# 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}):
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This leads to a \textsl{tangle duality} between an associative (unital algebra) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \A=(A,m)} , and an associative (unital) `co--gebra' ${\mathcal {C}}=(C,\Delta )$ :
where the Sweedler notation (Sweedler, 1996), with respect to an arbitrary basis is adopted: Failed to parse (syntax error): {\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:
   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \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:
The Grassmann-Hopf algebra ${\widehat {H}}$ thus consists of--is defined by-- the septet Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \widehat{H}=(\Lambda^*V, \wedge, \ID, \varepsilon, \hat{\tau},S)~} .
Its generalization to a Grassmann-Hopf algebroid is straightforward by considering a groupoid Failed to parse (unknown function "\grp"): {\displaystyle \grp} , and then defining a $H^{\wedge }-''Algebroid''$ as a quadruple Failed to parse (unknown function "\vep"): {\displaystyle (GH, \Delta, \vep, S)} by modifying the Hopf algebroid definition so that Failed to parse (unknown function "\ID"): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\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.