Andrius Kulikauskas

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  • 读物 书 影片 维基百科

Introduction E9F5FC

Questions FFFFC0


Bott periodicity

Understand the significance of the octonions.


  • Complex numbers describes rotations in two-dimensions, and quaternions can be used to describe rotations in three dimensions. Is there a connection between octonions and rotations in four dimensions?

Clifford algebras

  • Relate the octonions to Clifford algebras. Compare the (associative) Clifford algebra construction with the (nonassociative) Cayley-Dickson construction. Give combinatorial interpretations of both and see how they differ.
  • Is the Cayley-Dickson construction associative up to plus or minus signs (up to reflection)?

Modeling cognition

  • How are the (nonassociative) octonions relate to the (associative) split-biquaternions?
  • Compare the learning three-cycle (for the quaternions) and Fano's plane (eightfold way) for the octonions.
  • Is there a way that the octonions get identified with the reals? The eightsome = nullsome gets understood as a onesome? And the identification of nullsome with onesome is related to the field with one element. And we are left with exceptional Lie structures.


Facts about octonions

  • John Baez, based on Dixon: The group of symmetries (or technically, "automorphisms") of the octonions is the exceptional group {$G_2$}, which contains {$SU(3)$}. To get {$SU(3)$}, we can take the subgroup of {$G_2$} that preserves a given unit imaginary octonion... say {$e_1$}.

Modeling with octonions

  • The octonions can model the nonassociativity of perspectives.
  • The Clifford algebra {$\textrm{Cl}_{0,7}$} with seven generators (squaring to {$-1$}) and {$2^7$} basis elements models the sevensome. The Clifford algebra {$\textrm{Cl}_{0,3}$}, the split-biquaternions, with three generators (squaring to {$-1$}) and {$2^3$} basis elements models the threesome. The octonions have three generators and eight basis elements.


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This page was last changed on May 30, 2024, at 07:06 PM