Bol loop

In mathematics, a Bol loop is an algebraic structure generalizing the notion of group. Specifically, 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".

A loop is both left Bol and right Bol if and only if it is a Moufang loop. The unmodified term "Bol loop" can refer to either a left Bol or a right Bol loop, depending on author preferences.

Bruck loops

A Bol loop satisfying 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. The example in the preceding section is a Bruck loop. Left Bruck loops are equivalent to A. A. Ungar's gyrocommutative gyrogroups, though the latter are defined differently; see Ungar (2002).

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 B"² "A")^1/2, where the superscript 1/2 indicates the unique positive definite Hermitian square root.

Applications

Bol loops, especially Bruck loops, have applications in special relativity; see Ungar (2002).

References

* H. Kiechle (2002), "Theory of K-Loops", Springer. ISBN 978-3-540-43262-3.
* H. O. Pflugfelder (1990), "Quasigroups and Loops: Introduction", Heldermann. ISBN 978-3-88538-007-8 . Chapter VI is about Bol loops.
* D. A. Robinson, Bol loops, "Trans. Amer. Math. Soc." 123 (1966) 341-354.
* A. A. Ungar (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.