PlanetPhysics/Topic on Algebraic Foundations of Quantum Algebraic Topology

This is a contributed topic on Quantum Algebraic Topology (QAT) introducing mathematical concepts of QAT based on algebraic topology (AT), category theory (CT) and their non-Abelian extensions in higher dimensional algebra (HDA) and supercategories.

Introduction
Quantum algebraic topology (QAT) is an area of physical mathematics and mathematical physics concerned with the foundation and study of general theories of quantum algebraic structures from the standpoint of algebraic topology, category theory, as well as non-Abelian extensions of AT and CT in higher dimensional algebra and supercategories.

The following are examples of QAT topics:

 * 1) Poisson algebras, quantization methods and Hamiltonian algebroids


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


 * 1) Logic lattice algebras and many-valued (MV) logic algebras


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

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


 * 1) Quantum von Neumann algebra and subfactors


 * 1) Kac-Moody and K-algebras


 * 1) quantum groups, quantum group algebras and Hopf 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, and 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) Non-Abelian quantum algebraic topology (NAQAT)
 * 2) noncommutative geometry, quantum geometry, and non-Abelian quantum algebraic geometry
 * 3) Kochen-Specker theorem (K-S theorem)
 * 4) Other -- Miscellaneous

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