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\begin{titlepage}
\begin{center}{\Huge \textbf{\textsf{LieAlgDB}}}\\[1cm]
\hypersetup{pdftitle=\textsf{LieAlgDB}}
\markright{\scriptsize \mbox{}\hfill \textsf{LieAlgDB} \hfill\mbox{}}
{\Large \textbf{A database of Lie algebras}}\\[1cm]
{Version 1.0}\\[1cm]
{October 2006}\\[1cm]
\mbox{}\\[2cm]
{\large \textbf{ Willem de Graaf    }}\\
{\large \textbf{ Csaba Schneider    }}\\
\hypersetup{pdfauthor= Willem de Graaf    ;  Csaba Schneider    }
\end{center}\vfill

\mbox{}\\
{\mbox{}\\
\small \noindent \textbf{ Willem de Graaf    } --- Email: \href{mailto://degraaf@science.unitn.it}{\texttt{degraaf@science.unitn.it}}\\
 --- Homepage: \href{http://www.science.unitn.it/~degraaf/}{\texttt{http://www.science.unitn.it/\~{}degraaf/}}\\
 --- Address: \begin{minipage}[t]{8cm}\noindent
 Dipartimento di Matematica\\
 Via Sommarive 14\\
 I-38050 Povo (Trento)\\
 Italy\\
 \end{minipage}
}\\
{\mbox{}\\
\small \noindent \textbf{ Csaba Schneider    } --- Email: \href{mailto://csaba.schneider@sztaki.hu}{\texttt{csaba.schneider@sztaki.hu}}\\
 --- Homepage: \href{http://www.sztaki.hu/~schneider}{\texttt{http://www.sztaki.hu/\~{}schneider}}\\
 --- Address: \begin{minipage}[t]{8cm}\noindent
 Informatics Research Laboratory\\
 Computer and Automation Research Institute\\
 1518 Budapest Pf. 63.\\
 Hungary \end{minipage}
}\\
\end{titlepage}

\newpage\setcounter{page}{2}
{\small 
\section*{Abstract}
\logpage{[ 0, 0, 1 ]}
 This package provides access to the the classification of the following
families of Lie algebras: 
\begin{itemize}
\item solvable Lie algebras of dimension at most 4; 
\item nilpotent Lie algebras of dimension at most 5, and those of dimension~6 over a
field of characteristic different from 2; 
\item  non-solvable Lie algebras of dimension at most 6 over finite fields; 
\item  simple Lie algebras of dimension at most 6 over finite fields and those of
dimension at most 9 over \mbox{\texttt{GF(2)}}. 
\end{itemize}
 }\\[1cm]
{\small 
\section*{Copyright}
\logpage{[ 0, 0, 2 ]}
 {\copyright} 2006 Willem de Graaf and Csaba Schneider }\\[1cm]
{\small 
\section*{Acknowledgements}
\logpage{[ 0, 0, 3 ]}
 We are grateful to Andrea Caranti, Marco Costantini, Bettina Eick, Helmut
Strade, Michael Vaughan-Lee. Without their help, interest, and encouragement,
this package would not have been completed. }\\[1cm]
\newpage

\def\contentsname{Contents\logpage{[ 0, 0, 4 ]}}

\tableofcontents
\newpage

 
\chapter{\textcolor{Chapter }{Introduction}}\logpage{[ 1, 0, 0 ]}
{
  This is the manual for the \textsf{GAP} package \textsf{LieAlgDB}, for accessing and working with several classifications of Lie algebras. 

 In the mathematics literature many classifications of Lie algebras of various
types have been published (we refer to the bibliography for a few examples).
However, working with these classifications from paper is not always easy.
This package aims at making a few classifications of small dimensional Lie
algebras that have appeared in recent years more accessible. For each
classification that is contained in the package, functions are provided that
construct Lie algebras from that classification inside \textsf{GAP}. This allows the user to obtain easy access to the often rather complicated
data contained in a classification, and to directly interface the Lie algebras
to the functionality for Lie algebras which is already contained in \textsf{GAP}. 

 The package contains the following classifications: 
\begin{itemize}
\item  non-solvable Lie algebras over finite fields up to dimension 6 (from \cite{Strade}), 
\item  nilpotent Lie algebras of dimension upto 9 over \mbox{\texttt{F=GF(2)}}, of dimension upto 7 over \mbox{\texttt{F=GF(3)}} or \mbox{\texttt{F=GF(5)}} (from \cite{sch}),
\item  simple Lie algebras of dimensions between 7 and 9 over \mbox{\texttt{F=GF(2)}} (from \cite{VL}), 
\item  the classification of solvable Lie algebras of dimensions 2,3, and 4 (from \cite{wdg05}), 
\item  the classification of nilpotent Lie algebras of dimensions 5 and 6 (from \cite{wdg07}). 
\end{itemize}
 This manual is structured as follows. The next chapter contains a description
of the main functions of the package. Subsequently, the appendices contain
descriptions of the classifications used in the package. These are of course
also contained in published papers, but for the convenience of the user they
have been added here. }

 
\chapter{\textcolor{Chapter }{The families of Lie algebras included in the database}}\label{Intro}
\logpage{[ 2, 0, 0 ]}
{
  
\section{\textcolor{Chapter }{Non-solvable Lie algebras}}\label{nonsolv}
\logpage{[ 2, 1, 0 ]}
{
  The package contains the list of non-solvable Lie algebras over finite fields
up to dimension 6. The classification follows the one in \cite{Strade}. 

