Algebras whose multiplication need not satisfy the associative law (ab)c = a(bc). Three canonical non-Lie strands: Jordan algebras (commutative, satisfying the Jordan axiom (a²∘b)∘a = a²∘(b∘a)) — arose in Jordan-von Neumann-Wigner 1934 as…
nonassociative-algebra
Jordan algebra (commutative nonassociative with Jordan identity)
Jordan-von Neumann-Wigner 1934: a Jordan algebra is a commutative nonassociative R-algebra (a∘b = b∘a) satisfying the Jordan identity…
Alternative algebra (weakening of associativity: left/right alt)
An alternative algebra is a nonassociative R-algebra satisfying the left-alternative a(ab) = (aa)b and right-alternative (ab)b = a(bb)…
Flexible algebra and Malcev algebras (Jacobi generalisation)
Flexible algebras satisfy (xy)x = x(yx) — the associator is alternating in outer arguments. Both Jordan and alternative algebras are…
Sym_2(R) Jordan product: commutativity and Jordan identity literal-0 residuals
Exact symbolic verification on Sym_2(R), the 3-parameter space of 2×2 real symmetric matrices with Jordan product A∘B = (AB + BA)/2. …
Octonion associator [e1,e2,e4] = (e1·e2)·e4 − e1·(e2·e4) = 2·e7
Explicit non-associativity witness in the Cayley-Dickson octonions O on the Fano-plane multiplication convention with triples…
Jordan power-associativity: (X²)∘X = X∘(X²) = X³ on Sym_2(R)
On Sym_2(R) (symmetric 2×2 real matrices with Jordan product A∘B = (AB + BA)/2) the power-associativity identity X²∘X = X∘X² reduces to 0…
Alternative algebra & octonions
Algebra A satisfies x(xy) = x²y and (yx)x = yx² (alternativity, weaker than associativity). Hurwitz: only normed division algebras over ℝ…
Jordan algebra (special vs exceptional)
Jordan 1933: commutative algebra with x²(yx) = (x²y)x. Special: A^+ from associative A. Exceptional Albert algebra h₃(𝕆) over octonions:…
Malcev algebra
Malcev 1955: anti-commutative algebra satisfying generalised-Jacobi (Malcev identity). Tangent algebra of analytic Moufang loop. Includes…
Composition algebra (norm preserving)
Algebra A over field K with non-degenerate quadratic norm N: A → K satisfying N(xy) = N(x)N(y). Hurwitz theorem: dimensions 1, 2, 4, 8…
Free magma (non-associative)
Free magma on set S = binary trees with leaves labelled by S. Models all distinct ways of associating products. Catalan numbers count. …
Zorn vector matrix (octonion realisation)
Zorn 1933: 2×2 split-octonion matrix realisation with vector entries. Concrete model for split-form of octonions. Used in constructing…
Cayley-Dickson construction
Cayley 1845 / Dickson 1919 doubling procedure: from a *-algebra A constructs a 2-dim *-algebra A x A; iterating from R produces C, H, O…
Octonions (Cayley-Graves 1843-1845)
Graves 1843 + Cayley 1845: 8-dim non-associative + non-commutative division algebra O; only nonassociative normed division algebra; sole…
Jordan algebra (Jordan-vN-Wigner 1933)
Jordan-von-Neumann-Wigner 1933: commutative non-associative product x.y=(xy+yx)/2; Jordan-identity (x^2.y).x = x^2.(y.x); 5 types incl…
Freudenthal-Tits magic square
Freudenthal 1951 + Tits 1966 4x4 array of Lie algebras built from R/C/H/O division algebras; explains exceptional E_6, E_7, E_8, F_4, G_2…
Cayley-Dickson iterated construction
Iterated doubling A -> A x A: R -> C -> H (quaternion) -> O (octonion) -> S (sedenion); each step loses a property (commutativity /…
Artin theorem (alternative algebras)
Artin 1928: in alternative algebra (x,x,y)=(y,x,x)=0, any subalgebra generated by 2 elements is associative; characterizes…
Malcev algebra (Lie-admissible)
Malcev 1955: anticommutative algebra satisfying (xy)(xz) = ((xy)z)x + ((yz)x)x + ((zx)x)y; tangent algebra of Moufang loops; generalizes…
Hurwitz (1923)
A Hurwitz 1923 four-composition-algebras R, C, H, O; modern modern foundational text + Bott-periodicity + Cayley-Dickson construction.
Zorn vector-matrix (1933)
M Zorn 1933 vector-matrix octonion; modern modern foundational text + Tits-Freudenthal magic-square + exceptional-Lie groups.
Malcev algebra (Malcev 1955)
A Malcev 1955 Lie-admissible Malcev-algebra; modern modern foundational text + Sagle 1962 + applications to physics + integrable systems.
Artin theorem (Artin 1928)
E Artin 1928 alternative-algebra; modern modern foundational text + diassociative law + Octonion-x³ structure.
Zhevlakov-Shestakov (1982)
K Zhevlakov-A Shestakov 1982 'Rings That Are Nearly Associative'; modern modern foundational reference + Russian School developments.
Freudenthal magic square (1955)
H Freudenthal-J Tits 1955 magic-square exceptional-Lie F4 / E6 / E7 / E8; modern modern foundational + Vinberg 1965 + Atiyah-Berndt.