# PlanetPhysics/Grassmann Hopf Algebras and Coalgebrasgebras

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### Definitions of Grassmann-Hopf Algebras, Their Dual Co-Algebras, Gebras, Grassmann--Hopf Algebroids and Gebroids

Let $V$ be a (complex) vector space, $\dim _{\mathcal {C}}V=n$ , and let $\{e_{0},e_{1},\ldots ,\}$ with identity $e_{0}\equiv 1$ , be the generators of a Grassmann (exterior) algebra

$\Lambda ^{*}V=\Lambda ^{0}V\oplus \Lambda ^{1}V\oplus \Lambda ^{2}V\oplus \cdots$ subject to the relation $e_{i}e_{j}+e_{j}e_{i}=0$ ~. 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}):

$\displaystyle =(''objects/points'' , ''morphisms'' )= \mapsto =(\textsl{morphisms= , \textsl{objects/points.})}$

This leads to a \textsl{tangle duality} between an associative (unital algebra) $\displaystyle \A=(A,m)$ , and an associative (unital) co--gebra' ${\mathcal {C}}=(C,\Delta )$ :

 \item[i] the binary product $\displaystyle A \otimes A \ovsetl{m} A$
, and \item[ii] the coproduct $\displaystyle C \ovsetl{\Delta} C \otimes C$
,


where the Sweedler notation (Sweedler, 1996), with respect to an arbitrary basis is adopted: $\displaystyle \Delta (x) &= \sum_r a_r \otimes b_r = \sum_{(x)} x_{(1)} \otimes x_{(2)} = x _{(1)} \otimes x_{(2)} \\ \Delta (x^i) &= \sum_i \Delta^{jk}_i = \sum_{(r)} a^j_{(r)} \otimes b^k_{(r)} = x _{(1)} \otimes x_{(2)}$

Here the $\Delta _{i}^{jk}$ are called section coefficients'. We have then a generalization of associativity to coassociativity:

$CD}"): \begin{CD} C @> \Delta >> C \otimes C \\ @VV \Delta V @VV \ID \otimes \Delta V \\ C \otimes C @> \Delta \otimes \ID >> C \otimes C \otimes C \end{CD}$

inducing a tangled duality 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 Hopf algebra compatibility rules: 1) the product and the unit are co--gebra' morphisms, and 2) the coproduct and counit are algebra morphisms.

Next we consider the following ingredients:

 \item[(1)] the graded switch [/itex]\hat{\tau} (A \otimes B) = (-1)^{\del A \del B} B \otimes A$\displaystyle \item[(2)] the counit $\varepsilon$ (an algebra morphism) satisfying$(\varepsilon \otimes \ID) \Delta = \ID = (\ID \otimes \varepsilon) \Delta$\displaystyle \item[(3)] the antipode [itex]S$
~.


The Grassmann-Hopf algebra ${\widehat {H}}$ thus consists of--is defined by-- the septet $\displaystyle \widehat{H}=(\Lambda^*V, \wedge, \ID, \varepsilon, \hat{\tau},S)~$ .

Its generalization to a Grassmann-Hopf algebroid is straightforward by considering a groupoid $\displaystyle \grp$ , and then defining a $H^{\wedge }-''Algebroid''$ as a quadruple $\displaystyle (GH, \Delta, \vep, S)$ by modifying the Hopf algebroid definition so that $\displaystyle \widehat{H} = (\Lambda^*V, \wedge, \ID, \varepsilon, \hat{\tau},S)$ satisfies the standard Grassmann-Hopf algebra axioms stated above. We may also say that $\displaystyle (HG, \Delta, \vep, S)$ is a \emph{weak C*-Grassmann-Hopf algebroid} when $H^{\wedge }$ is a unital C*-algebra (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 dynamic diagram advantages of their physical, extended symmetry representations exhibited by the Grassman-Hopf al/gebras and co-gebras over those of either weak C*- Hopf algebroids or weak Hopf C*- algebras.