note-math

$(ℝ, +, \cdot)$ algebraic structure or

\begin{matrix} ℝ \in Set \\ + & ℝ^{2} \to ℝ \\ \cdot & ℝ^{2} \to ℝ \\ property-real-algebra \end{matrix}

There are two ways to extend to $ℝ^{𝑛}$

Linear Algebra

\begin{matrix} ℝ^{𝑛} \in Set \\ + & {(ℝ^{𝑛})}^{2} \to ℝ^{𝑛} \\ \cdot & ℝ \times ℝ^{𝑛} \to ℝ^{𝑛} \\ property-linear-algebra \end{matrix}

Example Real 2-dimensional space. $(\begin{matrix} 𝑥 \\ 𝑦 \end{matrix}), (\begin{matrix} 𝑥^{'} \\ 𝑦^{'} \end{matrix}) \in ℝ^{2}, 𝑎 \in ℝ$

\begin{matrix} (\begin{matrix} 𝑥 \\ 𝑦 \end{matrix}) + (\begin{matrix} 𝑥^{'} \\ 𝑦^{'} \end{matrix}) & = & (\begin{matrix} 𝑥 + 𝑥^{'} \\ 𝑦 + 𝑦^{'} \end{matrix}) \\ 𝑎 (\begin{matrix} 𝑥 \\ 𝑦 \end{matrix}) & = & (\begin{matrix} 𝑎 𝑥 \\ 𝑎 𝑦 \end{matrix}) \end{matrix}

and the distributive law. The construction of $ℝ^{𝑛}$ property-linear-algebra uses the property-real-algebra of $ℝ$

Algebra

\begin{matrix} (ℝ^{𝑛}) \in Set \\ + & {(ℝ^{𝑛})}^{2} \to ℝ^{𝑛} \\ \cdot & {(ℝ^{𝑛})}^{2} \to ℝ^{𝑛} \\ property-algebra \end{matrix}

Example complex-number_(tag) Complex number. $𝑥 + 𝑦 i, 𝑥^{'} + 𝑦^{'} i \in ℂ$

Addition is the same as $ℝ^{2}$ . Multiplication uses $i^{2} = - 1$ or $\frac{1}{i} = - i$ and the distributive law

\begin{matrix} (𝑥 + 𝑦 i) (𝑥^{'} + 𝑦^{'} i) & = & (𝑥 𝑥^{'} - 𝑦 𝑦^{'}) + (𝑥 𝑦^{'} + 𝑦 𝑥^{'}) i \end{matrix}

one of motivation of complex-number or $𝑧^{2} = - 1$ is the characteristic polynomial equation of #link(<harmonic-oscillator>)[] $𝜉^{2} + 𝜔^{2} = 0$

Example split-complex-number_(tag) Split-complex number. $𝑥 + 𝑦 i_{split}, 𝑥^{'} + 𝑦^{'} i_{split} \in ℂ_{split}$

Addition is the same as $ℝ^{2}$ . Multiplication uses ${i_{split}}^{2} = 1$ or $\frac{1}{i_{split}} = i_{split}$ and the distributive law

\begin{matrix} (𝑥 + 𝑦 i_{split}) (𝑥^{'} + 𝑦^{'} i_{split}) & = & (𝑥 𝑥^{'} + 𝑦 𝑦^{'}) + (𝑥 𝑦^{'} + 𝑦 𝑥^{'}) i_{split} \end{matrix}

linear_(tag) $ℝ^{𝑛}$ linear structure

struct homomorphism := a mapping $𝑓$ that preserves struct

Example linear struct hom $𝑓 : ℝ^{𝑛} \to ℝ^{𝑚}$

$𝑥 \in ℝ^{𝑛}$ map to $𝑓 (𝑥) \in ℝ^{𝑚}$
$+ : {(ℝ^{𝑛})}^{2} \to ℝ^{𝑛}$ map to $𝑓 (+) : {(ℝ^{𝑚})}^{2} \to ℝ^{𝑚}$

so $𝑓 (𝑥 + 𝑥^{'})$ map to $𝑓 (𝑥) 𝑓 (+) 𝑓 (𝑥^{'})$ or abbreviated as $𝑓 (𝑥) + 𝑓 (𝑥^{'})$

linear struct hom is also called linear mapping

bijection to itself + $𝑓, 𝑓^{- 1}$ hom = isomorphism

linear isomorphism of $ℝ^{𝑛}$ := $GL (𝑛, ℝ)$

$ℝ^{𝑛}$ algebraic structure

Example

$ℂ$ complex numbers, $ℍ$ quaternions, $𝕆$ octonions
Matrix algebra. But conceptually and meaningfully, it doesn't seem like a good generalization of $ℝ$ algebra. So we need other restrictions, e.g. normed algebra, composition algebra

normed-algebra_(tag)

