Bol loop

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In mathematics, a Bol loop is an algebraic structure generalizing the notion of group. Bol loops, especially Bruck loops, have applications in special relativity; see Ungar (2002).

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. Different authors use the term "Bol loop" to refer to either a left Bol or a right Bol loop.

[edit] 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).

[edit] 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.

[edit] 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.