PlanetPhysics/Grassmann Hopf Algebras and Coalgebrasgebras
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Definitions of Grassmann-Hopf Algebras, Their Dual Co-Algebras, Gebras, Grassmann--Hopf Algebroids and Gebroids
editLet be a (complex) vector space, , and let with identity , be the generators of a Grassmann (exterior) algebra
subject to the relation ~. 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' :
\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 (unknown function "\ovsetl"): {\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 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 , and an associative (unital) `co--gebra' ~. The idea is to take this structure and combine the Grassmann algebra with the `co-gebra' (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 thus consists of--is defined by-- the septet 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 \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 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 (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 (HG, \Delta, \vep, S)} is a \emph{weak C*-Grassmann-Hopf algebroid} when is a unital C*-algebra (with ). We thus set . 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.
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References
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- ↑ 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).
- ↑ 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).
- ↑ I.C.Baianu, R. Brown J.F. Glazebrook, and G. Georgescu, Towards Quantum Non--Abelian Algebraic Topology , (2008).
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- ↑ 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)
- ↑ M. Chaician and A. Demichev: Introduction to Quantum Groups , World Scientific (1996).
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- ↑ 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).
- ↑ 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
- ↑ V. G. Drinfel'd: Quantum groups, In \emph{Proc. Int. Congress of Mathematicians, Berkeley, 1986}, (ed. A. Gleason), Berkeley, 798-820 (1987).
- ↑ G. J. Ellis: Higher dimensional crossed modules of algebras, J. of Pure Appl. Algebra 52 : 277-282 (1988), .
- ↑ 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).
- ↑ P. I. Etingof and A. N. Varchenko: Exchange dynamical quantum groups, Commun. Math. Phys. 205 (1): 19-52 (1999)
- ↑ 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.
- ↑ B. Fauser: A treatise on quantum Clifford Algebras . Konstanz, Habilitationsschrift. \\ arXiv.math.QA/0202059 (2002).
- ↑ B. Fauser: Grade Free product Formulae from Grassmann--Hopf Gebras. Ch. 18 in R. Ablamowicz, Ed., Clifford Algebras: Applications to Mathematics, Physics and Engineering , Birkh\"{a}user: Boston, Basel and Berlin, (2004).
- ↑ J. M. G. Fell.: The Dual Spaces of C*--Algebras., \emph{Transactions of the American Mathematical Society}, 94 : 365--403 (1960).
- ↑ F.M. Fernandez and E. A. Castro.: (Lie) Algebraic Methods in Quantum Chemistry and Physics. , Boca Raton: CRC Press, Inc (1996).
- ↑ R. P. Feynman: Space--Time Approach to Non--Relativistic Quantum Mechanics, {\em Reviews of Modern Physics}, 20: 367--387 (1948). [It is also reprinted in (Schwinger 1958).]
- ↑ A.~Fr{\"o}hlich: Non-Abelian Homological Algebra. {I}.{D}erived functors and satellites.\/, Proc. London Math. Soc. , 11 (3): 239--252 (1961).
- ↑ R. Gilmore: Lie Groups, Lie Algebras and Some of Their Applications. , Dover Publs., Inc.: Mineola and New York, 2005.
- ↑ P. Hahn: Haar measure for measure groupoids., Trans. Amer. Math. Soc . 242 : 1--33(1978).
- ↑ P. Hahn: The regular representations of measure groupoids., Trans. Amer. Math. Soc . 242 :34--72(1978).
- ↑ R. Heynman and S. Lifschitz. 1958. Lie Groups and Lie Algebras ., New York and London: Nelson Press.
- ↑ C. Heunen, N. P. Landsman, B. Spitters.: A topos for algebraic quantum theory, (2008) \\ arXiv:0709.4364v2 [quant--ph]