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Bol loop

Algebraic structure

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Source updated July 13, 2025

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 (Bol 1937).

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.

01Properties

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.

02Bruck 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.

03Example

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.

04Bol 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.

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Sources and credits

This article is adapted from the Wikipedia article “Bol loop”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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