Moufang Loops and Groups with Triality are Essentially the Same Thing

Abstract

In 1925 Élie Cartan introduced the principal of triality specifically for the Lie groups of type D 4 D_4 , and in 1935 Ruth Moufang initiated the study of Moufang loops. The observation of the title was made by Stephen Doro in 1978 who was in turn motivated by work of George Glauberman from 1968. Here we make the statement precise in a categorical context. In fact the most obvious categories of Moufang loops and groups with triality are not equivalent, hence the need for the word “essentially.”

Transposition Algebras

“…we finish with G(P(D), L(D)) isomorphic to G/ Z(G), not always isomorphic to G. To remedy this and other problems, we must replace each 3-transposition group by an appropriate "universal central extension" and consider the subcategory of such universal groups. This is similar to the situation encountered in [21].…”

Moufang patterns and geometry of information

“…Moufang loops and Latin square designs, We recall here briefly some notions from combinatorial designs and the geometry of buildings, closely related to loops. We refer the reader to [Cam03], [Hall19], [MeiStWe13] for more details.…”

“…Given a loop L, the Thomsen design D(L) has set of points P = L 1 ⊔ L 2 ⊔ L 3 , three copies of L labelled i = 1, 2, 3, and set of lines A = {(x 1 , x 2 , x 3 ) | (x 1 ⋆x 2 )⋆x 3 = 1 ∈ L}. Conversely, given any Latin square design D, there is a loop L(D) with this property, the Thomsen loop of D. The Thomsen loop assignment D → L(D) is functorial and gives an equivalence of categories between the category of Latin square designs and the category of loops, where objects are loops L and morphisms are isotopisms, namely triples of maps (α, β, γ) : L → L ′ satisfying α(x) ⋆ ′ β(y) = γ(x ⋆ y) for all x, y ∈ L, see Theorem 3.4 of [Hall19].…”

“…An automorphism of a Latin square design D = (P, A) is a permutation of P that sends lines to lines. A central automorphism τ x of D, centered at a point x ∈ P is an automorphism that fixes x and exchanges the remaining two points on each line in A containing x (see Section 3.2 of [Hall19]).…”

“…The Thomsen functor restricted to this subcategory gives an equivalence between the category of central Latin square designs and the category of Moufang loops (Theorem 3.11 of [Hall19]).…”