PlanetPhysics/Grassmann Hopf Algebras and Coalgebrasgebras

\newcommand{\sqdiagram}[9]{Failed to parse (unknown function "\diagram"): {\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}} }

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}):

Failed to parse (unknown function "\textsl"): {\displaystyle =(''objects/points'' , ''morphisms'' )= \mapsto =(\textsl{morphisms= , \textsl{objects/points.})}}

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

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

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 (unknown function "\begin{CD}"): {\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:

 \item[(1)] the graded switch </math>\hat{\tau} (A \otimes B) = (-1)^{\del A \del B} B \otimes AFailed to parse (unknown function "\item"): {\displaystyle  \item[(2)] the counit <math>\varepsilon}
 (an algebra morphism) satisfying </math>(\varepsilon \otimes \ID) \Delta = \ID = (\ID \otimes \varepsilon) \DeltaFailed to parse (unknown function "\item"): {\displaystyle    \item[(3)] the antipode <math>S}
~.

The Grassmann-Hopf algebra ${\widehat {H}}$ thus consists of--is defined by-- the septet Failed to parse (unknown function "\ID"): {\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 (unknown function "\vep"): {\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.