Exploring a Lie 3-Algebra: Does a=-g Mean a*x^1=-g*x^2?

[Digs]

Hi!

Im messing around with a Lie 3-Algebra at the moment(Im not sure how widespread these are. They obey similar rules to a Lie Algebra).

I have a bracket that looks like this:
[x1,x2,x3]=c*x0+a*x1+g*x2+h*x3, with c,a,g,h, some structure constants.

I also have the relation a=-g.
Am I allowed to then say [x1,x2,x3]=c*x0+a*x2-g*x2+h*x3
or [x1,x2,x3]=c*x0=h*x3?

I suppose my question amounts to does a=-g also mean a*x^1=-g*x^2?

thanks so much

 

[HallsofIvy]

Hi!

Im messing around with a Lie 3-Algebra at the moment(Im not sure how widespread these are. They obey similar rules to a Lie Algebra).

I have a bracket that looks like this:
[x1,x2,x3]=c*x0+a*x1+g*x2+h*x3, with c,a,g,h, some structure constants.

I also have the relation a=-g.
Am I allowed to then say [x1,x2,x3]=c*x0+a*x2-g*x2+h*x3

Why would you ask? you have replaced x1 by x2 and g by -g. What right do you have to do that?

or [x1,x2,x3]=c*x0=h*x3?

I suppose my question amounts to does a=-g also mean a*x^1=-g*x^2?

No, of course not. Knowing that a=-g tells you nothing about ax1. Knowing that a= -g tells you that ax1= -gx1, not -gx2.

thanks so much

 

[Digs]

in hindsight that was a silly question
thanks for your help though

 

[Digs]

Bumping this for some more structure constant help.
I'm attempting to figure out what a particular Lie 3-algebra is by classifying it's structure constants, which are constrained by what's called the Fundamental Identity(Which is the 3-analog of the Jacobi Identity for normal Lie algebras).

I'm convinced I'm not doing something right as I've actually never computed an algebra this way before(I've only previously studied matrix lie algebras): could someone take a look at my equations?
I've unwound the identity into a system of equations that looks like

1. a=b
2. cd=ea
3. cf=eg
4.cb=eh
5. -g=j
6.ck=la
7.cm=lg
8.cj=lh
9.k=-f
10.ld=ek
11.lf=em
12.lb=ej

where {a,b,c,d,e,f,g,h,j,k,l,m} are all 4 indexed structure constants. I was proceeding before trying to divide by various things, but now I've realized that's probably not a legal move as these are indexed objects. Any advice would be wonderful!