The dot: commutators and the Jacobi identity in the clifford package

Robin K. S. Hankin

To cite the freealg package in publications, please use Hankin (2022). This short document introduces the dot object and shows how it can be used to work with commutators and verify the Jacobi identity. In the clifford package, we define \(\left[A,B\right] := (AB-BA)/2\). The factor of \(\frac{1}{2}\) is to consistentify the Lie bracket with the cross product \(A\times B\). The prototypical dot.Rmd is that of the freealg package (Hankin 2022). The dot object is a (trivial) S4 object of class dot:

`.` <- new("dot")

The point of the dot (!) is that it allows one to calculate the Lie bracket \([x,y]=(xy-yx)/2\) using R idiom .[x,y] in the clifford package. Thus:

(x <- 1 + 3*e(2))

## Element of a Clifford algebra, equal to
## + 1 + 3e_2

(y <- 5*e(3) - 7*e(1:3)) 

## Element of a Clifford algebra, equal to
## + 5e_3 - 7e_123

.[x,y]

## Element of a Clifford algebra, equal to
## + 15e_23

We see that these two clifford objects do not commute. It is possible to apply the dot construction .[x,y] to more complicated examples. Here I show that the Lie bracket is nonassociative:

z <- 3 - e(1:4)
.[x,.[y,z]]

## Element of a Clifford algebra, equal to
## - 21e_24

.[.[x,y],z]

## Element of a Clifford algebra, equal to
## the zero clifford element (0)

.[x,.[y,z]] == .[.[x,y],z]

## [1] FALSE

However, it does satisfy the Jacobi identity \(\left[x,\left[y,z\right]\right]+\left[y,\left[z,x\right]\right]+ \left[z,\left[x,y\right]\right]=0\):

.[x,.[y,z]] + .[y,.[z,x]] + .[z,.[x,y]]

## Element of a Clifford algebra, equal to
## the zero clifford element (0)

Bivectors

It is an interesting, useful, and nontrivial fact that the commutator of two bivectors is a bivector:

(a <- rcliff(d=9,g=2,include.fewer=FALSE))

## Element of a Clifford algebra, equal to
## - 8e_13 - 5e_23 + 9e_25 + 5e_17 - 7e_37 - 6e_49 - 4e_59 + 2e_69

(b <- rcliff(d=9,g=2,include.fewer=FALSE))

## Element of a Clifford algebra, equal to
## + 7e_14 - 9e_24 + 2e_16 + 3e_36 - 1e_67 - 3e_48 + 6e_49 + 8e_79

.[a,b]

## Element of a Clifford algebra, equal to
## + 11e_34 - 105e_45 - 19e_16 - 15e_26 + 9e_36 + 12e_46 + 83e_47 + 32e_57 -
## 27e_67 + 78e_19 - 54e_29 - 62e_39 - 2e_79 + 18e_89

grades(.[a,b])

##  [1] 2 2 2 2 2 2 2 2 2 2 2 2 2 2

Package dataset

Following lines create dot.rda, residing in the data/ directory of the package.

save(`.`,file="dot.rda")

References

Hankin, Robin K. S. 2022. “The Free Algebra in R.” arXiv. https://doi.org/10.48550/ARXIV.2211.04002.