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}):

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

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

\item[i] the binary product $$A \otimes A \ovsetl{m} A$$, and \item[ii] the coproduct $$C \ovsetl{\Delta} C \otimes C$$ , where the Sweedler notation (Sweedler, 1996), with respect to an arbitrary basis is adopted: $$

\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 $$\Delta^{jk}_i$$ are called `section coefficients'. We have then a generalization of associativity to coassociativity: $$ \begin{CD} C @> \Delta >> C \otimes C \\ @VV \Delta V   @VV \ID \otimes \Delta V  \\ C \otimes C @> \Delta \otimes \ID >> C \otimes C \otimes C \end{CD} $$ 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 \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 $$S$$~.

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

Its generalization to a Grassmann-Hopf algebroid is straightforward by considering a groupoid $$\grp$$, and then defining a $$H^{\wedge}- Algebroid $$ as a quadruple $$(GH, \Delta, \vep, S)$$ by modifying the Hopf algebroid 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 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.