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 an arbitrary 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 structure and 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 product and the unit are ‘co–gebra’ morphisms, and 2) the coproduct and counit 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– the septet $\widehat{H}=({\mathrm{\Lambda }}^{*}V,\wedge ,\mathrm{id},\epsilon ,\widehat{\tau },S)$.

Its generalization to a Grassmann-Hopf algebroid is straightforward by considering a groupoid $𝖦$, and then defining a $H^{\wedge }-\text{𝐴𝑙𝑔𝑒𝑏𝑟𝑜𝑖𝑑}$ as a quadruple $(GH,\mathrm{\Delta },\epsilon ,S)$ by modifying the Hopf algebroid definition so that $\widehat{H}=({\mathrm{\Lambda }}^{*}V,\wedge ,\mathrm{id},\epsilon ,\widehat{\tau },S)$ satisfies the standard Grassmann-Hopf algebra axioms stated above. We may also say that $(HG,\mathrm{\Delta },\epsilon ,S)$ is a weak C*-Grassmann-Hopf algebroid when $H^{\wedge }$ is a unital C*-algebra (with $\mathrm{𝟏}$). We thus set $𝔽=ℂ$. 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.

Title	Grassmann-Hopf algebras and coalgebras
Canonical name	GrassmannHopfAlgebrasAndCoalgebrasgebras
Date of creation	2013-03-22 18:10:55
Last modified on	2013-03-22 18:10:55
Owner	bci1 (20947)
Last modified by	bci1 (20947)
Numerical id	52
Author	bci1 (20947)
Entry type	Topic
Classification	msc 81T18
Classification	msc 81T13
Classification	msc 55Q25
Classification	msc 81T10
Classification	msc 16W30
Classification	msc 81T05
Classification	msc 57T05
Classification	msc 15A75
Synonym	tangled-dual Grassmann-Hopf co-algebra
Related topic	QED
Related topic	WeakHopfCAlgebra2
Related topic	WeakHopfCAlgebra
Related topic	DualOfACoalgebraIsAnAlgebra
Related topic	CAlgebra3
Related topic	TopicEntryOnTheAlgebraicFoundationsOfQuantumAlgebraicTopology
Related topic	AlgebraicFoundationsOfQuantumAlgebraicTopology
Related topic	QuantumAlgebraicTopologyOfCWComplexRepresentationsNewQATResult
Defines	Grassmann-Hopf algebra
Defines	dual Grassmann-Hopf co-algebra and gebra or tangled algebra
Defines	observable operator algebra
Defines	Grassman-Hopf algebroid