PlanetPhysics/Topic on Foundations of Quantum Algebraic Topology

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topic on the Algebraic Foundations of Quantum Algebraic Topology
This is a contributed topic on the algebraic foundations of Quantum Algebraic Topology (QAT)

(A.) Quantum Algebraic Topology (QAT)  is defined as the mathematical and physical study of general theories of quantum algebraic structures from the standpoint of algebraic topology, category theory and their non-Abelian extensions in higher dimensional algebra and supercategories in relation to, or petinent to, quantum theories, Quantum Field Theories, general relativity and its Quantum extensions, quantum gravity.

'''(B). Several suggested new QAT topics are:'''


 * 1) Poisson algebras, quantization methods and Hamiltonian algebroids


 * 1) K-S theorem and its Quantum algebraic consequences in QAT


 * 1) Logic Lattice algebras or Many-Valued (MV) Logic algebras


 * 1) Quantum MV-Logic algebras and $$\L{}-M_n$$-noncommutative algebras


 * 1) quantum operator algebras ( such as : involution, *-algebras, or $$*$$-algebras, von Neumann algebras, JB- and JL- algebras, $$C^*$$ - or C*- algebras, etc.


 * 1) Quantum von Neumann algebra and subfactors


 * 1) Kac-Moody and K-algebras


 * 1) Hopf algebras, Quantum groups and quantum group algebras


 * 1) Quantum groupoids and weak Hopf $$C^*$$-algebras


 * 1) groupoid C*-convolution algebras and *-Convolution algebroids
 * 2) quantum spacetimes and Quantum Fundamental Groupoids


 * 1) Quantum Double Algebras


 * 1) Quantum Gravity, supersymmetries, supergravity, superalgebras and graded `Lie' algebras
 * 2) Quantum categorical algebra and Higher Dimensional, $$\L{}-M_n$$- Toposes


 * 1) Quantum R-categories, R-supercategories and Symmetry Breaking

algebroids, double algebroids, categorical algebroids, double groupoids, \\ convolution algebroids, groupoid $$C^*$$ -convolution algebroids
 * 1) extended quantum symmetries in Higher Dimensional Algebras (HDA), such as: \\


 * 1) Universal algebras in R-Supercategories


 * 1) Supercategorical algebras (SA) as concrete interpretations of the Theory of Elementary Abstract Supercategories (ETAS).


 * 1) Quantum non-Abelian algebraic topology (QNAAT)


 * 1) noncommutative geometry, quantum geometry, and Non-Abelian Quantum Algebraic Geometry

remarks, proofs, conjectures, and so on...]}
 * 1) Other -- Miscellaneous \textbf{[please add here your additions, changes, editing,

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