Grassmann-Hopf algebras and coalgebras

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

Let $V$ be a (complex) vector space, ${dim}_{𝒞}V=n$, and let $\{e_0,e_1,\mathrm{\dots },\}$ with identity $e_0\equiv 1$, be the generators of a Grassmann (exterior) algebra

$${\mathrm{\Lambda }}^{*}V={\mathrm{\Lambda }}^{0}V\oplus {\mathrm{\Lambda }}^{1}V\oplus {\mathrm{\Lambda }}^{2}V\oplus \mathrm{\cdots }$$(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’):

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

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

• the binary product $A\otimes A\stackrel{𝑚}{⟶}A$, and

• the coproduct $C\stackrel{\Delta }{⟶}C\otimes C$

,where the Sweedler notation (Sweedler, 1996), with respect to anarbitrary basis is adopted:

$\mathrm{\Delta }(x)$$={\displaystyle \sum _r}{a}_r\otimes b_r={\displaystyle \sum _{(x)}}{x}_{(1)}\otimes x_{(2)}=x_{(1)}\otimes x_{(2)}$

$\mathrm{\Delta }(x^i)$$={\displaystyle \sum _i}{\mathrm{\Delta }}_i^{jk}={\displaystyle \sum _{(r)}}{a}_{(r)}^j\otimes b_{(r)}^k=x_{(1)}\otimes x_{(2)}$

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

$$\begin{array}{ccc}\hfill C\hfill & \hfill \stackrel{\mathrm{\Delta }}{\to }\hfill & \hfill C\otimes C\hfill \\ \hfill \downarrow \mathrm{\Delta }\hfill & & \hfill \downarrow \mathrm{id}\otimes \mathrm{\Delta }\hfill & & \\ \hfill C\otimes C\hfill & \hfill \stackrel{\mathrm{\Delta }\otimes \mathrm{id}}{\to }\hfill & \hfill C\otimes C\otimes C\hfill \end{array}$$(0.2)

inducing a tangled duality between an associative (unital algebra$𝒜=(A,m)$, and an associative (unital) ‘co–gebra’$𝒞=(C,\mathrm{\Delta })$ . The idea is to take this structureand combine the Grassmann algebra $({\mathrm{\Lambda }}^{*}V,\wedge )$ with the‘co-gebra’ $({\mathrm{\Lambda }}^{*}V,{\mathrm{\Delta }}_{\wedge })$ (the ‘tangled dual’)along with the Hopf algebra compatibility rules: 1) the productand the unit are ‘co–gebra’ morphisms, and 2) the coproduct andcounit are algebra morphisms.

Next we consider the following ingredients:

• the graded switch $\widehat{\tau }(A\otimes B)={(-1)}^{\partial A\partial B}B\otimes A$

• the counit $\epsilon$ (an algebra morphism) satisfying$(\epsilon \otimes \mathrm{id})\mathrm{\Delta }=\mathrm{id}=(\mathrm{id}\otimes \epsilon )\mathrm{\Delta }$

• the antipode $S$ .

The Grassmann-Hopf algebra $\widehat{H}$ thus consists of–is defined by– theseptet $\widehat{H}=({\mathrm{\Lambda }}^{*}V,\wedge ,\mathrm{id},\epsilon ,\widehat{\tau },S)$.

Its generalization to a Grassmann-Hopf algebroid isstraightforward by considering a groupoid $𝖦$, and thendefining a $H^{\wedge }-\text{𝐴𝑙𝑔𝑒𝑏𝑟𝑜𝑖𝑑}$ as aquadruple $(GH,\mathrm{\Delta },\epsilon ,S)$ by modifying the Hopfalgebroid definition so that$\widehat{H}=({\mathrm{\Lambda }}^{*}V,\wedge ,\mathrm{id},\epsilon ,\widehat{\tau },S)$ satisfies the standardGrassmann-Hopf algebra axioms stated above. We may also say that$(HG,\mathrm{\Delta },\epsilon ,S)$ is a weak C*-Grassmann-Hopfalgebroid when $H^{\wedge }$ is a unital C*-algebra (with $\mathrm{𝟏}$).We thus set $𝔽=ℂ$. Note howeverthat the tangled-duals of Grassman-Hopf algebroids retain both theintuitive interactions and the dynamic diagram advantages of theirphysical, extended symmetry representations exhibited by theGrassman-Hopf al/gebras and co-gebras over those of either weakC*- Hopf algebroids or weak Hopf C*- algebras.