The multiplication of $ℝ$ has the property $| 𝑥 𝑦 | = | 𝑥 | | 𝑦 |$

The $ℝ^{𝑛}$ spatial quadratic form has the property ${⟨ 𝑎 𝑥 ⟩}^{2} = 𝑎^{2} {⟨ 𝑥 ⟩}^{2}$

For $ℝ^{𝑛}$ algebra, we expect the property ${⟨ 𝑥 𝑦 ⟩}^{2} = {⟨ 𝑥 ⟩}^{2} {⟨ 𝑦 ⟩}^{2}$

Example

$ℂ$

def complex conjugate $i^{*} = - i$

${(𝑥 + 𝑦 i)}^{*} ≔ 𝑥 + 𝑦 i^{*} = 𝑥 - 𝑦 i$

$𝑧 𝑧^{*} = 𝑧^{*} 𝑧 = | 𝑧 |^{2} = 𝑥^{2} + 𝑦^{2}$ . This is $ℝ^{2}$ spatial

$| 𝑧 𝑧^{'} |^{2} = | 𝑧 |^{2} | 𝑧^{'} |^{2}$ by

\begin{matrix} {(𝑥 𝑥^{'} - 𝑦 𝑦^{'})}^{2} + {(𝑥 𝑦^{'} + 𝑦 𝑥^{'})}^{2} & = & 𝑥^{2} 𝑥^{' 2} + 𝑦^{2} 𝑦^{' 2} + 𝑥^{2} 𝑦^{' 2} + 𝑦^{2} 𝑥^{' 2} \\ = & (𝑥^{2} + 𝑦^{2}) (𝑥^{' 2} + 𝑦^{' 2}) \end{matrix}

$ℂ_{split}$

$𝑧 𝑧^{*} = 𝑧^{*} 𝑧 = | 𝑧 |^{2} = 𝑥^{2} - 𝑦^{2}$ . This is $ℝ^{1, 1}$ spacetime

$| 𝑧 𝑧^{'} |^{2} = | 𝑧 |^{2} | 𝑧^{'} |^{2}$ by

\begin{matrix} {(𝑥 𝑥^{'} + 𝑦 𝑦^{'})}^{2} - {(𝑥 𝑦^{'} + 𝑦 𝑥^{'})}^{2} & = & 𝑥^{2} 𝑥^{' 2} + 𝑦^{2} 𝑦^{' 2} - 𝑥^{2} 𝑦^{' 2} - 𝑦^{2} 𝑥^{' 2} \\ = & (𝑥^{2} - 𝑦^{2}) (𝑥^{' 2} - 𝑦^{' 2}) \end{matrix}

null elements have no multiplicative inverse

$exp (Im (ℂ))$ give $U (1, ℂ) ≃ 𝕊 ≃ SO (2)$
$exp (Im (ℂ_{split}))$ give $U (1, ℂ_{split}) ≃ ℍ 𝕪 ≃ SO (1, 1)$

New imaginary unit construction method

Example

Use a new imaginary unit $i_{2}$ in the complex number $𝑥_{0} + 𝑥_{1} i_{1}$ with ${(i_{1})}^{2} = - 1$

Define other imaginary units $i_{3} ≔ i_{1} i_{2}$
Different imaginary units anticommute $i_{2} i_{1} ≔ - i_{1} i_{2}$
Invert the imaginary unit conjugate or $\frac{1}{i} = - i$
- $\begin{matrix} {(i_{2})}^{*} & ≔ & - & i_{2} \end{matrix}$
- $\begin{matrix} {(i_{3})}^{*} & ≔ & - & i_{3} \end{matrix}$

Anti-commutation + conjugate inversion makes $𝑥^{*} 𝑥 = 𝑥 𝑥^{*} = | 𝑥 |^{2}$ , and also gives ${(𝑥 𝑥^{'})}^{*} = 𝑥^{' *} 𝑥^{*}$

Imaginary unit associativity $i_{𝑖^{″}} (i_{𝑖^{'}} i_{𝑖}) = (i_{𝑖^{″}} i_{𝑖^{'}}) i_{𝑖}$

