Talk:PlanetPhysics/Topic on Algebraic Foundations of Quantum Algebraic Topology

Original TeX Content from PlanetPhysics Archive
%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: topic on algebraic foundations of quantum algebraic topology %%% Primary Category Code: 03. %%% Filename: TopicOnAlgebraicFoundationsOfQuantumAlgebraicTopology.tex %%% Version: 8 %%% Owner: bci1 %%% Author(s): bci1 %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

\setlength{\topmargin}{0.00in} \setlength{\headsep}{0.00in} \setlength{\headheight}{0.00in} \setlength{\evensidemargin}{0.00in} \setlength{\oddsidemargin}{0.00in} \setlength{\textwidth}{6.5in} \setlength{\textheight}{9.00in} \setlength{\voffset}{0.00in} \setlength{\hoffset}{0.00in} \setlength{\marginparwidth}{0.00in} \setlength{\marginparsep}{0.00in} \setlength{\parindent}{0.00in} \setlength{\parskip}{0.15in}

\usepackage{html}

\begin{document}

This is a contributed topic on \htmladdnormallink{Quantum Algebraic Topology}{http://planetphysics.us/encyclopedia/TriangulationMethodsForQuantizedSpacetimes2.html} (QAT) introducing mathematical \htmladdnormallink{concepts}{http://planetphysics.us/encyclopedia/PreciseIdea.html} of QAT based on \htmladdnormallink{algebraic topology}{http://planetphysics.us/encyclopedia/ModuleAlgebraic.html} (AT), \htmladdnormallink{category theory}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} (CT) and their \htmladdnormallink{non-Abelian}{http://planetphysics.us/encyclopedia/NonAbelianQuantumAlgebraicTopology3.html} extensions in \htmladdnormallink{higher dimensional algebra}{http://planetphysics.us/encyclopedia/InfinityGroupoid.html} (\htmladdnormallink{HDA}{http://planetphysics.us/encyclopedia/2Groupoid2.html}) and \htmladdnormallink{supercategories}{http://planetphysics.us/encyclopedia/SuperCategory6.html}.

\subsection{Introduction}

\emph{Quantum algebraic topology (QAT)} is an area of \htmladdnormallink{physical mathematics}{http://planetphysics.us/encyclopedia/IHES.html} and \htmladdnormallink{mathematical physics}{http://planetphysics.us/encyclopedia/PhysicalMathematics2.html} concerned with the foundation and study of \htmladdnormallink{general theories}{http://planetphysics.us/encyclopedia/GeneralTheory.html} of quantum \htmladdnormallink{algebraic structures}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} from the standpoint of algebraic topology, category theory, as well as non-Abelian extensions of AT and CT in higher dimensional algebra and supercategories.

\subsubsection{The following are examples of QAT topics:}

\begin{enumerate}

\item \htmladdnormallink{Poisson algebras}{http://planetphysics.us/encyclopedia/PoissonRing.html}, \htmladdnormallink{quantization methods}{http://planetphysics.us/encyclopedia/QuantizationMethods.html} and \htmladdnormallink{Hamiltonian algebroids}{http://planetphysics.us/encyclopedia/Algebroids.html}

\item K-S theorem and its quantum \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} consequences in QAT

\item Logic lattice algebras and many-valued (MV) logic algebras

\item Quantum MV-logic algebras and $\L{}-M_n$-noncommutative algebras

\item \htmladdnormallink{quantum operator algebras}{http://planetphysics.us/encyclopedia/Groupoid.html} ( such as : involution, *-algebras, or $*$-algebras, \htmladdnormallink{von Neumann algebras}{http://planetphysics.us/encyclopedia/VonNeumannAlgebra2.html},, JB- and JL- algebras, $C^*$ - or C*- algebras,

\item Quantum von Neumann algebra and subfactors

\item Kac-Moody and K-algebras

\item \htmladdnormallink{quantum groups}{http://planetphysics.us/encyclopedia/ComultiplicationInAQuantumGroup.html}, quantum group algebras and \htmladdnormallink{Hopf algebras}{http://planetphysics.us/encyclopedia/Groupoid.html}

\item \htmladdnormallink{quantum groupoids}{http://planetphysics.us/encyclopedia/WeakHopfAlgebra.html} and weak Hopf $C^*$-algebras

