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Grassmann-Hopf algebras and coalgebras\gebras (Topic)

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

$\displaystyle \Lambda^*V = \Lambda^0 V \oplus \Lambda^1 V \oplus \Lambda^2 V \oplus \cdots$ (0.1)
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'http://physicslibrary.org/encyclopedia/TrivialGroupoid.html):

(objects/points, morphisms)$\displaystyle \mapsto$   (morphisms, objects/points.)

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

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

\begin{equation*} \begin{aligned} \Delta (x) &= \sum_r a_r \otimes b_r = \sum_{(... ..._{(r)} \otimes b^k_{(r)} = x _{(1)} \otimes x_{(2)} \end{aligned}\end{equation*}

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

$\displaystyle \begin{CD} C @> \Delta >> C \otimes C \\ @VV \Delta V @VV {\rm id... ...ta V \\ C \otimes C @> \Delta \otimes {\rm id}>> C \otimes C \otimes C \end{CD}$ (0.2)
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:

(1)  the graded switch $\hat{\tau} (A \otimes B) = (-1)^{\partial A \partial B} B \otimes A$
(2)  the counit $\varepsilon$ (an algebra morphism) satisfying $(\varepsilon \otimes {\rm id}) \Delta = {\rm id}= ({\rm id}\otimes \varepsilon) \Delta$
(3)  the antipode $S$ .

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

Its generalization to a Grassmann-Hopf algebroidhttp://physicslibrary.org/encyclopedia/Algebroids.html is straightforward by considering a groupoid ${\mathsf{G}}$, and then defining a $H^{\wedge}- \textit{Algebroid}$ as a quadruple $(GH, \Delta, \varepsilon , S)$ by modifying the Hopf algebroid definition so that $\widehat{H} = (\Lambda^*V, \wedge, {\rm id}, \varepsilon, \hat{\tau},S)$ satisfies the standard Grassmann-Hopf algebra axioms stated above. We may also say that $(HG, \Delta, \varepsilon , S)$ 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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Keywords:  Grassmann-Hopf algebras, coalgebras\gebras

Cross-references: representations, dynamic diagram, C*-algebra, Hopf algebroid, groupoid, algebroid, morphisms, Hopf algebra, tangled duality, section, coproduct, duality, relation, generators, identity, vector space

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Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
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