dot: Class "dot"

In RobinHankin/onion: Octonions and Quaternions

Class “dot”

Description

\loadmathjax

The dot object is defined so that idiom like .[x,y] returns the commutator, that is, xy-yx or the Lie bracket \mjseqn[x,y]. It would have been nice to use [x,y] (that is, without the dot) but although this is syntactically consistent, it cannot be done in R.

The “meat” of the package is:

setClass("dot", slots = c(ignore='numeric'))
`.` <- new("dot")
setMethod("[",signature(x="dot",i="ANY",j="ANY"),function(x,i,j,drop){i*j-j*i})

The package code includes other bits and pieces such as informative error messages for idiom such as .[]. The package defines a matrix method for the dot object. This is because “*” returns (incorrectly, in my view) the elementwise product, not the matrix product.

The Jacobi identity, satisfied by any associative algebra, is

\mjdeqn \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]] = 0

Function ad() returns the adjoint operator. The adjoint vignette provides details and examples of the adjoint operator.

The dot object is generated by running script inst/dot.Rmd, which includes some further discussion and technical documentation, and creates file dot.rda which resides in the data/ directory.

Value

Always returns an object of the same class as xy

Slots

ignore:

Object of class "numeric", just a formal placeholder

Methods

[

signature(x = "dot", i = "ANY", j = "ANY"): ...

[

signature(x = "dot", i = "ANY", j = "missing"): ...

[

signature(x = "dot", i = "function", j = "function"): ...

[

signature(x = "dot", i = "matrix", j = "matrix"): ...

[

signature(x = "dot", i = "missing", j = "ANY"): ...

[

signature(x = "dot", i = "missing", j = "missing"): ...

Author(s)

Robin K. S. Hankin

See Also

adjoint

Examples

x <- rquat()
y <- rquat()
z <- rquat()
.[x,y]


.[x,.[y,z]] + .[y,.[z,x]] + .[z,.[x,y]]  # Jacobi, expanded

RobinHankin/onion documentation built on April 20, 2024, 2:05 p.m.