\item \htmladdnormallink{groupoid C*-convolution algebras}{http://planetphysics.us/encyclopedia/GroupoidCConvolutionAlgebra.html} and *-convolution \htmladdnormallink{algebroids}{http://planetphysics.us/encyclopedia/Algebroids.html} \item \htmladdnormallink{Quantum spacetimes}{http://planetphysics.us/encyclopedia/QuantumSpaceTimes.html} and \htmladdnormallink{quantum fundamental groupoids}{http://planetphysics.us/encyclopedia/QuantumFundamentalGroupoid4.html}

\item Quantum double algebras

\item \htmladdnormallink{quantum gravity}{http://planetphysics.us/encyclopedia/LQG2.html}, \htmladdnormallink{supersymmetries}{http://planetphysics.us/encyclopedia/Supersymmetry.html}, \htmladdnormallink{supergravity}{http://planetphysics.us/encyclopedia/AntiCommutationRelations.html}, \htmladdnormallink{superalgebras}{http://planetphysics.us/encyclopedia/MathematicalFoundationsOfQuantumTheories.html} and graded `\htmladdnormallink{Lie' algebras}{http://planetphysics.us/encyclopedia/BilinearMap.html} \item Quantum \htmladdnormallink{categorical algebra}{http://planetphysics.us/encyclopedia/CategoryOfLogicAlgebras.html} and higher dimensional, $\L{}-M_n$- toposes

\item Quantum \htmladdnormallink{R-categories}{http://planetphysics.us/encyclopedia/RCategory.html}, \htmladdnormallink{R-supercategories}{http://planetphysics.us/encyclopedia/RDiagram.html} and symmetry breaking

\item \htmladdnormallink{extended quantum symmetries}{http://planetphysics.us/encyclopedia/ExtendedQuantumSymmetries.html} in higher dimensional algebras (HDA), such as: algebroids, \htmladdnormallink{double algebroids}{http://planetphysics.us/encyclopedia/GeneralizedSuperalgebras.html}, categorical algebroids, \htmladdnormallink{double groupoids}{http://planetphysics.us/encyclopedia/WeakHomotopy.html}\htmladdnormallink{,convolution}{http://planetphysics.us/encyclopedia/AssociatedGroupoidAlgebraRepresentations.html} algebroids, and \htmladdnormallink{groupoid}{http://planetphysics.us/encyclopedia/LocallyCompactGroupoid.html} $C^*$ -convolution algebroids

\item Universal algebras in R-supercategories

\item Supercategorical algebras (SA) as concrete interpretations of the theory of elementary abstract supercategories (\htmladdnormallink{ETAS}{http://planetphysics.us/encyclopedia/ETACAxioms.html}).

\item \htmladdnormallink{Non-Abelian quantum algebraic topology (NAQAT)}{http://planetmath.org/?op=getobj&from=papers&id=410} \item \htmladdnormallink{noncommutative geometry}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html}, quantum geometry, and non-Abelian quantum algebraic geometry \item Kochen-Specker theorem (K-S theorem) \item Other -- Miscellaneous \end{enumerate}

\begin{thebibliography} {9} \bibitem{AS} Alfsen, E.M. and F. W. Schultz: \emph{Geometry of State Spaces of Operator Algebras}, Birk\"auser, Boston--Basel--Berlin (2003).

\bibitem{AMF56} Atiyah, M.F. 1956. On the Krull-Schmidt theorem with applications to sheaves. \emph{Bull. Soc. Math. France}, \textbf{84}: 307--317.

\bibitem{AS2k6} Awodey, S., 2006, Category Theory, Oxford: Clarendon Press.

\bibitem{BAJ-DJ98a} Baez, J. \& Dolan, J., 1998a, Higher-Dimensional Algebra III. n-Categories and the Algebra of Opetopes, Advances in Mathematics, 135, 145--206.

\bibitem{BAJ-DJ2k1} Baez, J. \& Dolan, J., 2001, From Finite Sets to Feynman Diagrams, Mathematics Unlimited -- 2001 and Beyond, Berlin: Springer, 29--50.

\bibitem{BAJ-DJ97} Baez, J., 1997, An Introduction to n-Categories, Category Theory and Computer Science, Lecture Notes in Computer Science, 1290, Berlin: Springer-Verlag, 1--33.

\bibitem{ICB4} Baianu, I.C.: 1971b, Categories, Functors and Quantum Algebraic Computations, in P. Suppes (ed.), \emph{Proceed. Fourth Intl. Congress Logic-Mathematics-Philosophy of Science}, September 1--4, 1971, Bucharest.

