%%%% 2005Athens/slides.tex
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\title{Lie-rings}
\author{David Pierce}
\date{Athens, 2005, August}

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\newtheorem*{thmA}{Theorem A}
\newtheorem*{thmB}{Theorem B}
%\newtheorem*{thm}{Theorem}

\newcommand{\End}[1]{\operatorname{End}(#1)}
\newcommand{\Der}[1]{\operatorname{Der}(#1)}
\newcommand{\Mult}[1]{\operatorname{Mult}(#1)}
\newcommand{\Hom}[2]{\operatorname{Hom}(#1,#2)}
\newcommand{\invo}[1]{\overset{\bullet}{#1}}

\newcommand{\myop}[1]{\mathsf{#1}} % my operation
\newcommand{\mult}{\myop m}
\newcommand{\genprdt}[3]{#1(#2,#3)}
\newcommand{\prdt}[2]{\genprdt{\mult}{#1}{#2}}

\newcommand{\genlmlt}[1]{\lambda^{#1}}
\newcommand{\lmlt}{\genlmlt{\mult}}
\newcommand{\genlmby}[2]{\genlmlt{#1}(#2)}
\newcommand{\lmby}[1]{\lmlt(#1)}

\newcommand{\cmpt}{\myop c}
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\newcommand{\Th}[1]{\operatorname{Th}(#1)}

\newcommand{\included}{\subseteq}

\newcommand{\defn}[1]{\textbf{#1}}

\begin{document}\thispagestyle{empty}
  
    \begin{center}
  \textbf{\Huge Lie-rings}

{\huge David Pierce}

{\huge \emph{Athens, summer, 2005}}
    \end{center}

   \begin{huge}
(From Plato's door, according to Tzetzes [12th c.]:
\begin{center}\vspace{-0.75\baselineskip} 
  \begin{greektext}\sf
  Mhde`is >agewm'etrhtos e>is'itw mou t`hn st'eghn.
\end{greektext}
\end{center}\vspace{-0.5\baselineskip} 
Why did the AMS choose a version of this as its motto?)
    \end{huge}

\mbox{}

\begin{huge}
      $E$ is an abelian group, in signature $\{+,-,0\}$.

$\End E$ is its \emph{abelian group} of endomorphisms.

$\Mult E$ is the abelian group of \defn{multiplications} of $E$:
  bi-additive maps
\begin{equation*}
  (x,y)\mapsto\prdt xy:E\times E\to E.
\end{equation*}
If $\mult\in\Mult E$, then $(E,\mult)$ is a \defn{ring;} also,
$\Mult E$ contains $\invo{\mult}$, the \defn{converse} of $\mult$,
given by
\begin{equation*}
  \genprdt{\invo{\mult}}xy=\prdt yx.
\end{equation*}
$\Mult{\End E}$ contains \defn{composition,} $\cmpt$, and so
\begin{equation*}
  \langle\cmpt,\invo{\cmpt}\rangle\leqslant\Mult{\End E};
\end{equation*}
in particular, the \defn{bracket,} $\cmpt-\invo{\cmpt}$ or $\brkt$, is in
$\Mult{\End E}$.
\end{huge}

\newpage

\begin{huge}
%$(\End E, \cmpt)$ is an \emph{associative ring;}

%$(\End E,\brkt)$ is a \emph{Lie-ring.}


A ring $(E,\mult)$ is:
\begin{itemize}
  \item
\defn{associative,} if $\prdt{\prdt xy}z=\prdt x{\prdt yz}$;
\item
\defn{commutative,} if associative, and $\mult-\invo{\mult}=0$;
\item
a \defn{Lie-ring,} if $\mult+\invo{\mult}=0$ and
\begin{equation*}
  \prdt{\prdt xy}z=\prdt x{\prdt yz}-\prdt y{\prdt xz}
\end{equation*}
(the \defn{Jacobi identity}).
\end{itemize}
It is obvious that $(\End E,\cmpt)$ is associative;

to see that $(\End
E,\brkt)$ is a Lie-ring requires a computation.


\begin{thmA}
  Among non-associative rings, Lie-rings are the most ``natural'' in a
  precise sense.
\end{thmA}

\begin{thmB}
  There are Lie-rings $(E,\mult)$ with $t$ in $\End E$ such that
  $\Th{E,\mult,t}$ is model-complete and $\omega$-stable.
\end{thmB}
\end{huge}

\newpage
\begin{huge}
There is an isomorphism
\begin{equation*}
  \mult\mapsto\lmlt:\Mult E\to \Hom E{\End E},
\end{equation*}
\begin{itemize}
  \item
where $\lmlt$ is $x\mapsto\lmby x:E\to \End E$,
\item
where $\lmby x$ is
$y\mapsto\prdt xy:E\to E$.
\end{itemize}
We can now recast the Jacobi identity:
\begin{equation*}
  \prdt{\prdt xy}z=\prdt x{\prdt yz}-\prdt y{\prdt xz}
\end{equation*}
becomes
\begin{align*}
  \lmby{\prdt xy}&=\lmby x\circ\lmby y-\lmby y\circ\lmby x\\
%&=\genprdt{\cmpt}{\lmby x}{\lmby y}-\genprdt{\invo{\cmpt}}{\lmby x}{\lmby y}\\
&=\brkt(\lmby x,\lmby y),
\end{align*}
This says $\lmlt$ is a \emph{ring-}homomorphism from $(E,\mult)$ to
$(\End E,\brkt)$.

