Bol loop
Algebraic structure
title: "Bol loop" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["non-associative-algebra", "group-theory"] description: "Algebraic structure" topic_path: "general/non-associative-algebra" source: "https://en.wikipedia.org/wiki/Bol_loop" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary Algebraic structure ::
In mathematics and abstract algebra, a Bol loop is an algebraic structure generalizing the notion of group. Bol loops are named for the Dutch mathematician Gerrit Bol who introduced them in .
A loop, L, is said to be a left Bol loop if it satisfies the identity
:a(b(ac))=(a(ba))c, for every a,b,c in L,
while L is said to be a right Bol loop if it satisfies
:((ca)b)a=c((ab)a), for every a,b,c in L.
These identities can be seen as weakened forms of associativity, or a strengthened form of (left or right) alternativity.
A loop is both left Bol and right Bol if and only if it is a Moufang loop. Alternatively, a right or left Bol loop is Moufang if and only if it satisfies the flexible identity a(ba) = (ab)a . Different authors use the term "Bol loop" to refer to either a left Bol or a right Bol loop.
Properties
The left (right) Bol identity directly implies the left (right) alternative property, as can be shown by setting b to the identity.
It also implies the left (right) inverse property, as can be seen by setting b to the left (right) inverse of a, and using loop division to cancel the superfluous factor of a. As a result, Bol loops have two-sided inverses.
Bol loops are also power-associative.
Bruck loops
A Bol loop where the aforementioned two-sided inverse satisfies the automorphic inverse property, (ab)−1 = a−1 b−1 for all a,b in L, is known as a (left or right) Bruck loop or K-loop (named for the American mathematician Richard Bruck). The example in the following section is a Bruck loop.
Bruck loops have applications in special relativity; see Ungar (2002). Left Bruck loops are equivalent to Ungar's (2002) gyrocommutative gyrogroups, even though the two structures are defined differently.
Example
Let L denote the set of n x n positive definite, Hermitian matrices over the complex numbers. It is generally not true that the matrix product AB of matrices A, B in L is Hermitian, let alone positive definite. However, there exists a unique P in L and a unique unitary matrix U such that AB = PU; this is the polar decomposition of AB. Define a binary operation * on L by A * B = P. Then (L, *) is a left Bruck loop. An explicit formula for * is given by A * B = (A B2 A)1/2, where the superscript 1/2 indicates the unique positive definite Hermitian square root.
Bol algebra
A (left) Bol algebra is a vector space equipped with a binary operation [a,b]+[b,a]=0 and a ternary operation {a,b,c} that satisfies the following identities:
:{a, b, c} + {b, a, c} = 0 and :{a, b, c} + {b, c, a} + {c, a, b}= 0 and :[{a, b, c}, d] - [{a, b, d}, c] + {c, d, [a, b]} - {a, b, [c, d]}+ [a, b],c, d = 0 and :{a, b, {c, d, e}} - {{a, b, c}, d, e} - {c, {a, b, d}, e} - {c, d, {a, b, e}} = 0. Note that {.,.,.} acts as a Lie triple system. If A is a left or right alternative algebra then it has an associated Bol algebra A**b, where [a,b]=ab-ba is the commutator and {a,b,c}=\langle b,c,a\rangle is the Jordan associator.
References
- Chapter VI is about Bol loops.
References
- Irvin R. Hentzel, Luiz A. Peresi, "[https://www.researchgate.net/publication/251484095_Special_identities_for_Bol_algebras Special identities for Bol algebras]", ''Linear Algebra and its Applications'' '''436'''(7) · April 2012
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