Division algebra ladder — flashcards
Every card the map can derive about Division algebra ladder — definitions, statements, attribution, and the edges it sits on.
Theorems 1
State: Hurwitz's theorem on normed division algebras.
The only normed division algebras over ℝ are ℝ, ℂ, ℍ and 𝕆, of dimensions 1, 2, 4 and 8. This is why the ladder stops.
derived from st/hurwitz by the statement rule
Names and dates 1
Hurwitz's theorem on normed division algebras — who, and when?
Adolf Hurwitz, 1898.
derived from st/hurwitz by the attribution rule
Connections 7
Under the Information cost lens, how do Division algebra ladder and Groups & symmetry relate? [Information cost]
Climbing the ladder surrenders order, then commutativity, then associativity — and the amounts are not equal. This edge is the same observation as the What breaks edge; it is here because how much is forgotten and what is surrendered are different questions.
derived from information-cost:division-algebras→group-theory by the connection rule
Under the Degeneracy lens, how do Division algebra ladder and Rings & fields relate? [Degeneracy]
The commutative, associative, ordered case is one rung of a short ladder that provably stops at eight dimensions. It is the rung everybody starts on and the least general one.
derived from degeneracy:division-algebras→rings-fields by the connection rule
Division algebra ladder and Groups & symmetry look like one thing. What separates them? [Forks]
Descending ℍ → ℂ means choosing a U(1) inside SU(2) — a whole sphere of equally valid choices. Descending 𝕆 → ℍ offers three. Descending ℂ → ℝ offers one. You noticed the descent is non-uniform before I did.
derived from forks:division-algebras→group-theory by the connection rule
What breaks on the way from Division algebra ladder to Order? [What breaks]
The first rung of the ladder. Adjoining i buys algebraic closure and costs the ability to compare any two elements — which is why there is no sensible ordering on ℂ.
derived from what-breaks:division-algebras→prop/order by the connection rule
What breaks on the way from Division algebra ladder to Commutativity? [What breaks]
Doubling again costs ab = ba. The loss is productive rather than merely regrettable: non-commutativity is exactly what lets quaternions encode rotation in three dimensions.
derived from what-breaks:division-algebras→prop/commutativity by the connection rule
What breaks on the way from Division algebra ladder to Associativity? [What breaks]
The last rung costs (ab)c = a(bc). Hurwitz says there is nothing further to surrender, which is why the ladder stops at eight dimensions rather than continuing.
derived from what-breaks:division-algebras→prop/associativity by the connection rule
Under the Correspondence test lens, how do Division algebra ladder and Topological spaces relate? [Correspondence test]
REFUTED as proposed. The instinct that a complex-analytic proof of the necklace theorem wouldn't generalize was right; the reason given (a quaternion/division-algebra obstruction) was wrong. The actual obstruction is Adams' Hopf-invariant-one theorem — non-parallelisability of spheres — related territory, wrong mechanism.
Recorded as refuted: this edge was drawn and then withdrawn.
derived from correspondence-test:division-algebras→topology by the connection rule
Concepts 1
What is Division algebra ladder, in one sentence?
ℝ → ℂ → ℍ → 𝕆. Each rung doubles dimension and surrenders a property: order, then commutativity, then associativity. Hurwitz's theorem says it stops at eight.
derived from division-algebras by the concept rule