\subsection{\textcolor{Chapter }{NonSolvableLieAlgebra}}
\logpage{[ 2, 1, 1 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{NonSolvableLieAlgebra( F, pars )\index{NonSolvableLieAlgebra@\texttt{NonSolvableLieAlgebra}}
\label{NonSolvableLieAlgebra}
}\hfill{\scriptsize (method)}}\\


 \mbox{\texttt{F}} is a finite field, \mbox{\texttt{pars}} is a list of parameters with length between 1 and 4. The first entry of \mbox{\texttt{pars}} is the dimension of the algebra, and the possible additional entries of \mbox{\texttt{pars}} describe the algebra if there are more algebras with dimension \mbox{\texttt{pars[1]}}. }

 The possible values of pars are as follows. 
\subsection{\textcolor{Chapter }{Dimension 1 and 2}}\logpage{[ 2, 1, 2 ]}
{
  There are no non-solvable Lie algebras with dimension less than 3, and so if \mbox{\texttt{pars[1]}} is less than 3 then \mbox{\texttt{NonSolvableLieAlgebra}} returns an error message. }

 
\subsection{\textcolor{Chapter }{Dimension 3}}\logpage{[ 2, 1, 3 ]}
{
  There is just 1 non-solvable Lie algebra over an arbitrary finite field \mbox{\texttt{F}} (see Section~\ref{appdim3}) which is returned by \mbox{\texttt{NonSolvableLieAlgebra( F, [3] )}}. }

 
\subsection{\textcolor{Chapter }{Dimension 4}}\logpage{[ 2, 1, 4 ]}
{
  If \mbox{\texttt{F}} has odd characteristic then there is a unique non-solvable Lie algebra with
dimension 4 over \mbox{\texttt{F}} and this algebra is returned by \mbox{\texttt{NonSolvableLieAlgebra( F, [4] )}}. If \mbox{\texttt{F}} has characteristic 2, then there are two distinct Lie algebras and they are
returned by \mbox{\texttt{NonSolvableLieAlgebra( F, [4,i] )}} for \mbox{\texttt{i=1, 2}}. See Section~\ref{appdim4} for a description of the algebras. }

 
\subsection{\textcolor{Chapter }{Dimension 5}}\logpage{[ 2, 1, 5 ]}
{
  If \mbox{\texttt{F}} has characteristic 2 then there are 5 isomorphism classes of non-solvable Lie
algebras over \mbox{\texttt{F}} and they are described in Sections~\ref{appdim5char2}. The possible values of \mbox{\texttt{pars}} are as follows. 
\begin{itemize}
\item  \mbox{\texttt{[5,1]}}: the Lie algebra in \ref{appdim5char2}(1).
\item  \mbox{\texttt{[5,2,a]}}: \mbox{\texttt{a=0, 1}}; the Lie algebras in \ref{appdim5char2}(2).
\item  \mbox{\texttt{[5,3,a]}}: \mbox{\texttt{a=0, 1}}; the Lie algebras in \ref{appdim5char2}(3).
\end{itemize}
 If the characteristic of \mbox{\texttt{F}} is odd, then the list of Lie algebras is as follows (see Section~\ref{appdim5charodd}). 
\begin{itemize}
\item  \mbox{\texttt{[5,1,a]}}: \mbox{\texttt{a=1, 0}}; the Lie algebras that occur in \ref{appdim5charodd}(1). 
\item  \mbox{\texttt{[5,2]}}: the Lie algebra in \ref{appdim5charodd}(2). 
\item  \mbox{\texttt{[5,3]}}: this algebra only exists if the characteristic of \mbox{\texttt{F}} is 3 or 5. In the former case the algebra is the one in \ref{appdim5charodd}(3), while in the latter it is in \ref{appdim5charodd}(4). 
\end{itemize}
 }

