Bol loop

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 ${\displaystyle [a,b]+[b,a]=0}$ and a ternary operation ${\displaystyle \{a,b,c\}}$ that satisfies the following identities:

${\displaystyle \{a,b,c\}+\{b,a,c\}=0}$

and

${\displaystyle \{a,b,c\}+\{b,c,a\}+\{c,a,b\}=0}$

and

${\displaystyle [\{a,b,c\},d]-[\{a,b,d\},c]+\{c,d,[a,b]\}-\{a,b,[c,d]\}+[[a,b],[c,d]]=0}$

and

${\displaystyle \{a,b,\{c,d,e\}\}-\{\{a,b,c\},d,e\}-\{c,\{a,b,d\},e\}-\{c,d,\{a,b,e\}\}=0}$.

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 ${\displaystyle [a,b]=ab-ba}$ is the commutator and ${\displaystyle \{a,b,c\}=⟨b,c,a⟩}$ is the Jordan associator.