In RobinHankin/freealg: The Free Algebra

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

![](`r system.file("help/figures/freealg.png", package = "freealg")`){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 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 $[x,y]$ using R idiom .[x,y]. Thus:

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

It would have been nice to use \code{[x,y]} (that is, without the dot) but although this is syntactically consistent, it cannot be done in R.

It is possible to apply the dot construction .[x,y] to more complicated examples. Here I show that the Lie bracket is nonassociative:

x <- as.freealg("1+a")
y <- as.freealg("3 - 2a + 7b")
z <- as.freealg("2 - 5b + 7x")
.[x,.[y,z]]
.[.[x,y],z]
.[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]]

We can see this more directly using jacobi():

jacobi(x,y,z)

The dot S4 class and matrices {-}

The dot S4 class defines

setMethod("[", signature(x="dot",i="ANY",j="ANY"),function(x, i, j, drop){i*j - j*i})

(slightly simplified). The assumption is that * and - are defined appropriately, so the commutator makes sense for a wide range of classes. Special dispensation is made if the arguments are matrices. This is because in standard R idiom, A*B refers to Hadamard (elementwise) multiplication, which is an abomination; with this definition the Lie bracket is identically zero. The package therefore includes a matrix method:

setMethod("[", signature(x="dot",i="matrix",j="matrix"),function(x, i, j, drop){i%*%j-j%*%i})

So we have

A <- matrix(1:4,2,2)
B <- matrix(c(6,7,3,3),2,2)
C <- matrix(c(0,2,1,6),2,2)
.[A,B]

With this method, we can again verify Jacobi:

jacobi(A,B,C)

Actually there is typically a small numeric roundoff error:

rM <- function(n=4){matrix(rnorm(n^2),n,n)}
options(digits=3)
jacobi(rM(),rM(),rM())

Package dataset {-}

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

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

RobinHankin/freealg documentation built on Dec. 24, 2024, 3:16 a.m.