% -*- mode: noweb; noweb-default-code-mode: R-mode; -*-
\documentclass[nojss]{jss}
\usepackage{amssymb}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% declarations for jss.cls %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% just as usual
\author{Robin K. S. Hankin\\University of Stirling}
\title{The adjoint operator in the freealg package}
%\VignetteIndexEntry{adjoint}
%% for pretty printing and a nice hypersummary also set:
\Plainauthor{Robin K. S. Hankin}
\Plaintitle{The adjoint operator}
\Shorttitle{The adjoint operator}
%% an abstract and keywords
\Abstract{ In this very short document I discuss the adjoint operator
\code{ad()} and illustrate some of its properties.
}
\Keywords{Adjoint operator, free algebra}
\Plainkeywords{Adjoint operator, free algebra}
%% publication information
%% NOTE: This needs to filled out ONLY IF THE PAPER WAS ACCEPTED.
%% If it was not (yet) accepted, leave them commented.
%% \Volume{13}
%% \Issue{9}
%% \Month{September}
%% \Year{2004}
%% \Submitdate{2004-09-29}
%% \Acceptdate{2004-09-29}
%% \Repository{https://github.com/RobinHankin/freegroup} %% this line for Tragula
%% The address of (at least) one author should be given
%% in the following format:
\Address{
Robin K. S. Hankin\\%\orcid{https://orcid.org/0000-0001-5982-0415}\\
University of Stirling\\
E-mail: \email{hankin.robin@gmail.com}\hfill\includegraphics[width=1in]{\Sexpr{system.file("help/figures/freealg.png",package="freealg")}}
}
%% It is also possible to add a telephone and fax number
%% before the e-mail in the following format:
%% Telephone: +43/1/31336-5053
%% Fax: +43/1/31336-734
%% for those who use Sweave please include the following line (with % symbols):
%% need no \usepackage{Sweave.sty}
%% end of declarations %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\SweaveOpts{}
\begin{document}
<>=
library("freealg")
@
\hfill\includegraphics[width=1in]{\Sexpr{system.file("help/figures/freealg.png",package="freealg")}}
<<>>=
ad
@
\subsection*{The adjoint operator: definition}
Given an associative algebra ${\mathcal A}$ and $X,Y\in{\mathcal A}$,
we define the \emph{Lie Bracket} $[X,Y]$ as $XY-YX$. In the {\tt
freealg} package this is implemented with the{\tt .[]} construction:
<<>>=
X <- as.freealg("X")
Y <- as.freealg("Y")
.[X,Y]
@
\subsection*{The Jacobi identity}
The Lie bracket is bilinear and satisfies the Jacobi condition:
<<>>=
X <- rfalg(3)
Y <- rfalg(3)
Z <- rfalg(3)
X # Y and Z are similar objects
.[X,Y] # quite complicated
.[X,.[Y,Z]] + .[Y,.[Z,X]] + .[Z,.[X,Y]] # Zero by Jacobi
@
\subsection*{The adjoint map: definition}
Now we define the adjoint as follows. Given a Lie algebra
$\mathfrak{g}$, and $X\in{\mathcal A}$, we define a linear map
$\mathrm{ad}_X\colon\mathfrak{g}\longrightarrow\mathfrak{g}$ with
\[
\mathrm{ad}_X(Y)=\left[X,Y\right]
\]
In the {\tt freealg} package, this is implemented using the {\tt ad()} function:
<<>>=
ad(X)
@
See how function {\tt ad()} returns a {\em function}. We can play with this:
<<>>=
f <- ad(X)
f(Y)
f(Y) == X*Y-Y*X
@
The first thing to note is that $\mathrm{ad}_X$ is NOT a Lie
homomorphism, for any particular (non-constant) value of $X$. If
$\phi$ is a Lie homomorphism then $\phi([x,y]) =
\left[\phi(x),\phi(y)\right]$. There is no reason to expect the
adjoint to be a Lie homomorphism, but it does not hurt to check:
<<>>=
phi <- ad(Z)
phi(.[X,Y]) == .[phi(X),phi(Y)]
@
With this definition, it is easy to calculate, say,
$[Z,[Z,[Z,[Z,[Z,X]]]]]$:
<