 
\subsection{\textcolor{Chapter }{Dimension 6}}\logpage{[ 2, 1, 6 ]}
{
  The 6-dimensional non-solvable Lie algebras are described in Section~\ref{appdim6}. If \mbox{\texttt{F}} has characteristic 2, then the possible values of \mbox{\texttt{pars}} is as follows. 
\begin{itemize}
\item  \mbox{\texttt{[6,1]}}: the Lie algebra in \ref{appdim6char2}(1). 
\item  \mbox{\texttt{[6,2]}}: the Lie algebra in \ref{appdim6char2}(2). 
\item  \mbox{\texttt{[6,3,a]}}: \mbox{\texttt{a=0, 1}}; the two Lie algebras \ref{appdim6char2}(3). 
\item  \mbox{\texttt{[6,4,a]}}: \mbox{\texttt{a=0, 1, 2, 3}} or \mbox{\texttt{a}} is a field element. In this case \mbox{\texttt{NonSolvableLieAlgebras}} returns one of the Lie algebras in \ref{appdim6char2}(4). If \mbox{\texttt{a=0, 1, 2, 3}} then the Lie algebra corresponding to the \mbox{\texttt{a+1}}-th matrix of \ref{appdim6char2}(4) is returned. If \mbox{\texttt{a}} is a field element, then a Lie algebra is returned which corresponds to the
4th matrix in \ref{appdim6char2}(4).
\item  \mbox{\texttt{[6,5]}}: the Lie algebra in \ref{appdim6char2}(5). 
\item \mbox{\texttt{[6,6,1], [6,6,2], [6,6,3,a], [6,6,4,a]}}: \mbox{\texttt{a}} is a field element; the Lie algebras in \ref{appdim6char2}(6). The third and fourth entries of \mbox{\texttt{pars}} determine the isomorphism type of the radical as a solvable Lie algebra. More
precisely, if the third argument \mbox{\texttt{pars[3]}} is 1 or 2 then the radical is isomorphic to \mbox{\texttt{SolvableLieAlgebra( F,[3,pars[3]] )}}. If the third argument \mbox{\texttt{pars[3]}} is 3 or 4 then the radical is isomorphic to \mbox{\texttt{SolvableLieAlgebra( F,[3,pars[3],pars[4]] )}}; see \texttt{SolvableLieAlgebra} (\ref{SolvableLieAlgebra}). 
\item  \mbox{\texttt{[6,7]}}: the Lie algebra in \ref{appdim6char2}(7). 
\item  \mbox{\texttt{[6,8]}}: the Lie algebra in \ref{appdim6char2}(8).
\end{itemize}
 If the characteristic of \mbox{\texttt{F}} is odd, then the possible values of \mbox{\texttt{pars}} are the following (see Sections~\ref{appdim6charodd}, \ref{appdim6char3}, and \ref{appdim6char5}). 
\begin{itemize}
\item  \mbox{\texttt{[6,1]}}: the Lie algebra in \ref{appdim6charodd}(1).
\item  \mbox{\texttt{[6,2]}}: the Lie algebra in \ref{appdim6charodd}(2).
\item \mbox{\texttt{[6,3,1], [6,3,2], [6,3,3,a], [6,3,4,a]}}: \mbox{\texttt{a}} is a field element; the Lie algebras in \ref{appdim6charodd}(3). The third and fourth entries of \mbox{\texttt{pars}} determine the isomorphism type of the radical as a solvable Lie algebra. More
precisely, if the third argument \mbox{\texttt{pars[3]}} is 1 or 2 then the radical is isomorphic to \mbox{\texttt{SolvableLieAlgebra( F,[3,pars[3]] )}}. If the third argument \mbox{\texttt{pars[3]}} is 3 or 4 then the radical is isomorphic to \mbox{\texttt{SolvableLieAlgebra( F,[3,pars[3],pars[4]] )}}; see \texttt{SolvableLieAlgebra} (\ref{SolvableLieAlgebra}). 
\item \mbox{\texttt{[6,4]}}: the Lie algebra in \ref{appdim6charodd}(4).
\item \mbox{\texttt{[6,5]}}: the Lie algebra in \ref{appdim6charodd}(5).
\item \mbox{\texttt{[6,6]}}: the Lie algebra in \ref{appdim6charodd}(6).
\item \mbox{\texttt{[6,7]}}: the Lie algebra in \ref{appdim6charodd}(7).
\end{itemize}
 If the characteristic is 3 or 5 then there are additional families. In
characteristic 3, these families are as follows. 
\begin{itemize}
\item  \mbox{\texttt{[6,8,a]}}: \mbox{\texttt{a}} is a field element; returns one of the Lie algebras in \ref{appdim6char3}(1).
\item \mbox{\texttt{[6,9]}}: the Lie algebra in \ref{appdim6char3}(2).
\item \mbox{\texttt{[6,10]}}: the Lie algebra in \ref{appdim6char3}(3).
\item \mbox{\texttt{[6,11,a]}}: \mbox{\texttt{a=0, 1}}; one of the two Lie algebras in \ref{appdim6char3}(4). 
\item \mbox{\texttt{[6,12]}}: the first Lie algebra in \ref{appdim6char3}(5).
\item \mbox{\texttt{[6,13]}}: the second Lie algebra \ref{appdim6char3}(5).
\end{itemize}
 If the characteristic is 5, then the additional Lie algebras are the
following. 
\begin{itemize}
\item \mbox{\texttt{[6,8]}}: the Lie algebra in \ref{appdim6char5}(1).
\item \mbox{\texttt{[6,9]}}: the Lie algebra in \ref{appdim6char5}(2).
\end{itemize}
 }

 

\subsection{\textcolor{Chapter }{NonSolvableLieAlgebras}}
\logpage{[ 2, 1, 7 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{NonSolvableLieAlgebras( F, dim )\index{NonSolvableLieAlgebras@\texttt{NonSolvableLieAlgebras}}
\label{NonSolvableLieAlgebras}
}\hfill{\scriptsize (method)}}\\


 Here \mbox{\texttt{F}} is a finite field, and \mbox{\texttt{dim}} is at least 6. The list of non-solvable Lie algebras over \mbox{\texttt{F}} of dimension \mbox{\texttt{dim}} is returned. }

 

\subsection{\textcolor{Chapter }{SimpleLieAlgebras}}
\logpage{[ 2, 1, 8 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{SimpleLieAlgebras( F, dim )\index{SimpleLieAlgebras@\texttt{SimpleLieAlgebras}}
\label{SimpleLieAlgebras}
}\hfill{\scriptsize (method)}}\\


