneous components.

Lie superalgebras and graded Lie superalgebras have consistently turned out to be the ones arising from classical constructions. This is covered in the note by Deligne, discussed earlier. So far, general bicharacters are not usually needed, except in some limited uses in Fermi-Bose statistical mechanics. But Lie color algebras are of interest to algebraists, and future applications may arise. Recall the resurgence of Hopf algebras when quantum groups emerged.

Here's the definition. A map ε:G×G→k is called a skew-symmetric bicharacter on an abelian group G if it satisfies (1), (2), and (3) below, for any f,g,h∈G.

1.ε(f,g+h)=ε(f,g)ε(f,h), 2. ε(g+h,f)=ε(g,f)ε(h,f), & 3.ε(g,h)ε(h,g)=1

A skew-symmetric bicharacter ε satisfies ε(g,g)=±1 for each g∈G. Set G_{±}={g∈G:ε(g,g)=±1} so that G_+ is a subgroup with [G:G_+]≤2 and G_-=G\G_+.

For example, if G is a cyclic group then either ε is identically 1 or G=Z/2mZ for some m∈N and ε(a,b)=(-1)^{ab}. When G=Z/2Z and ε(a+2Z,b+2Z)=(-1)^{ab} for any a,b∈Z, then G_+={0+2Z}, G_-={1+2Z}, and ε is called the super bicharacter.

A (G,ε)-Lie color algebra is a G-graded vector space L equipped with a graded bilinear map [,]:L×L→L, called the bracket of L, which satisfies the following for any x,y,z∈L with x∈L_f, y∈L_g, z∈L_h, and f,g,h∈G.

[x,y]=-ε(f,g)[y,x] (ε-skew-symmetry) & ε(h,f)[x,[y,z]]+ε(g,h)[z,[x,y]]+ε(f,g)[y,[z,x]]=0 (ε-Jacobi identity)

The Lie color algebra can be decomposed into L=L_+ ⊕ L_- with L_{±}=⊕_{g∈G_{±}}L_{g}. This is analogous to the "super duper" gradings.

If A is a G-graded algebra, then it becomes a (G,ε)-Lie color algebra when equipped with the bracket given by [a,b] = ab - ε(f,g)ba for all x∈L_f, y∈L_g, and f,g∈G.

Lie superalgebras are (Z/2Z,ε)-Lie color algebras where ε is the super bicharacter defined above. Graded Lie algebras are defined analogously, with G=Z.