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Bott periodicity, Topological invariants

Todd Trimble. The Super Brauer Group and Super Division Algebras

• Super division algebras can be understood to relate the raw mind (even part of division super algebra), the contextual mind (isomorphic odd part of division super algebra) and consciousness (the automorphism of the even part). These are three ways of manifesting the same information.

The Tenfold Way of Super Division Algebras

• Gregory Moore. Quantum Symmetries and K-Theory
  • Gregory Moore. Quantum Symmetries and Compatible Hamiltonians. Extended version.
  • Daniel S. Freed, Gregory W. Moore. Twisted equivariant matter. Original source. Describes the 1:1 correspondence between the 10 CT groups and the 10 Morita classes of real and complex Clifford algebras. There is a natural equivalence of categories of representations.
• 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.

The collapse of the eightsome is modeled by the fact that H = H^{-1} in the Super Brauer Algebra where these are equivalence classes. So I need to understand how to interpret H H^{-1}=I.

Tensoring by Cl(0,1) yields the effect of the linear complex structure J. Tensoring H with Cl(0,1) yields H+H where H is identified with the even part and so H+H = H tensor Cl(1,0). Then the fact that Cl(1,0) is the inverse of Cl(0,1) means that next we get H from a Morita equivalency point of view. Moreover this H is purely bosonic. That means conjugating it (inverting it)(with regard to even and odd) does not change it. So we have for this H that H=H^{-1}.

I should explore what this means as regards how Cl(0,1) and Cl(1,0) fit together to give the 2x2 real matrices. And how Cl(0,4) and Cl(4,0) fit together.

Morita equivalence of superalgebras can be understood as: {$A_1\cong A_2\hat{\otimes} \textrm{End}(V)$} - one is a matrix superalgebra over the other - where Koszul sign rule {$(a_1\hat{\otimes}b_1)(a_2\hat{\otimes}b_2)=(-1)^{|a_2||b_1|}a_1a_2\hat{\otimes}b_1b_2$} The Koszul sign rule keeps the tensor product of matrix superalgebras from being a matrix superalgebra directly.

{$(1+e)(1-e)=0$} for {$\mathbb{C}\oplus e\mathbb{C}$}

{$\mathcal{A}\hat{\otimes}\mathcal{A}^{opp}\cong\textrm{End}_K(\mathcal{A})$} is equivalent to {$\mathcal{A}$} being a central simple algebra. And {$\textrm{End}_K(\mathcal{A})\cong_{Morita} K$} can be understood as the identity. So we have {$\mathcal{A}$} and {$\mathcal{A}^{opp}$} as inverses, just as with the orthogonal, unitary, compact symplectic groups. What does that mean? How is it that the algebra and its opposite express orthonormality? A central simple algebra over {$K$} is simple and its center is the field {$K$}.

Math StackExchange. Reason to apply the Koszul sign rule everywhere in graded contexts Keeps track of the orientation.