 Here \mbox{\texttt{F}} is a finite field, and \mbox{\texttt{dim}} is either an integer not greater than 6, or, if \mbox{\texttt{F=GF(2)}}, then \mbox{\texttt{dim}} is not greater than 9. The output is the list of simple Lie algebras over \mbox{\texttt{F}} of dimension \mbox{\texttt{dim}}. If \mbox{\texttt{dim}} is at most 6, then the classification by Strade~\cite{Strade} is used. If \mbox{\texttt{F=GF(2)}} and \mbox{\texttt{dim}} is between 7 and 9, then the Lie algebras in \cite{VL} are returned. }

 }

 
\section{\textcolor{Chapter }{Solvable and nilpotent Lie algebras}}\label{solvnilp}
\logpage{[ 2, 2, 0 ]}
{
  The package contains the classification of solvable Lie algebras of dimensions
2,3, and 4 (taken from \cite{wdg05}), and the classification of nilpotent Lie algebras of dimensions 5 and 6
(from \cite{wdg07}). The classification of nilpotent Lie algebras of dimension 6 is only
complete for base fields of characteristic not 2. The classifications are
complemented by a function for identifying a given Lie algebra as a member of
the list. This function also returns an explicit isomorphism. In Appendix \ref{listdescr} the list is given of the multiplication tables of the solvable and nilpotent
Lie algebras, corresponding to the functions in this section. 

\subsection{\textcolor{Chapter }{SolvableLieAlgebra}}
\logpage{[ 2, 2, 1 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{SolvableLieAlgebra( F, pars )\index{SolvableLieAlgebra@\texttt{SolvableLieAlgebra}}
\label{SolvableLieAlgebra}
}\hfill{\scriptsize (method)}}\\


 Here \mbox{\texttt{F}} is a field, \mbox{\texttt{pars}} is a list of parameters with length between \texttt{2} and \texttt{4}. The first entry of \mbox{\texttt{pars}} is the dimension of the algebra, which has to be 2, 3, or 4. If the dimension
is 3, or 4, then the second entry of \mbox{\texttt{pars}} is the number of the Lie algebra with which it appears in the list of \cite{wdg05}. If the dimension is 2, then there are only two (isomorphism classes of)
solvable Lie algebras. In this case, if the second entry is 1, then the
abelian Lie algebra is returned, if it is 2, then the unique non-abelian
solvable Lie algebra of dimension 2 is returned. A Lie algebra in the list of \cite{wdg05} can have one or two parameters. In that case the list \mbox{\texttt{pars}} also has to contain the parameters. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  gap> SolvableLieAlgebra( Rationals, [4,6,1,2] );
  <Lie algebra of dimension 4 over Rationals>
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{NilpotentLieAlgebra}}
\logpage{[ 2, 2, 2 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{NilpotentLieAlgebra( F, pars )\index{NilpotentLieAlgebra@\texttt{NilpotentLieAlgebra}}
\label{NilpotentLieAlgebra}
}\hfill{\scriptsize (method)}}\\


 Here \mbox{\texttt{F}} is a field, \mbox{\texttt{pars}} is a list of parameters with length between \texttt{2} and \texttt{3}. The first entry of \mbox{\texttt{pars}} is the dimension of the algebra, which has to be 5 or 6. The second entry of \mbox{\texttt{pars}} is the number of the Lie algebra with which it appears in the list of \cite{wdg07}. A Lie algebra in the list of \cite{wdg07} can have one parameter. In that case the list \mbox{\texttt{pars}} also has to contain the parameter. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  gap> NilpotentLieAlgebra( GF(3^7), [ 6, 24, Z(3^7)^101 ] );
  <Lie algebra of dimension 6 over GF(3^7)>
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{SolvableLieAlgebras}}
\logpage{[ 2, 2, 3 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{SolvableLieAlgebras( F, dim )\index{SolvableLieAlgebras@\texttt{SolvableLieAlgebras}}
\label{SolvableLieAlgebras}
}\hfill{\scriptsize (method)}}\\


 Here \mbox{\texttt{F}} is a finite field, and \mbox{\texttt{dim}} is one of 2,3,4. The list of all solvable Lie algebras over \mbox{\texttt{F}} of dimension \mbox{\texttt{dim}} is returned. }

 

\subsection{\textcolor{Chapter }{NilpotentLieAlgebras}}
\logpage{[ 2, 2, 4 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{NilpotentLieAlgebras( F, dim )\index{NilpotentLieAlgebras@\texttt{NilpotentLieAlgebras}}
\label{NilpotentLieAlgebras}
}\hfill{\scriptsize (method)}}\\


 Here \mbox{\texttt{F}} is a finite field, and \mbox{\texttt{dim}} not greater than 9 if \mbox{\texttt{F=GF(2)}}, \mbox{\texttt{dim}} is not greater than 7 if \mbox{\texttt{F=GF(3)}} or \mbox{\texttt{F=GF(5)}}, and \mbox{\texttt{dim}} is not greater than 6 otherwise. The list of all nilpotent Lie algebras over \mbox{\texttt{F}} of dimension \mbox{\texttt{dim}} is returned. If \mbox{\texttt{dim}} is not greater than 6, then the list of nilpotent Lie algebras is determined
by \cite{wdg07}, otherwise the classification can be found in \cite{sch}. }

 

\subsection{\textcolor{Chapter }{NumberOfNilpotentLieAlgebras}}
\logpage{[ 2, 2, 5 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{NumberOfNilpotentLieAlgebras( F, dim )\index{NumberOfNilpotentLieAlgebras@\texttt{NumberOfNilpotentLieAlgebras}}
\label{NumberOfNilpotentLieAlgebras}
}\hfill{\scriptsize (method)}}\\


