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%%% Primary Title: Grassmann-Hopf algebras and coalgebras\gebras
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\begin{document}
\subsection{Definitions of Grassmann-Hopf Algebras, Their Dual Co-Algebras, Gebras, Grassmann--Hopf Algebroids and Gebroids}
Let $V$ be a (complex) \htmladdnormallink{vector space}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html}, $\dim_{\mathcal C} V = n$, and let $\{e_0, e_1, \ldots, \}$ with \htmladdnormallink{identity}{http://planetphysics.us/encyclopedia/Cod.html} $e_0 \equiv 1$, be the \htmladdnormallink{generators}{http://planetphysics.us/encyclopedia/Generator.html} of a Grassmann (exterior) algebra
\begin{equation}
\Lambda^*V = \Lambda^0 V \oplus \Lambda^1 V \oplus \Lambda^2 V
\oplus \cdots
\end{equation}
subject to the \htmladdnormallink{relation}{http://planetphysics.us/encyclopedia/Bijective.html} $e_i e_j + e_j e_i = 0$~. Following Fauser
(2004) we append this algebra with a Hopf structure to obtain a
`co--gebra' based on the interchange (or \textsl{`tangled \htmladdnormallink{duality'}}{http://planetphysics.us/encyclopedia/GroupoidSymmetries.html}):
$$\text{(\textit{objects/points}, \textit{morphisms})} \mapsto \text{(\textsl{morphisms}, \textsl{objects/points.})}$$
This leads to a \textsl{tangle duality} between an associative (unital algebra)
$\A=(A,m)$, and an associative (unital) `co--gebra' $\mathcal{C}=(C,\Delta)$ :
\begin{itemize}
\item[i] the binary product $A \otimes A \ovsetl{m} A$, and
\item[ii] the \htmladdnormallink{coproduct}{http://planetphysics.us/encyclopedia/Coproduct.html} $C \ovsetl{\Delta} C \otimes C$ \end{itemize},
where the Sweedler notation (Sweedler, 1996), with respect to an
arbitrary basis is adopted: $$
\begin{aligned}
\Delta (x) &= \sum_r a_r \otimes b_r = \sum_{(x)} x_{(1)} \otimes
x_{(2)} = x _{(1)} \otimes x_{(2)} \\ \Delta (x^i) &= \sum_i
\Delta^{jk}_i = \sum_{(r)} a^j_{(r)} \otimes b^k_{(r)} = x _{(1)}
\otimes x_{(2)}
\end{aligned}
$$
Here the $\Delta^{jk}_i$ are called `\htmladdnormallink{section}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html} coefficients'. We have then a generalization of associativity to coassociativity:
\begin{equation}
\begin{CD}
C @> \Delta >> C \otimes C
\\ @VV \Delta V @VV \ID \otimes \Delta V \\ C \otimes C
@> \Delta \otimes \ID >> C \otimes C \otimes C
\end{CD}
\end{equation}
inducing a \htmladdnormallink{tangled duality}{http://planetphysics.us/encyclopedia/DualityAndTriality.html} between an associative (unital algebra
$\mathcal A = (A,m)$, and an associative (unital) `co--gebra'
$\mathcal C = (C, \Delta)$~. The idea is to take this structure
and combine the Grassmann algebra $(\Lambda^*V, \wedge)$ with the
`co-gebra' $(\Lambda^*V, \Delta_{\wedge})$ (the `tangled dual')
along with the \htmladdnormallink{Hopf algebra}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html} compatibility rules: 1) the product
and the unit are `co--gebra' \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, and 2) the coproduct and
counit are algebra morphisms.
Next we consider the following ingredients:
\begin{itemize}
\item[(1)]
the graded switch $\hat{\tau} (A \otimes B) = (-1)^{\del
A \del B} B \otimes A$
\item[(2)]
the counit $\varepsilon$ (an algebra morphism) satisfying
$(\varepsilon \otimes \ID) \Delta = \ID = (\ID \otimes
\varepsilon) \Delta$
\item[(3)] the antipode $S$~.
\end{itemize}
The \textit{Grassmann-Hopf algebra} $\widehat{H}$ thus consists of--is defined by-- the
\textit{septet} $\widehat{H}=(\Lambda^*V, \wedge, \ID, \varepsilon, \hat{\tau},S)~$.
Its generalization to a \textit{Grassmann-Hopf \htmladdnormallink{algebroid}}{http://planetphysics.us/encyclopedia/Algebroids.html} is
straightforward by considering a \htmladdnormallink{groupoid}{http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html} $\grp$, and then
defining a $H^{\wedge}- \textit{Algebroid}$ as a
\textit{quadruple} $(GH, \Delta, \vep, S)$ by modifying the \htmladdnormallink{Hopf algebroid}{http://planetphysics.us/encyclopedia/HilbertBundle.html} definition so that
$\widehat{H} = (\Lambda^*V, \wedge, \ID, \varepsilon, \hat{\tau},S)$ satisfies the standard
Grassmann-Hopf algebra axioms stated above. We may also say that
$(HG, \Delta, \vep, S)$ is a \emph{weak C*-Grassmann-Hopf
algebroid} when $H^{\wedge}$ is a unital \htmladdnormallink{C*-algebra}{http://planetphysics.us/encyclopedia/VonNeumannAlgebra2.html} (with $\mathbf 1$).
We thus set $\mathbb F = \mathbb C~$. Note however
that the tangled-duals of Grassman-Hopf algebroids retain both the
intuitive interactions and the \htmladdnormallink{dynamic diagram}{http://planetphysics.us/encyclopedia/GenericityInOpenSystems.html} advantages of their
physical, extended symmetry \htmladdnormallink{representations}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} exhibited by the
Grassman-Hopf al/gebras and co-gebras over those of either weak
C*- Hopf algebroids or weak Hopf C*- algebras.
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\end{document}
