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

In freegroup: The Free Group

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

![](`r system.file("help/figures/freegroup.png", package = "freegroup")`){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 $[x,y]=x^{-1}y^{-1}xy$ using R idiom .[x,y]. Thus:

x <- as.free("x")
y <- as.free("y")
.[x, y]

We see that x and y do not commute (if they did, .[x,y] would be the identity). It is possible to apply the dot construction .[x,y] to more complicated examples. Here I show that the Lie bracket is nonassociative:

set.seed(0)
z <- as.free("z")
(LHS <- .[x, .[y, z]])
(RHS <- .[.[x, y], z])
LHS == RHS

We can have some fun verifying elementary properties. First, observe that the abelianization of the commutator is the identity:

abelianize(.[x, y])

With this definition, the Jacobi identity does not hold:

is.id(.[x, .[y, z]] + .[y, .[z, x]] + .[z, .[x, y]])

[the Jacobi identity does hold for $[x,y] = xy-yx$, but here we have $[x,y]=x^{-1}y^{-1}xy$]. However, the Hall-Witt identity does hold:

all(is.id(.[.[x, -y], z]^y + .[.[y, -z], x]^z + .[.[z, -x], y]^x))

Package dataset {-}

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

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

freegroup documentation built on July 15, 2026, 5:07 p.m.