 Here \mbox{\texttt{F}} is a finite field, and \mbox{\texttt{dim}} not greater than 9 if \mbox{\texttt{F=GF(2)}}, \mbox{\texttt{dim}} is not greater than 7 if \mbox{\texttt{F=GF(3)}} or \mbox{\texttt{F=GF(5)}}, and \mbox{\texttt{dim}} is not greater than 6 otherwise. The number of nilpotent Lie algebras over \mbox{\texttt{F}} of dimension \mbox{\texttt{dim}} is returned. }

 

\subsection{\textcolor{Chapter }{LieAlgebraIdentification}}
\logpage{[ 2, 2, 6 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{LieAlgebraIdentification( L )\index{LieAlgebraIdentification@\texttt{LieAlgebraIdentification}}
\label{LieAlgebraIdentification}
}\hfill{\scriptsize (method)}}\\


 Here \mbox{\texttt{L}} is a solvable Lie algebra of dimension 2,3, or 4, or it is a nilpotent Lie
algebra of dimension 5 or 6 (in the latter case it has to be of characteristic
not 2). This function returns a record with three fields. 
\begin{itemize}
\item \mbox{\texttt{name}} This is a string containing the name of the Lie algebra. It starts with a
capital L if it is a solvable Lie algebra of dimension 2,3,4. It starts with a
capital N if it is a nilpotent Lie algebra of dimension 5 or 6. A name like 
\begin{verbatim}  "N6_24( GF(3^7), Z(3^7) )"
\end{verbatim}
 means that the input Lie algebra is isomorphic to the Lie algebra with number
24 in the list of 6-dimensional nilpotent Lie algebras. Furthermore the field
is given and the value of the parameters (if there are any). 
\item \mbox{\texttt{parameters}} This contains the parameters that appear in the name of the algebra.
\item \mbox{\texttt{isomorphism}} This is an isomorphism of the input Lie algebra to the Lie algebra from the
classification with the given name.
\end{itemize}
 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  gap> L:= SolvableLieAlgebra( Rationals, [4,7,1/2,4/9] );
  <Lie algebra of dimension 4 over Rationals>
  gap> LieAlgebraIdentification( L );
  rec( name := "L4_7( Rationals, 256/729, 256/729 )",
    parameters := [ 256/729, 256/729 ],
    isomorphism := CanonicalBasis( <Lie algebra of dimension
      4 over Rationals> ) ->
      [ (10)*v.1+v.2+(4)*v.3, (16/9)*v.1+(848/81)*v.2+(8/9)*v.3,
        (32/81)*v.1+(1408/729)*v.2+(6784/729)*v.3, (8/9)*v.4 ] )
\end{Verbatim}
 In the example we see that the program finds different parameters, than the
ones with which the Lie algebra was constructed. The explanation is that some
parametric classes of Lie algebras contain isomorphic Lie algebras, for
different values of the parameters. In that case the identification function
makes its own choice. }

 }

 }

 

