Grassmann-Hopf algebras and coalgebras\gebras

0.1 Definitions of Grassmann-Hopf Algebras, Their Dual Co-Algebras, Gebras, Grassmann–Hopf Algebroids and Gebroids

Let V be a (complex) vector space, dim _𝒞V = n, and let {e₀,e₁,…,} with identity e₀ ≡ 1, be the generators of a Grassmann (exterior) algebra

Λ ∗V = Λ0V ⊕ Λ1V ⊕ Λ2V ⊕ ⋅ ⋅⋅

(0.1)

subject to the relation e_ie_j + e_je_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’https://physicslibrary.org/encyclopedia/GroupoidSymmetries.html):

(objects/points, morphisms ) ↦→ (morphisms, objects/points.)

This leads to a tangle duality between an associative (unital algebra) 𝒜 = (A,m), and an associative (unital) ‘co–gebra’ 𝒞 = (C, Δ) :

the binary product A ⊗ AA, and
the coproduct CC ⊗ C

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

∑ ∑ Δ (x) = a ⊗ b = x ⊗ x = x ⊗ x r r (1) (2) (1) (2) ∑r ∑ (x) Δ (xi) = Δjk = aj ⊗ bk(r) = x (1) ⊗ x(2) i i (r) (r)

Here the Δ_i^jk are called ‘section coefficients’. We have then a generalization of associativity to coassociativity:

Δ C − −−→ C ⊗ C ↓Δ Δ ⊗id ↓id⊗Δ C ⊗ C − −−→ C ⊗ C ⊗ C

(0.2)

inducing a tangled duality between an associative (unital algebra 𝒜 = (A,m), and an associative (unital) ‘co–gebra’ 𝒞 = (C, Δ) . The idea is to take this structure and combine the Grassmann algebra (Λ^∗V,∧) with the ‘co-gebra’ (Λ^∗V, Δ_∧) (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 τ(A ⊗ B) = (−1)^∂A∂BB ⊗ A
the counit 𝜀 (an algebra morphism) satisfying (𝜀 ⊗ id)Δ = id = (id ⊗ 𝜀)Δ
the antipode S .

The Grassmann-Hopf algebra H thus consists of–is defined by– the septet H = (Λ^∗V,∧,id,𝜀,τ,S) .

Its generalization to a Grassmann-Hopf algebroidhttps://physicslibrary.org/encyclopedia/Algebroids.html is straightforward by considering a groupoid G, and then defining a H^∧− Algebroid as a quadruple (GH, Δ,𝜀,S) by modifying the Hopf algebroid definition so that H = (Λ^∗V,∧,id,𝜀,τ,S) satisfies the standard Grassmann-Hopf algebra axioms stated above. We may also say that (HG, Δ,𝜀,S) is a weak C*-Grassmann-Hopf algebroid when H^∧ is a unital C*-algebra (with 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.

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]

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