Satisfies norm multiplication $| 𝑥 𝑦 |^{2} = | 𝑥 |^{2} | 𝑦 |^{2}$

${(i_{3})}^{2} = i_{1} i_{2} i_{1} i_{2} = - i_{1}^{2} i_{2}^{2} = i_{2}^{2}$

\begin{matrix} {(i_{2})}^{2} & = & - 1 & give {(i_{3})}^{2} = - 1 and ℍ \\ {(i_{2})}^{2} & = & + 1 & give {(i_{3})}^{2} = + 1 and ℍ_{split} with (2, 2) signature \end{matrix}

Example If split complex $i_{1}^{2} = 1$ is used, then ${(i_{3})}^{2} = - i_{1}^{2} i_{2}^{2} = - i_{2}^{2}$ , so $i_{2}^{2} = \pm 1$ both give split quaternion

$exp (Im (ℍ))$ give $U (1, ℍ) ≃ 𝕊^{3} ↠ SO (3)$
$exp (Im (ℍ_{split}))$ give $U (1, ℍ_{split}) ≃ ℚ^{2, 2} (1) ↠ SO (1, 2)$

Example Using a new imaginary unit $i_{4}$ in quaternion $𝑥_{0} + 𝑥_{1} i_{1} + 𝑥_{2} i_{2} + 𝑥_{3} i_{3}$

Define other imaginary units

\begin{matrix} i_{5} & ≔ & i_{1} i_{4} \\ i_{6} & ≔ & i_{2} i_{4} \\ i_{7} & ≔ & i_{3} i_{4} \end{matrix}

Anti-commutation of different imaginary units $i_{𝑖^{'}} i_{𝑖} ≔ - i_{𝑖} i_{𝑖^{'}}$

Anti-associativity of different imaginary units $i_{𝑖^{″}} (i_{𝑖^{'}} i_{𝑖}) ≔ - (i_{𝑖^{″}} i_{𝑖^{'}}) i_{𝑖}$ if $i_{𝑖^{″}} \neq \pm i_{𝑖^{'}} i_{𝑖}$

Imaginary unit conjugate inversion ${(i_{𝑖})}^{*} = - i_{𝑖}$

Anti-commutation + conjugate inversion makes $𝑥^{*} 𝑥 = 𝑥 𝑥^{*} = | 𝑥 |^{2}$ , and also gives ${(𝑥 𝑥^{'})}^{*} = 𝑥^{' *} 𝑥^{*}$

Question Anti-associativity is needed for norm multiplication $| 𝑥 𝑦 |^{2} = | 𝑥 |^{2} | 𝑦 |^{2}$

${(i_{4})}^{2} = - 1$ gives octonion $𝕆$ . split octonion similarly, with $(4, 4)$ signature

$exp (Im (𝕆))$ give $U (1, 𝕆) ≃ 𝕊^{7} ↪ SO (7)$ (Question)
$exp (Im (𝕆_{split}))$ give $U (1, 𝕆_{split}) ≃ ℚ^{4, 4} (1) ↪ SO (3, 4)$

What is obtained from $ℍ$ and the associative law of imaginary units is another algebra $ℍ \oplus ℍ$ , which does not satisfy $| 𝑥 𝑦 |^{2} = | 𝑥 |^{2} | 𝑦 |^{2}$

Question Anti-combination cannot be further extended to sixteen dimensions and beyond

The new imaginary unit construction method is not coordinate-free, so we need to consider the automorphism of imaginary units $𝑓 : Im (𝕂) \to Im (𝕂)$ with $𝑓 (𝑥 𝑦) = 𝑓 (𝑥) 𝑓 (𝑦)$ . Since it preserves multiplication, it automatically preserves distance

Example for $ℂ$ it's $ℤ_{2} = O (1)$ symmetric $i \to - i$

Question

$SO (3)$ for $ℍ$
$𝐺_{2}$ for $𝕆$ . $dim 𝐺_{2} = 14 < 21 = dim (SO (7))$

$𝐺_{2}$ as automorphism of $𝕆$ illustrates that, without additional structure, such as multiplication $𝑥 \cdot 𝑦$ and $| 𝑥 \cdot 𝑦 |^{2} = | 𝑥 |^{2} | 𝑦 |^{2}$ , only a simple linear space structure, it is impossible to give special groups like $𝐺_{2}$ . (Although it is said that all compact groups can have matrix representations.)

affine_(tag) $ℝ^{𝑛}$ affine structure

Change the origin, translate

hom additionally keeps $𝑓 (𝑥 - 𝑦) = 𝑓 (𝑥) 𝑓 (-) 𝑓 (𝑦)$ abbreviated as $𝑓 (𝑥) - 𝑓 (𝑦)$