\appendix


\chapter{\textcolor{Chapter }{Description of the solvable and nilpotent Lie algebras}}\label{listdescr}
\logpage{[ "A", 0, 0 ]}
{
  In this appendix we list the multiplication tables of the nilpotent and
solvable Lie algebras contained in the package. Some parametric classes
contain isomorphic Lie algebras, for different values of the parameters. For
exact descriptions of these isomorphisms we refer to \cite{wdg05}, \cite{wdg07}. In dimension 2 there are just two classes of solvable Lie algebras: 
\begin{itemize}
\item  $L_2^1$: The Abelian Lie algebra. 
\item  $L_2^2$: $[x_2,x_1]=x_1$. 
\end{itemize}
 We have the following solvable Lie algebras of dimension 3: 
\begin{itemize}
\item  $L_3^1$ The Abelian Lie algebra. 
\item  $L_3^2$ $[x_3,x_1]=x_1, [x_3,x_2]=x_2$. 
\item  $L_3^3(a)$ $[x_3,x_1]=x_2, [x_3,x_2]=ax_1+x_2$. 
\item  $L_3^4(a)$ $[x_3,x_1]=x_2, [x_3,x_2]=ax_1. $ 
\end{itemize}
 And the following solvable Lie algebras of dimension 4: 
\begin{itemize}
\item  $L_4^1$ The Abelian Lie algebra. 
\item  $L_4^2$ $ [x_4,x_1]=x_1, [x_4,x_2]=x_2, [x_4,x_3]=x_3.$ 
\item  $L_4^3(a)$ $[x_4,x_1]=x_1, [x_4,x_2]=x_3, [x_4,x_3]=-ax_2 +(a+1)x_3$ .
\item  $L_4^4$ $[x_4,x_2]=x_3, [x_4,x_3]= x_3$ .
\item  $L_4^5$ $[x_4,x_2]=x_3$ .
\item  $L_4^6(a,b)$ $[x_4,x_1] = x_2, [x_4,x_2]=x_3, [x_4,x_3] = ax_1+bx_2+x_3$ .
\item  $L_4^7(a,b)$ $[x_4,x_1] = x_2, [x_4,x_2]=x_3, [x_4,x_3] = ax_1+bx_2.$ 
\item  $L_4^8$ $[x_1,x_2]=x_2, [x_3,x_4]=x_4$ .
\item  $L_4^9(a)$ $[x_4,x_1] = x_1+ax_2, [x_4,x_2]=x_1, [x_3,x_1]=x_1, [x_3,x_2]=x_2$ .
\item  $L_4^{10}(a)$ $[x_4,x_1] = x_2, [x_4,x_2]=ax_1, [x_3,x_1]=x_1, [x_3,x_2]=x_2$ .
\item  $L_4^{11}(a,b)$ $[x_4,x_1] = x_1, [x_4,x_2] = bx_2, [x_4,x_3]=(1+b)x_3, [x_3,x_1]=x_2,
[x_3,x_2]=ax_1$. Condition on F: the characteristic of F is 2. 
\item  $L_4^{12}$ $ [x_4,x_1] = x_1, [x_4,x_2]=2x_2, [x_4,x_3] = x_3, [x_3,x_1]=x_2$ .
\item  $L_4^{13}(a)$ $[x_4,x_1] = x_1+ax_3, [x_4,x_2]=x_2, [x_4,x_3] = x_1, [x_3,x_1]=x_2$ .
\item  $L_4^{14}(a)$ $[x_4,x_1] = ax_3, [x_4,x_3]=x_1, [x_3,x_1]=x_2$ .
\end{itemize}
 Nilpotent of dimension 5: 
\begin{itemize}
\item  $N_{5,1}$ Abelian. 
\item  $N_{5,2}$ $[x_1,x_2]=x_3$ .
\item  $N_{5,3}$ $ [x_1,x_2]=x_3, [x_1,x_3]=x_4$ .
\item  $N_{5,4}$ $[x_1,x_2] = x_5, [x_3,x_4]=x_5$ .
\item  $N_{5,5}$ $[x_1,x_2]=x_3, [x_1,x_3]= x_5, [x_2,x_4] = x_5$ .
\item  $N_{5,6}$ $[x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5, [x_2,x_3]=x_5$ .
\item  $N_{5,7}$ $[x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5$ .
\item  $N_{5,8}$ $[x_1,x_2]=x_4, [x_1,x_3]=x_5$ .
\item  $N_{5,9}$ $[x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_2,x_3]=x_5$ .
\end{itemize}
 We get nine 6-dimensional nilpotent Lie algebras denoted $N_{6,k}$ for k=1,...,9 that are the direct sum of $N_{5,k}$ and a 1-dimensional abelian ideal. Subsequently we get the following Lie
algebras. 
\begin{itemize}
\item  $N_{6,10}$ $[x_1,x_2]=x_3, [x_1,x_3]=x_6, [x_4,x_5]=x_6.$ 
\item  $N_{6,11}$ $[x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_6, [x_2,x_3]=x_6, [x_2,x_5]=x_6$ .
\item  $N_{6,12}$ $[x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_6, [x_2,x_5]=x_6$ .
\item  $N_{6,13}$ $[x_1,x_2]=x_3, [x_1,x_3]=x_5, [x_2,x_4]=x_5, [x_1,x_5]=x_6, [x_3,x_4]=x_6$ .
\item  $N_{6,14}$ $[x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5, [x_2,x_3]=x_5,
[x_2,x_5]=x_6,[x_3,x_4]=-x_6$ .
\item  $N_{6,15}$ $[x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5, [x_2,x_3]=x_5,
[x_1,x_5]=x_6,[x_2,x_4]=x_6$ .
\item  $N_{6,16}$ $[x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5, [x_2,x_5]=x_6, [x_3,x_4]=-x_6$ .
\item  $N_{6,17}$ $[x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5, [x_1,x_5]=x_6, [x_2,x_3]= x_6$ .
\item  $N_{6,18}$ $[x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5, [x_1,x_5]=x_6$ .
\item  $N_{6,19}(a)$ $[x_1,x_2]=x_4, [x_1,x_3]=x_5, [x_2,x_4]=x_6, [x_3,x_5]=a x_6$ .
\item  $N_{6,20}$ $[x_1,x_2]=x_4, [x_1,x_3]=x_5, [x_1,x_5]=x_6, [x_2,x_4]=x_6$ .
\item  $N_{6,21}(a)$ $[x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_2,x_3]=x_5, [x_1,x_4]=x_6, [x_2,x_5]= a x_6$ .
\item  $N_{6,22}(a)$ $[x_1,x_2]=x_5, [x_1,x_3]=x_6, [x_2,x_4]= a x_6, [x_3,x_4]=x_5$ .
\item  $N_{6,23}$ $[x_1,x_2]=x_3, [x_1,x_3]=x_5, [x_1,x_4]=x_6, [x_2,x_4]= x_5$ .
\item  $N_{6,24}(a)$ $[x_1,x_2]=x_3, [x_1,x_3]=x_5, [x_1,x_4]=a x_6, [x_2,x_3]=x_6, [x_2,x_4]= x_5$ .
\item  $N_{6,25}$ $[x_1,x_2]=x_3, [x_1,x_3]=x_5, [x_1,x_4]=x_6$ .
\item  $N_{6,26}$ $[x_1,x_2]=x_4, [x_1,x_3]=x_5, [x_2,x_3]=x_6$ .
\end{itemize}
 }


