Book: Rotations

Understand Lie groups in terms of rotations.

Understand numbers

• Understand quaternions and complex numbers of complex numbers, and relate that description to the i, j, k three-cycle.
• In category theory, what does or could play the role of the conjugates i and -i in labeling the direction of morphisms?
• Consider what the Jacob identity tells us in terms of distinguishing marked opposites and unmarked opposites (conjugates).

Understand matrices

• Why are the transpose, conjugate transpose, etc. the only "path reversals" possible for continuity?
• Relate the combinatorial interpretation of the symplectic inverse to the conjugate transpose. Is the conjugation applied twice and thus it disappears? (Consider Vogan).
• Why does the determinant have to equal 1 so that Cramer's rule works out nicely?

Understand how real rotations and their axes chain together real dimensions.

• Study three dimensional rotations, how they add, and how they relate to spinors.
• Try to imagine rotations in four real dimensions.

Understand rotations in complex dimensions and quaternionic dimensions.

• Understand how {$SO(n,\mathbb{C})$} relates to rotations in real dimensions.
• Understand how {$SL(n,\mathbb{C})$} relates to rotations in complex dimensions.
• Understand how {$Sp(n,\mathbb{C})$} relates to rotations in quaternionic dimensions.
• Give a combinatorial intepretation of {$O^TO = OO^T = I$}.
• Give a combinatorial intepretation of {$U^*U = UU^* = I$}.
• Understand how rotations in complex dimensions relate to rotations in pairs of real dimensions.
• Study how rotations work in two complex dimensions.
• Understand how quaternions model real rotations in three dimensions.

Understand how, in general, rotations and their axes chain together dimensions.

• Look for the symmetry that can be interpreted in the chains of dimensions.
• What is the role of the determinant in the special linear group? As the product of the eigenvalues?

Linear algebra

• Understand polar decomposition.
• Can the eigenvectors of a matrix be understood as axes of rotation?
• Given a complex matrix, what are the possibilities for its inverse, if we want continuity?

Geometry

• Relate geometry as the regularity of choice and as rotations.
• Understand how the six transformations relate to rotations.
• Consider a triangle as offering four ways of looking at rotations.

• Understand rotation in one octonion dimension.
• Understand Bott periodicity in terms of rotations.
• Think of exact sequences as relating a sequence of axes or eigenvectors.

Perspectives

• Consider rotations around rotations as an analogy of perspectives on perspectives, and likewise rotations around rotations around rotations.

Mathematical facts

Numbers

• Unmarked (ambiguous) opposites (conjugates): left x left x left = right and right x right x right = left
• Marked (unambiguous) opposites (1 and -1): 1 x 1 x 1 = 1 and -1 x -1 x -1 = -1
• Thus cubing unmarked opposites yield the opposite, whereas cubing marked opposites yields themselves.
• Thus the three-cycle distinguishes between marked and unmarked opposites - if you apply three times, then you can tell if you stay or switch.
• One dimension: real numbers - reflection, complex numbers - rotation.
• Explore the significance of the fact that reflection is not continuous.
• Complex numbers express in two dimensions the asymmetrical choice (+1 or -1) and the symmetrical choice (this i or that j).
• Complex numbers combine conjugates (rotations left and right) and positive-negative distinction (amplification by positive R and reflection by -R).

Norm-preserving matrices

• Orthogonal matrices are the real matrices that preserve the real norm.
• Unitary matrices are the complex matrices that preserves the complex norm.
• Symplectic group. The compact symplectic group {$\textrm{Sp}(n)=U(n,\mathbb{H})$} is the quaternionic unitary group or hyperunitary group. It consists of the quaternionic matrices that preserve the standard Hermitian form on {$H^n$}.

Rotation matrices

Transpose

• {$U^*U = UU^* = I$} where {$U^*$}, the conjugate tranpose of {$U$}, is also its inverse.

The relevant complex matrices.

• Given a complex matrix, its inverse can be its transpose, so that it is an orthogonal (complex) matrix. What are its properties? Why does it matter if the dimension is odd or even?
• Given a complex matrix, its inverse can be its conjugate transpose.
• Given a complex matrix {$M$}, its inverse can be {$\Omega^{-1}M^\text{T} \Omega = - \Omega M^\text{T} \Omega$} where

{$\Omega = \begin{bmatrix} 0 & I_n \\ -I_n & 0 \\ \end{bmatrix}$}

Combinatorics

• The transpose reverses the arrows (as in the opposite category).
• The conjugate transpose reverses the arrows and also switches the conjugate (thus matching the conjugate to a direction of the arrow).
• Only the unitary case makes use of the conjugate. Both the real and the quaternionic cases lose this and are degenerate in that sense. That degeneracies is a loss of the duality of counting forwards and backwards - the two directions get connected and distinguished.

Ideas

• Rotations are perspectives.
• Rotations distinguish a constraint (the fixed length) and a freedom (to rotate).
• The constraint can be understood as a continuum of positive lengths and their negatives.
• Thus the Lie group needs to relate all possible axes.
• Rotations chain together dimensions. And there are symmetries inherent in those chains.

Geometry

• Geometry (rotations) relates circles (analysis) and triangles (algebra).

Polytopes define axes of rotations and related geometrical frameworks. They define different ways of describing opposite directions of rotation.

• A simplex defines vectors which specify directed axes of rotations.
• A cross-polytope defines lines which specify undirected axes of rotations.
• A hypercube defines angles, and the angles are partial rotations.
• A coordinate system defines directed axes in pairs.

Oriented areas are swept out by rotations and the axis can be placed anywhere inside the area of the triangle.