Bol loop

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
while L is said to be a right Bol loop if it satisfies
These identities can be seen as weakened forms of associativity, or a strengthened form of 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 = a. Different authors use the term "Bol loop" to refer to either a left Bol or a right Bol loop.

Properties

The left Bol identity directly implies the left 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 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, ⁻¹ = a⁻¹ b⁻¹ for all a,b in L, is known as a Bruck loop or K-loop. The example in the following section is a Bruck loop.
Bruck loops have applications in special relativity; see Ungar. Left Bruck loops are equivalent to Ungar's 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 is a left Bruck loop. An explicit formula for * is given by A * B = ^1/2, where the superscript 1/2 indicates the unique positive definite Hermitian square root.

Bol algebra

A Bol algebra is a vector space equipped with a binary operation and a ternary operation that satisfies the following identities:
and
and
and
Note that acts as a Lie triple system.
If is a left or right alternative algebra then it has an associated Bol algebra, where is the commutator and is the Jordan associator.