\chapter{\textcolor{Chapter }{Description of the non-solvable Lie algebras}}\label{nonsolvdescr}
\logpage{[ "B", 0, 0 ]}
{
  In this appendix we list the non-solvable Lie algebras contained in the
package. Our notation follows \cite{Strade}, where a more detailed description can also be found. In particular if $L$ is a Lie algebra over $F$ then $C(L)$ denotes the center of $L$. Further, if $x_1,\ldots,x_k$ are elements of $L$, then $F{\textless}x_1,\ldots,x_k{\textgreater}$ denotes the linear subspace generated by $x_1,\ldots,x_k$, and we also write $Fx_1$ for $F{\textless}x_1{\textgreater}$ 
\section{\textcolor{Chapter }{Dimension 3}}\label{appdim3}
\logpage{[ "B", 1, 0 ]}
{
  There are no non-solvable Lie algebras with dimension 1 or 2. Over an
arbitrary finite field \mbox{\texttt{F}}, there is just one isomorphism type of non-solvable Lie algebras: 
\begin{enumerate}
\item  If \mbox{\texttt{char F=2}} then the algebra is $W(1;\underline 2)^{(1)}$. 
\item  If \mbox{\texttt{char F>2}} then the algebra is $\mbox{sl}(2,F)$.
\end{enumerate}
 See Theorem~3.2 of \cite{Strade} for details. }

 
\section{\textcolor{Chapter }{Dimension 4}}\label{appdim4}
\logpage{[ "B", 2, 0 ]}
{
  Over a finite field \mbox{\texttt{F}} of characteristic 2 there are two isomorphism classes of non-solvable Lie
algebras with dimension 4, while over a finite field \mbox{\texttt{F}} of odd characteristic the number of isomorphism classes is one (see
Theorem~4.1 of \cite{Strade}). The classes are as follows: 
\begin{enumerate}
\item  characteristic 2: $W(1;\underline 2)$ and $W(1;\underline 2)^{(1)}\oplus F$. 
\item  odd characteristic: $\mbox{gl}(2,F)$.
\end{enumerate}
 }

 
\section{\textcolor{Chapter }{Dimension 5}}\label{appdim5}
\logpage{[ "B", 3, 0 ]}
{
  
\subsection{\textcolor{Chapter }{Characteristic 2}}\label{appdim5char2}
\logpage{[ "B", 3, 1 ]}
{
  Over a finite field \mbox{\texttt{F}} of characteristic 2 there are 5 isomorphism classes of non-solvable Lie
algebras with dimension 5: 
\begin{enumerate}
\item  $\mbox{Der}(W(1;\underline 2)^{(1)})$;
\item  $W(1;\underline 2)\ltimes Fu$ where $[W(1;\underline 2)^{(1)},u]=0$, $[x^{(3)}\partial,u]=\delta u$ and $\delta\in\{0,1\}$ (two algebras);
\item  $W(1;\underline 2)^{(1)}\oplus(F\left{\textless} h,u\right{\textgreater})$, $[h,u]=\delta u$, where $\delta\in\{0,1\}$ (two algebras).
\end{enumerate}
 See Theorem 4.2 of \cite{Strade} for details. }

 
\subsection{\textcolor{Chapter }{Odd characteristic}}\label{appdim5charodd}
\logpage{[ "B", 3, 2 ]}
{
 Over a field $F$of odd characteristic the number of isomorphism types of 5-dimensional
non-solvable Lie algebras is $3$ if the characteristic is at least 7, and it is 4 otherwise (see Theorem 4.3 of \cite{Strade}). The classes are as follows. 
\begin{enumerate}
\item $\mbox{sl}(2,F)\oplus F{\textless}x,y{\textgreater}$, $[x,y]=\delta y$ where $\delta\in\{0,1\}$.
\item $\mbox{sl}(2,F)\ltimes V(1)$ where $V(1)$ is the irreducible 2-dimensional $\mbox{sl}(2,F)$=module.
\item If $\mbox{char }F=3$ then there is an additional algebra, namely the non-split extension $0\rightarrow V(1)\rightarrow L\rightarrow\mbox{sl}(2,F)\rightarrow 0$.
\item If $\mbox{char }F=5$ then there is an additional algebra: $W(1;\underline 1)$. 
\end{enumerate}
 }

 }

 
\section{\textcolor{Chapter }{Dimension 6}}\label{appdim6}
\logpage{[ "B", 4, 0 ]}
{
 
\subsection{\textcolor{Chapter }{Characteristic 2}}\label{appdim6char2}
\logpage{[ "B", 4, 1 ]}
{
 Over a field $F$ of characteristic 2, the isomorphism classes of non-solvable Lie algebras are
as follows. 
\begin{enumerate}
\item $W(1;\underline 2)^{(1)}\oplus W(1;\underline 2)^{(1)}$.
\item $W(1;\underline 2)^{(1)}\otimes F_{q^2}$ where $F=F_q$.
\item $\mbox{Der}(W(1;\underline 2)^{(1)})\ltimes Fu$, $[W(1;\underline 2),u]=0$, $[\partial^2,u]=\delta u$ where $\delta=\{0,1\}$.
\item $W(1;\underline 2)\ltimes (F{\textless}h,u{\textgreater})$, $[W(1;\underline 2)^{(1)},(F{\textless}h,u{\textgreater}]=0$, $[h,u]=\delta u$, and if $\delta=0$, then the action of $x^{(3)}\partial$ on $F{\textless}h,u{\textgreater}$ is given by one of the following matrices: 
\[ \left(\begin{array}{cc} 0 & 0\\ 0 & 0\end{array}\right),\
\left(\begin{array}{cc} 0 & 1\\ 0 & 0\end{array}\right),\
\left(\begin{array}{cc} 1 & 0\\ 0 & 1\end{array}\right),\
\left(\begin{array}{cc} 1 & 1\\ 0 & 1\end{array}\right),\
\left(\begin{array}{cc} 0 & \xi\\ 1 & 1\end{array}\right)\mbox{ where }\xi\in
F^*.\]

