dot-class {freegroup}

Class “dot”

Description

The dot object is defined in the freealg package, and imported here, so that idiom like .[x,y] returns the commutator, that is, x^-1 y^-1 xy.

Arguments

x

Object of any class

i, j

elements to commute

...

Further arguments to dot_error(), currently ignored

Value

Always returns an object of the same class as xy.

Author(s)

Robin K. S. Hankin

Examples

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

.[abc(1:6),"z"]

x <- rfree()
y <- rfree()
z <- rfree()

.[x,y] == -x-y+x+y   # should be TRUE

abelianize(.[x,y])

## Jacobi identity _not_ satisfied with this definition:
is.id(.[x,.[y,z]] + .[y,.[z,x]] + .[z,.[x,y]])

## But the Hall-Witt identity is:
all(is.id(.[.[x,-y],z]^y + .[.[y,-z],x]^z + .[.[z,-x],y]^x))