Bol loop

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.

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.

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.

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:[1]

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

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 ${\displaystyle \{a,b,c\}=\langle b,c,a\rangle }$ is the Jordan associator.

• Bol, G. (1937), "Gewebe und gruppen", Mathematische Annalen, 114 (1): 414–431, doi:10.1007/BF01594185, ISSN 0025-5831, JFM 63.1157.04, MR 1513147, Zbl 0016.22603
• Kiechle, H. (2002). Theory of K-Loops. Springer. ISBN 978-3-540-43262-3.
• Pflugfelder, H.O. (1990). Quasigroups and Loops: Introduction. Heldermann. ISBN 978-3-88538-007-8. Chapter VI is about Bol loops.
• Robinson, D.A. (1966). "Bol loops". Trans. Amer. Math. Soc. 123 (2): 341–354. doi:10.1090/s0002-9947-1966-0194545-4. JSTOR 1994661.
• Ungar, A.A. (2002). Beyond the Einstein Addition Law and Its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces. Kluwer. ISBN 978-0-7923-6909-7.