\item the algebra is as in (4.), but $\delta=1$. Note that Theorem 5.1(3/b) of \cite{Strade} lists two such algebras but they turn out to be isomorphic. We take the one
with $[x^{(3)}\partial,h]=[x^{(3)}\partial,u]=0$. 
\item $W(1;\underline 2)^{(1)}\oplus K$ where $K$ is a 3-dimensional solvable Lie algebra.
\item $W(1;\underline 2)^{(1)}\ltimes \mathcal O(1;\underline 2)/F$.
\item the non-split extension $0\rightarrow \mathcal O(1;\underline 2)/F\rightarrow L\rightarrow
W(1;\underline 2)^{(1)}\rightarrow 0$.
\end{enumerate}
 See Theorem 5.1 of \cite{Strade}. }

 
\subsection{\textcolor{Chapter }{General odd characteristic}}\label{appdim6charodd}
\logpage{[ "B", 4, 2 ]}
{
 If the characteristic of the field is odd, then the 6-dimensional non-solvable
Lie algebras are described by Theorem 5.2--5.4 of \cite{Strade}. Over such a field $F$, let us define the following isomorphism classes of 6-dimensional
non-solvable Lie algebras. 
\begin{enumerate}
\item  $\mbox{sl}(2,F)\oplus\mbox{sl}(2,F) $.
\item $\mbox{sl}(2,F_{q^2})$ where $F=F_q$;
\item $\mbox{sl}(2,F)\oplus K$ where $K$ is a solvable Lie algebra with dimension 3;
\item $\mbox{sl}(2,F)\ltimes (V(0)\oplus V(1))$ where $V(i)$ is the $(i+1)$-dimensional irreducible $\mbox{sl}(2,F)$-module;
\item $\mbox{sl}(2,F)\ltimes V(2)$ where $V(2)$ is the $3$-dimensional irreducible $\mbox{sl}(2,F)$-module; 
\item $\mbox{sl}(2,F)\ltimes(V(1)\oplus C(L))\cong \mbox{sl}(2,F)\ltimes H$ where $H$ is the Heisenberg Lie algebra;
\item $\mbox{sl}(2,F)\ltimes K$ where $K=Fd\oplus K^{(1)}$, $K^{(1)}$ is 2-dimensional abelian, isomorphic, as an $\mbox{sl}(2,F)$-module, to $V(1)$, $[\mbox{sl}(2,F),d]=0$, and, for all $v\in K$, $[d,v]=v$;
\end{enumerate}
 If the characteristic of $F$ is at least 7, then these algebras form a complete and irredundant list of the
isomorphism classes of the 6-dimensional non-solvable Lie algebras. }

 
\subsection{\textcolor{Chapter }{Characteristic 3}}\label{appdim6char3}
\logpage{[ "B", 4, 3 ]}
{
 If the characteristic of the field $F$ is 3, then, besides the classes in Section \ref{appdim6charodd}, we also obtain the following isomorphism classes. 
\begin{enumerate}
\item $\mbox{sl}(2,F)\ltimes V(2,\chi)$ where $\chi$ is a 3-dimensional character of $\mbox{sl}(2,F)$. Each such character is described by a field element $a$ such that $T^3+T^2-a$ has a root in $F$; see Proposition 3.5 of \cite{Strade} for more details. 
\item $W(1;\underline 1)\ltimes\mathcal O(1;\underline 1)$ where $\mathcal O(1;\underline 1)$ is considered as an abelian Lie algebra. 
\item $W(1;\underline 1)\ltimes\mathcal O(1;\underline 1)^*$ where $\mathcal O(1;\underline 1)^*$ is the dual of $\mathcal O(1;\underline 1)$ and it is considered as an abelian Lie algebra.
\item One of the two 6-dimensional central extensions of the non-split extension $0\rightarrow V(1)\rightarrow L\rightarrow \mbox{sl}(2,F)\rightarrow 0$; see Proposition 4.5 of \cite{Strade}. We note that Proposition 4.5 of \cite{Strade} lists three such central extensions, but one of them is not a Lie algebra.
\item One of the two non-split extensions $0\rightarrow\mbox{rad } L\rightarrow L\rightarrow L/\mbox{rad } L\rightarrow 0$ with a 5-dimensional ideal; see Theorem~5.4 of \cite{Strade}.
\end{enumerate}
 We note here that \cite{Strade} lists one more non-solvable Lie algebra over a field of characteristic 3,
namely the one in Theorem~5.3(5). However, this algebra is isomorphic to the
one in Theorem~5.3(4). }

 
\subsection{\textcolor{Chapter }{Characteristic 5}}\label{appdim6char5}
\logpage{[ "B", 4, 4 ]}
{
 If the characteristic of the field $F$ is 5, then, besides the classes in Section~\ref{appdim6charodd}, we also obtain the following isomorphism classes. 
\begin{enumerate}
\item $W(1;\underline 1)\oplus F$.
\item The non-split central extension $0\rightarrow F\rightarrow L\rightarrow W(1;\underline 1)\rightarrow 0$.
\end{enumerate}
 }

 }

 }

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