Definitions of Grassmann-Hopf Al/gebras, Their Dual
Co-Algebras, and Grassmann-Hopf Al/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

\begin{equation} \Lambda^*V = \Lambda^0 V \oplus \Lambda^1 V \oplus \Lambda^2 V \oplus \cdots \end{equation}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 `tangled duality'):

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

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

i

the binary product $A \otimes A \ovsetl{m} A$ , and

ii

the coproduct

Here the are called `section coefficients'. We have then a generalization of associativity to coassociativity:

(0.1)

Next we consider the following ingredients:

(1)

the graded switch $\hat{\tau} (A \otimes B) = (-1)^{\del A \del B} B \otimes A$

(2)

the counit $\varepsilon$ (an algebra morphism) satisfying

(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 , and then defining a $H^{\wedge}- {Algebroid}$ as a quadruple 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 is a 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.

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