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Bott periodicity, Topological invariants

The Tenfold Way of Super Division Aalgebras

• John Baez. The Tenfold Way.
• John Baez. The Tenfold Way. AMS. Real and complex Bott periodicity. Ten super division algebras: R, C, H, two Rs, three Cs, two Hs.
• John Baez. The Ten-Fold Way - Part 1, Part 2, Part 3 (Brauer-Wall monoid and division superalgebras), Part 4
• The Tenfold Way John Baez. Real and complex Bott periodicity.
• Tenfold way. Time reversal expresses becoming (nonsymmetry) or not becoming (symmetry). Charge conjugation expresses being (nonsymmetry) or not being (symmetry). The change between not being and being (Mindey) could be modeled by charge conjugation.
• John Baez. The Tenfold Way. 2023.02.06
• https://math.ucr.edu/home/baez/tenfold.html
• https://math.ucr.edu/home/baez/week211.html

Division superalgebras

• nLab: Division superalgebra A division superalgebra is a superalgebra in which every nonzero homogeneous element is invertible.
• Freeman J. Dyson. The Threefold Way. Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechanics.

Concepts

• A super vector space is simply a vector space written as a direct sum of two parts, called its even and odd parts. In physics, these parts can be used to describe two fundamentally different classes of particles: bosons and fermions, or alternatively, particles and antiparticles. For the pure mathematician, super vector spaces are a fundamental part of the landscape of thought. For example any chain complex, exterior algebra, or Clifford algebra is automatically a super vector space.