Talk:Graded Lie algebra

message: 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)
replies: text: 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)
thread_title:  blank page 

message: 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. User:66.168.73.198 21:39, 14 July 2006‎
:Sounds great. Wish this content was added at the time. My pet beef is that all this grading stuff is not yet rationalized across multiple articles on multiple topics. "Classical" supersymmetry is "supercommutative", and thus has elements that square to zero, and supposedly this had something to do with spinors because Pauli exclusion principle. By contrast, Clifford algebras have elements that square to -1 and actually allow actual spinors to be constructed, instead of the hand-wavey supersymmetry things. On the other hand, the supersymmetry things now show up in deformation theory but also in statistical mechanics. Beats the pants offa me, but I'm seeing people writing about BRST quantization and claiming this solves generic stochastic differential equations or something like that. Because BRST is finite to all perturbative orders. Go figure. So talking about grading correctly remains an issue. 67.198.37.16 (talk) 00:23, 25 May 2024 (UTC)
thread_title:  Graded Lie Algebra Definition