- Bol loop
:, for every "a","b","c" in "L",
while "L" is said to be a right Bol loop if it satisfies
:, 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.
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).
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 decompositionof "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.
Bol loops, especially Bruck loops, have applications in special relativity; see Ungar (2002).
* 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.
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