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

In weyl: The Weyl Algebra

set.seed(0)
knitr::opts_chunk$set(echo = TRUE)
library("weyl")

![](`r system.file("help/figures/weyl.png", package = "weyl")`){width=10%}

This short document introduces the dot object and shows how it can be used to work with commutators and verify the Jacobi identity. The prototypical dot.Rmd is that of the freealg package. 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 $[x,y]=xy-yx$ using R idiom .[x,y]. Thus:

d
x
.[x,d]

We see that x and d do not commute and indeed $x\partial-\partial x=1$. It is possible to apply the dot construction .[x,y] to more complicated examples. Here I show that the Lie bracket is nonassociative:

x <- rweyl(1)
y <- rweyl(2)
z <- rweyl(1)
.[x,.[y,z]] == .[.[x,y],z]

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]]

Package dataset {-}

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

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

weyl documentation built on April 12, 2025, 1:15 a.m.