What is a scalar (under rotation) 1-chain ?

[ianhoolihan]

What is a "scalar (under rotation) 1-chain"?

Hi all,

I am trying to make sense of a paper involving differenital geometry and Lie algebras. Here's the part I am confused about:

...Lie product map $\mu: V\times V \rightarrow V$ that satisfies the Jacobi identity,
$$\mu(x,\mu(y,z))=\mu(\mu(x,y),z)+\mu(y,\mu(x,z)).$$
This is usually written as a cyclic sum, a form that, in the case at hand, obscures its content. To clarify the latter, take as an example the case where [\itex]x[/itex] is a Lorentz group generator, $J_{\mu\nu}$, and $y,z$ are other generators carrying Lorentz indices, say $Y_\rho, Z_\sigma$ respectively. Suppose $\mu(y,z)=\mu(Y_\rho,Z_\sigma)=W$. Substituting this into the l.h.s above, one finds that the Jacobi identity requires that the transformation properties of $W$ under the Lorentz group are derived solely from those of $Y_\rho,Z_\sigma$, i.e. in this case $W$ ought to transform as a second-rank covariant tensor. Another way of saying this is that $\mu$ itself is a Lorentz scalar, an observation that we use later on.

Now things begin with finding the cohomology of a Lie algebra. The galilean algebra is taken as an example, and the Lie product is given in terms of differential forms:

$$\mu=\frac{1}{2}\epsilon_{ab}^c\Pi^a\Pi^b\otimes J_c +\epsilon_{ab}^c\Pi^a\Pi^{\bar{b}}\otimes K_c$$

where barred indices refer to boosts. The paper then goes on to say:

By an argument based on the observation made [above]...we conclude that only scalar (under rotations) cochains need to be considered.

I do not see how this applies. I assume it somehow helps to simplify "the most general scalar 1-cochain":

...the most general scalar 1-cochain is given by
$$\phi = \alpha_1 \phi_{JJ} +\alpha_2 \phi_{KJ} +\alpha_3 \phi_{JK} +\alpha_4 \phi_{KK}$$
with $\phi_{JJ}=\Pi^a\otimes J_a$ etc.

Now if someone could clarify this all to me, that'd be great. More specifically:

1. does $\phi_{JJ}=\Pi^a\otimes J_a$ mean $\phi_{JJ}=\Pi^a\otimes J_a=\Pi^1 J_1 + \Pi^1 J_2+ ... +\Pi^3 J_3$ i.e. with nine terms (remember the unbarred indices are rotation only, so three generators)?

2. Are the $\alpha_i$ real coefficients, or arrays? I.e. I would have thought
$$\phi=\phi^A_B\Pi^B \otimes T_A = ... = \phi ^a_b\Pi^b\otimes T_a+\phi ^a_{\bar{b}}\Pi^{\bar{b}}\otimes T_a+\phi ^{\bar{a}}_b\Pi^b\otimes T_{\bar{a}}+\phi ^{\bar{a}}_{\bar{b}}\Pi^{\bar{b}}\otimes T_{\bar{a}}$$
where I have let $A=\{\{a\},\{\bar{a}\}\}$. This is the closest I can get to the given expression, but here I have $\alpha_1 \phi_{JJ} = \alpha_1 \Pi^a \otimes T_a = \phi ^a_b\Pi^b\otimes T_a$, which doesn't seem to work. I am assuming the the $\alpha_i$ are simple scalars, which somehow is to do with $\phi$ being a "scalar 1-cochain".

I have a few more questions, but that will suffice for now -- hopefully this gets the ball rolling, and I can work them out myself, once I understand what's going on here.

Cheers,

Ianhoolihan

 

[ianhoolihan]

Any help, or even incomplete hints in the right direction? Cheers