    \begin{thmA}
      Let $(p,q)\in\mathbb Z\times\mathbb Z$, and let $\mult$ be
      $p\cmpt-q\invo{\cmpt}$.  Call a ring $(E,*)$ an
      \emph{$\mult$-ring} if $\genlmlt*$ is a ring-homomorphism from
      $(E,*)$ to $(\End E,\mult)$.
The following are
      equivalent:
      \begin{itemize}
	\item
$(\End E,\mult)$ is an $\mult$-ring for all $E$.
\item
$\mult$ is $\cmpt$ or $\brkt$ or $0$.
\end{itemize}
    \end{thmA}
\end{huge}

\newpage
\begin{huge}
A \defn{derivation} of a ring $(E,\mult)$ is an element $D$ of $\End
E$ such that 
    \begin{equation*}
      D(\prdt xy)=\prdt{Dx}y+\prdt x{Dy};
    \end{equation*}
---rearranged,
\begin{equation*}
      D(\prdt xy)-\prdt x{Dy}=\prdt{Dx}y;
\end{equation*}
---with $y$ removed,
\begin{equation*}
  D\circ\lmby x-\lmby x\circ D=\lmby{Dx},
\end{equation*}
that is,
\begin{equation*}
  \brkt(D,\lmby x)=\lmby{Dx};
\end{equation*}
---with $x$ removed,
\begin{equation*}
  \genlmby{\brkt}D\circ\lmlt=\lmlt\circ D,
\end{equation*}
that is, the following commutes:
\begin{equation*}
  \begin{CD}
    E@>{\lmlt}>>\End E\\
@VDVV @VV{\genlmby{\brkt}D}V\\
E@>>{\lmlt}>\End E
  \end{CD}
\end{equation*}
The derivations of $(E,\mult)$ compose a subgroup 
\begin{equation*}
  \Der{E,\mult}
\end{equation*}
of $\End E$; this subgroup is closed under $\brkt$.
\end{huge}

\newpage
\begin{huge}
  For any abelian group $E$, there is a commutative diagram
  \begin{equation*}
\qquad\qquad    \xymatrix@=5ex{
(\End E,\brkt)\ar[r]^(.4){\genlmlt{\brkt}} \ar[dr]_(.4){\genlmlt{\brkt}}& 
(\Der{\End E,\cmpt},\brkt)\ar[d]^{\included}\\
&(\End{\End E},\brkt)
}
  \end{equation*}
Suppose now $(E,\mult)$ is a Lie-ring.  Then $\lmlt$ is the
\defn{adjoint representation:} there is a commutative diagram
  \begin{equation*}
    \xymatrix@=5ex{
(E,\mult)\ar[r]^(.4){\lmlt} \ar[dr]_(.4){\lmlt}& 
(\Der{E,\mult},\brkt)\ar[d]^{\included}\\
&(\End E,\brkt)
}\qquad\qquad\qquad\qquad
  \end{equation*}
Combining gives
\begin{equation*}
  \xymatrix@=5ex{
(E,\mult) \ar[dr]_(.4){\lmlt} \ar@/^2ex/[drr]^{\genlmlt{\brkt}\circ\lmlt} &&\\
&(\End E,\brkt)\ar[r]_(.4){\genlmlt{\brkt}} & 
(\Der{\End E,\cmpt},\brkt)
}
\end{equation*}
This means that
every $D$ in $E$ is now a derivation $f\mapsto Df$ of $(\End E,\cmpt)$
by the rule 
\begin{equation*}
  (Df)x=\prdt D{fx}-f(\prdt Dx).
\end{equation*}

\end{huge}

\newpage
\begin{huge}
Still $(E,\mult)$ is a Lie-ring, mapping into $(\Der{\End
E,\cmpt},\brkt)$.

Let $t\in \End E$.  Call the structure
\begin{equation*}
  (E,\mult,t)
\end{equation*}
a \defn{vector Lie-ring} if:
\begin{itemize}
  \item
$\{Dt:D\in E\}$ is the universe of a sub\emph{-field} $K$ of $(\End
E,\cmpt)$, and
\item
$E$ acts on $K$ as a \text{vector-space} (over $K$) of derivations:
\begin{equation*}
  (gD)f=g(Df).
\end{equation*}
\end{itemize}
\begin{thmB}
\mbox{}
\begin{itemize}
  \item
The class of vector Lie-rings is elementary.
\item
  If $n<\omega$, then the theory of vector
  Lie-rings of dimension $n$ is companionable.
\item
the model-companion of this theory becomes complete and $\omega$-stable when
  characteristic $0$ is specified.
\end{itemize}
\end{thmB}

(Related results are being worked out
independently by Martin Bays.)
\end{huge}
\end{document}