non-Abelian structuresNonAbelianStructures2008-07-14 18:49:432009-07-02 20:22:48Topicnon-Abelian structurenon-AbelianTheoriesnon-commutative structuresseveral examples of nonabelian structures% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\hr}{{\hookrightarrow}} \begin{definition} Any mathematical structure (algebraic or topological, etc.)--in the sense defined by either
C. Ehresmann \cite{EC65, EC66} or `N. Bourbaki' -- which is {\em not} commutative is usually called either {\em non-Abelian} ({\em non-abelian, nonabelian}) or {\em non-commutative}.
\end{definition}
\subsection{Examples}
Every non-commutative ring, non-commutative group, non-commutative groupoid, non-commutative monoid, non-commutative algebra, and so on, has a {\em non-Abelian structure}; a few specific examples of non-Abelian algebras are : Clifford algebras, matrix algebras, non-commutative $C^*$-algebras, quantum `groups' (non-commuative Hopf algebras), quantum `groupoids' (non-commutative weak Hopf algebras).
\subsection{Remarks}
The term `non-Abelian'--instead of noncommutative-- is often the preferred qualifier for theories,
with the possible exeception of `noncommutative geometry', which is a non-Abelian theory.
On the other hand, the term `anabelian' (in French, ``anabelienne'') was employed by Alexander Grothendieck to
describe a new field of research called ``Anabelian Geometry'' in his `Esquisse d'un Programme''
The commutativity property in all Abelian structures, such as Abelian groups and Abelian categories is a {\em global} rather than partial, or {\em local}, property. Thus, many categories or toposes/topoi may exhibit
local but not global commutativity properties; for example, a category is still non-Abelian
if $Hom_{Ab}(-, -)$ does not have the structure of a commutative (or {\em Abelian}) group;
alternatively, a category that does not satisfy one of the four $Ab$-axioms of Freyd, or one of the
$Ab1$ to $Ab6$ axioms in the current abelian category definition is non-Abelian.
\med
\subsection{Many-valued, algebraic logic examples}
The structures of several, n-valued logic algebras are represented as non-commutative lattices and are, therefore, non-Abelian. Specific examples are provided by the generally non-commutative $LM_n$-logic algebras, categories of $LM_n$-logic algebras and lattice morphisms. On the other hand, the Heyting (intuitionistic logic) algebra subobject classifier of a standard topos is {\em commutative} and thus, it is {\em Abelian}; this does {\em not} mean however that all toposes/topoi have Abelian structure-in fact, this is not generally case. \\
\subsection{Examples of {\em} Abelian categories}
\begin{enumerate}
\item The category of Abelian (or commutative) groups is Abelian;
\item The Grothendieck category is a special case of $\mathcal{\A}b5$ category;
\item Local Grothendieck categories are Abelian categories;
\item The Grassmann category is Abelian;
\item The category of (commutative) semi-noetherian rings is Abelian. (\cite{NP73});
\item The category of Heyting logic algebras, $Hy$ and the category of Boolean logic algebras, \textbf{Boole} are both categories of commutative lattices, and are Abelian categories;
\item If $\grp$ is a topological groupoid the {\em category of sheaves, $S_{hA}$} of Abelian groups over $\grp$ is an Abelian category.
\end{enumerate}
\textbf{Related results}
If $\mathcal{\G}$ is a Grothendieck catgeory and $\mathcal{\A}$ is a localizing subcategory of $\mathcal{\G}$, then $\mathcal{\G} \slash \mathcal{\A}$ is
Abelian, as it is also a Grothendieck category (\textbf{COROLLARY 6.2} on p. 186 in \cite{NP73}) .
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