# 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, $\dim_{\mathcal{C}}V=n$, and let $\{e_{0},e_{1},\ldots,\}$ with identity $e_{0}\equiv 1$, be the generators of a Grassmann (exterior) algebra

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

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

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:

 $\displaystyle\Delta(x)$ $\displaystyle=\sum_{r}a_{r}\otimes b_{r}=\sum_{(x)}x_{(1)}\otimes x_{(2)}=x_{(% 1)}\otimes x_{(2)}$ $\displaystyle\Delta(x^{i})$ $\displaystyle=\sum_{i}\Delta^{jk}_{i}=\sum_{(r)}a^{j}_{(r)}\otimes b^{k}_{(r)}% =x_{(1)}\otimes x_{(2)}$

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

 $\begin{CD}C@>{\Delta}>{}>C\otimes C\\ @V{}V{\Delta}V@V{}V{{\rm id}\otimes\Delta}V\\ C\otimes C@>{\Delta\otimes{\rm id}}>{}>C\otimes C\otimes C\end{CD}$ (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 algebroid 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.

## References

• 1 E. M. Alfsen and F. W. Schultz: Geometry of State Spaces of Operator Algebras, Birkhäuser, Boston–Basel–Berlin (2003).
• 2 I. Baianu : Categories, Functors and Automata Theory: A Novel Approach to Quantum Automata through Algebraic–Topological Quantum Computations., Proceed. 4th Intl. Congress LMPS, (August-Sept. 1971).
• 3 I. C. Baianu, J. F. Glazebrook and R. Brown.: A Non–Abelian, Categorical Ontology of Spacetimes and Quantum Gravity., Axiomathes 17,(3-4): 353-408(2007).
• 4 I.C.Baianu, R. Brown J.F. Glazebrook, and G. Georgescu, Towards Quantum Non–Abelian Algebraic Topology, (2008).
• 5 F.A. Bais, B. J. Schroers and J. K. Slingerland: Broken quantum symmetry and confinement phases in planar physics, Phys. Rev. Lett. 89 No. 18 (1–4): 181–201 (2002).
• 6 J.W. Barrett.: Geometrical measurements in three-dimensional quantum gravity. Proceedings of the Tenth Oporto Meeting on Geometry, Topology and Physics (2001). Intl. J. Modern Phys. A 18 , October, suppl., 97–113 (2003)
• 7 M. Chaician and A. Demichev: Introduction to Quantum Groups, World Scientific (1996).
• 8 Coleman and De Luccia: Gravitational effects on and of vacuum decay., Phys. Rev. D 21: 3305 (1980).
• 9 L. Crane and I.B. Frenkel. Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases. Topology and physics. J. Math. Phys. 35 (no. 10): 5136–5154 (1994).
• 10 W. Drechsler and P. A. Tuckey: On quantum and parallel transport in a Hilbert bundle over spacetime., Classical and Quantum Gravity, 13:611-632 (1996). doi: 10.1088/0264–9381/13/4/004
• 11 V. G. Drinfel’d: Quantum groups, In Proc. Int. Congress of Mathematicians, Berkeley, 1986, (ed. A. Gleason), Berkeley, 798-820 (1987).
• 12 G. J. Ellis: Higher dimensional crossed modules of algebras, J. of Pure Appl. Algebra 52: 277-282 (1988), .
• 13 P.. I. Etingof and A. N. Varchenko, Solutions of the Quantum Dynamical Yang-Baxter Equation and Dynamical Quantum Groups, Comm.Math.Phys., 196: 591-640 (1998).
• 14 P. I. Etingof and A. N. Varchenko: Exchange dynamical quantum groups, Commun. Math. Phys. 205 (1): 19-52 (1999)
• 15 P. I. Etingof and O. Schiffmann: Lectures on the dynamical Yang–Baxter equations, in Quantum Groups and Lie Theory (Durham, 1999), pp. 89-129, Cambridge University Press, Cambridge, 2001.
• 16 B. Fauser: A treatise on quantum Clifford Algebras. Konstanz, Habilitationsschrift.
arXiv.math.QA/0202059 (2002).
• 17 B. Fauser: Grade Free product Formulae from Grassmann–Hopf Gebras. Ch. 18 in R. Ablamowicz, Ed., Clifford Algebras: Applications to Mathematics, Physics and Engineering, Birkhäuser: Boston, Basel and Berlin, (2004).
• 18 J. M. G. Fell.: The Dual Spaces of C*–Algebras., Transactions of the American Mathematical Society, 94: 365–403 (1960).
• 19 F.M. Fernandez and E. A. Castro.: (Lie) Algebraic Methods in Quantum Chemistry and Physics., Boca Raton: CRC Press, Inc (1996).
• 20 R. P. Feynman: Space–Time Approach to Non–Relativistic Quantum Mechanics, Reviews of Modern Physics, 20: 367–387 (1948). [It is also reprinted in (Schwinger 1958).]
• 21 A. Fröhlich: Non-Abelian Homological Algebra. I.Derived functors and satellites., Proc. London Math. Soc., 11(3): 239–252 (1961).
• 22 R. Gilmore: Lie Groups, Lie Algebras and Some of Their Applications., Dover Publs., Inc.: Mineola and New York, 2005.
• 23 P. Hahn: Haar measure for measure groupoids., Trans. Amer. Math. Soc. 242: 1–33(1978).
• 24 P. Hahn: The regular representations of measure groupoids., Trans. Amer. Math. Soc. 242:34–72(1978).
• 25 R. Heynman and S. Lifschitz. 1958. Lie Groups and Lie Algebras., New York and London: Nelson Press.
• 26 C. Heunen, N. P. Landsman, B. Spitters.: A topos for algebraic quantum theory, (2008)
arXiv:0709.4364v2 [quant–ph]
 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