\bibitem{Bgg2} Baianu, I. C., Glazebrook, J. F. and G. Georgescu: 2004, Categories of Quantum Automata and N-Valued \L ukasiewicz Algebras in Relation to Dynamic Bionetworks, \textbf{(M,R)}--Systems and Their Higher Dimensional Algebra, \htmladdnormallink{Abstract and Preprint of Report}{http://www.ag.uiuc.edu/fs401/QAuto.pdf}

\bibitem{Bggb4} Baianu, I.C., R. Brown and J. F. Glazebrook: 2007b, A Non-Abelian, Categorical Ontology of Spacetimes and Quantum Gravity, Axiomathes, 17: 169-225.

\bibitem{Ba-We85} Barr, M. and Wells, C., 1985, Toposes, Triples and Theories, New York: Springer-Verlag.

\bibitem{BM-CW99} Barr, M. and Wells, C., 1999, Category Theory for Computing Science, Montreal: CRM.

\bibitem{BJL81} Bell, J. L., 1981, Category Theory and the Foundations of Mathematics, British Journal for the Philosophy of Science, 32, 349--358.

\bibitem{BJL82} Bell, J. L., 1982, Categories, Toposes and Sets, Synthese, 51, 3, 293--337.

\bibitem{BJL86} Bell, J. L., 1986, From Absolute to Local Mathematics, Synthese, 69, 3, 409--426.

\bibitem{BG-MCLS99} Birkoff, G. \& Mac Lane, S., 1999, Algebra, 3rd ed., Providence: AMS.

\bibitem{Borceux94} Borceux, F.: 1994, \emph{Handbook of Categorical Algebra}, vols: 1--3, in {\em Encyclopedia of Mathematics and its Applications} \textbf{50} to \textbf{52}, Cambridge University Press.

\bibitem{Bourbaki1} Bourbaki, N. 1961 and 1964: \emph{Alg\`{e}bre commutative.}, in \'{E}l\'{e}ments de Math\'{e}matique., Chs. 1--6., Hermann: Paris.

\bibitem{BJk4} Brown, R. and G. Janelidze: 2004, Galois theory and a new homotopy double groupoid of a map of spaces, \emph{Applied Categorical Structures} \textbf{12}: 63-80.

\bibitem{BHR2} Brown, R., Higgins, P. J. and R. Sivera,: 2008, \emph{Non-Abelian Algebraic Topology}, (vol.2 in preparation).

\bibitem{Br-Har-Ka-Po2k2} Brown, R., Hardie, K., Kamps, H. and T. Porter: 2002, The homotopy double groupoid of a Hausdorff space., \emph{Theory and Applications of Categories} \textbf{10}, 71-93.

\bibitem{Br-Hardy76} Brown, R., and Hardy, J.P.L.:1976, Topological groupoids I: universal constructions, \emph{Math. Nachr.}, 71: 273-286.

\bibitem{Br-Sp76} Brown, R. and Spencer, C.B.: 1976, Double groupoids and crossed modules, \emph{Cah. Top. G\'{e}om. Diff.} \textbf{17}, 343-362.

\bibitem{BR-SCB76} Brown R, Razak Salleh A (1999) Free crossed resolutions of groups and presentations of modules of identities among relations. {\em LMS J. Comput. Math.}, \textbf{2}: 25--61.

\bibitem{BDA55} Buchsbaum, D. A.: 1955, Exact categories and duality., Trans. Amer. Math. Soc. \textbf{80}: 1-34.

\bibitem{BDA55} Buchsbaum, D. A.: 1969, A note on homology in categories., Ann. of Math. \textbf{69}: 66-74.

\bibitem{BL2k3} Bunge, M. and S. Lack: 2003, Van Kampen theorems for toposes, \emph{Adv. in Math.} \textbf{179}, 291-317.

\bibitem{BM84} Bunge, M., 1984, Toposes in Logic and Logic in Toposes, \emph{Topoi}, 3, no. 1, 13-22.

\bibitem{BM-LS2k3} Bunge M, Lack S (2003) Van Kampen theorems for toposes. {\em Adv Math}, \textbf {179}: 291-317.

\bibitem{CH-ES56} Cartan, H. and Eilenberg, S. 1956. {\em Homological Algebra}, Princeton Univ. Press: Pinceton.

\bibitem{CPM65} Cohen, P.M. 1965. {\em Universal Algebra}, Harper and Row: New York, London and Tokyo.

\bibitem{CA94} Connes A 1994. \emph{Noncommutative geometry}. Academic Press: New York.

\bibitem{CR-LL63} Croisot, R. and Lesieur, L. 1963. \emph{Alg\`ebre noeth\'erienne non-commutative.}, Gauthier-Villard: Paris.

\end{thebibliography}

\end{document}