Mathematical Structures: Lie algebras

# Lie algebras

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Changed: 28,30c28,29
 \Large Name of class \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Template}{edit} % Note: replace "Template" with Name_of_class in previous line
 \Large Lie algebras \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Lie_algebras}{edit}

Changed: 32c31
 \abbreviation{Abbr}
 \abbreviation{LieA}

Changed: 35,38c34,35
 A ... is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle ...\rangle$ such that $\langle A,...\rangle$ is a \href{Name_of_class.pdf}{name of class}
 A Lie algebra is a \href{Bilinear_algebras.pdf}{bilinear algebra} $\mathbf{A}=\langle A,+,-,0,\cdot,s_r\ (r\in F)\rangle$ over a \href{Fields.pdf}{field} $\mathbf F$ such that

Changed: 40c37
 $op_1$ is (name of property): $axiom_1$
 $xx=0$ and

Changed: 42c39
 $op_2$ is ...: $...$
 $(xy)z + (yz)x + (zx)y = 0$.

Changed: 78c75
 Classtype & (value, see description) \cite{Ln19xx} \\\hline
 Classtype & variety \\\hline

Changed: 128,130c125
 \href{....pdf}{...} supervariety \href{....pdf}{...} subreduct
 \href{Bilinear_algebras.pdf}{Bilinear algebras}

http://mathcs.chapman.edu/structuresold/files/Lie_algebras.pdf
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\theoremstyle{definition}
\newtheorem{definition}{Definition}
\newtheorem*{morphisms}{Morphisms}
\newtheorem*{basic_results}{Basic Results}
\newtheorem*{examples}{Examples}
\newtheorem{example}{}
\newtheorem*{properties}{Properties}
\newtheorem*{finite_members}{Finite Members}
\newtheorem*{subclasses}{Subclasses}
\newtheorem*{superclasses}{Superclasses}
\newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
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\begin{document}
\textbf{\Large Lie algebras}
\quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Lie_algebras}{edit}

\abbreviation{LieA}

\begin{definition}
A \emph{Lie algebra} is a \href{Bilinear_algebras.pdf}{bilinear algebra} $\mathbf{A}=\langle A,+,-,0,\cdot,s_r\ (r\in F)\rangle$ over
a \href{Fields.pdf}{field} $\mathbf F$ such that

$xx=0$ and

$(xy)z + (yz)x + (zx)y = 0$.

Remark: This is a template.
If you know something about this class, click on the 'Edit text of this page' link at the bottom and fill out this page.

It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.
\end{definition}

\begin{morphisms}
Let $\mathbf{A}$ and $\mathbf{B}$ be ... . A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:
$h(x ... y)=h(x) ... h(y)$
\end{morphisms}

\begin{definition}
An \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle ...\rangle$ such that

$...$ is ...:  $axiom$

$...$ is ...:  $axiom$
\end{definition}

\begin{basic_results}
\end{basic_results}

\begin{examples}
\begin{example}
\end{example}
\end{examples}

\begin{table}[h]
\begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

\begin{tabular}{|ll|}\hline
Classtype                       & variety \\\hline
Equational theory               & \\\hline
Quasiequational theory          & \\\hline
First-order theory              & \\\hline
Locally finite                  & \\\hline
Residual size                   & \\\hline
Congruence distributive         & \\\hline
Congruence modular              & \\\hline
Congruence $n$-permutable       & \\\hline
Congruence regular              & \\\hline
Congruence uniform              & \\\hline
Congruence extension property   & \\\hline
Definable principal congruences & \\\hline
Equationally def. pr. cong.     & \\\hline
Amalgamation property           & \\\hline
Strong amalgamation property    & \\\hline
Epimorphisms are surjective     & \\\hline
\end{tabular}
\end{properties}
\end{table}

\begin{finite_members} $f(n)=$ number of members of size $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ \end{array}$\qquad
$\begin{array}{lr} f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$

\end{finite_members}

\begin{subclasses}\

\href{....pdf}{...} subvariety

\href{....pdf}{...} expansion

\end{subclasses}

\begin{superclasses}\

\href{Bilinear_algebras.pdf}{Bilinear algebras}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}
F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 \href{http://www.ams.org/mathscinet-getitem?mr=12a:08034}{MRreview}

\end{thebibliography}

\end{document}
%

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Last edited July 26, 2004 9:12 pm by Jipsen (diff)
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