Topic entry on the algebraic foundations of Quantum Algebraic Topology.

(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

(b). Several suggested new QAT topics are:

1. Poisson algebras, Quantization methods and Hamiltonian algebroids
2. K-S Theorem and its Quantum algebraic consequences in QAT
3. Logic Lattice algebras or Many-Valued (MV) Logic algebras
4. Quantum MV-Logic algebras and $\L{}-M_n$ -noncommutative lattice algebras
5. Quantum Operator Algebras ( such as : involution, *-algebras, or $*$ -algebras, von Neumann algebras, , JB- and JL- algebras, $C^*$ - or C*- algebras,
6. Quantum von Neumann algebra and subfactors
7. Kac-Moody and K-algebras
8. Hopf algebras, Quantum Groups and Quantum group algebras
9. Quantum Groupoids and weak Hopf $C^*$ -algebras
10. Groupoid C*-Convolution algebras and *-Convolution Algebroids
11. Quantum Spacetimes and Quantum Fundamental Groupoids
12. Quantum Double Algebras
13. Quantum Gravity, supersymmetries, supergravity, superalgebras and graded `Lie' algebras
14. Quantum Categorical algebra and Higher Dimensional, $\L{}-M_n$ - Toposes
15. Quantum R-categories, R-supercategories and Symmetry Breaking
16. Extended Quantum Symmetries in Higher Dimensional Algebras (HDA), such as:
    algebroids, double algebroids, categorical algebroids, double groupoids,
    convolution algebroids, groupoid $C^*$ -convolution algebroids
17. Universal algebras in R-Supercategories
18. Supercategorical algebras (SA) as concrete interpretations of the Theory of Elementary Abstract Supercategories (ETAS).
19. Quantum Non-Abelian Algebraic Topology (QNAAT)
20. Noncommutative Geometry, Quantum Geometry, and Non-Abelian Quantum Algebraic Geometry
21. Other - Miscellaneous [please add here your additions, changes, editing, remarks, proofs, conjectures, and so on...]

Bibliography

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29

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[This is an entry-in-progress for a new topic entry on the Algebraic Foundations of Quantum Algebraic Topology.]