Find Contraction of $$M_{ad}M^{be}M^{cf} f^d_{bc}f^a_{ef}$$

[JorisL]

Hi,

I was wondering if there exists some software to find the following contraction in an easy way
$$M_{ad}M^{be}M^{cf} f^d_{bc}f^a_{ef}$$

Here the 4x4 metric M has a block diagonal form
$$
\begin{pmatrix}
e^\phi & 0\\
0 & e^{-\phi/3}N
\end{pmatrix}$$

With $N$ a symmetric 3x3 matrix of determinant 1.
The $f^a_{bc}$ are structure constants of some algebra (I have to evaluate this for 13 distinct algebras).
A lot of them are zero, most of the others are 1.

I keep missing terms (due to antisymmetry $f^a_{bc} = -f^a_{cb}$) when doing any but the most trivial examples by hand.

Thanks,

Joris

 

[jedishrfu]

The best I can think of is Mathematica:

https://www.wolfram.com/mathematica/new-in-9/built-in-symbolic-tensors/

 

[JorisL]

Thanks.

Luckily there are some easy algebras that I can use to test the code.

 

[JorisL]

Well I looked at the tensor capabilities but so far it proved easier to hard code the stuff using tables. (More important, a lot faster to implement than to learn and use this new part of Mathematica)

I added a PDF showing how I did it. When I find the time I might make it more generally applicable. (more than 4 dimensions, less constraints on allowed algebras etc)