Introduction E9F5FC

Questions FFFFC0

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• Zig zag lemma
• Derived functors
• Jacobi identity
• Pauli matrix
• Quaternions
• Mayer-Vietoris sequence
• John Boyd's OODA loop Observe - Orient - Decide - Act
• https://en.wikipedia.org/wiki/Learning_cycle
• https://en.wikipedia.org/wiki/Double-loop_learning
• https://en.wikipedia.org/wiki/Decision_cycle
• https://en.wikipedia.org/wiki/DMAIC DMAIC Improvement cycle
• https://en.wikipedia.org/wiki/PDCA PDCA Improvement cycle

What do the Pauli matrices say about the Threefold way?

• Rotations: real, complex, quaternion
• http://en.wikipedia.org/wiki/Bloch_sphere rotation operators about the Bloch basis
• random matrices, ensembles
• Carlo Beenakker: In the context of Dyson's threefold way, the Pauli matrices produce two of the three ensembles of random Hamiltonians. A Hermitian matrix 𝐻 with normally distributed matrix elements belongs to the Gaussian Orthogonal Ensemble (GOE) if the matrix elements are real, to the Gaussian Unitary Ensemble (GUE) if the matrix elements are complex, and to the Gaussian Ensemble (GSE) if the matrix elements are linear combinations of Pauli matrices of the form 𝐻𝑛𝑚=𝑎(0)𝑛𝑚𝐼2+𝑖∑𝑝=13𝑎(𝑝)𝑛𝑚𝜎𝑝,𝑎(0),𝑎(1),𝑎(2),𝑎(3)∈ℝ. The restriction to real coefficients is essential, without it the Hamiltonian ensemble is the GUE instead of the GSE. The GOE cannot be obtained from Pauli matrices.

For John time reversal https://golem.ph.utexas.edu/category/2011/01/the_threefold_way_part_4_1.html

The Three-Fold Way actually says that every self-dual irreducible unitary complex representation of a group comes from either: • a real-unitary representation on a real Hilbert space, or • a quaternion-unitary representation on a quaternionic Hilbert space.

Jacobi identity is like a product rule. Think of x as differentiation (and y and z perhaps likewise). {$[x,[y,z]] = [[x,y],z] + [y,[x,z]]$}

• Vector cross product is an example of the three-cycle.
• Vector bundles. Jimmy told me about vector bundles having a threefold sense: Data, gluing, and 3 sets coming together. This happens in 1 degree, 2 degrees, 3 degrees.