Bol loop

In mathematics and abstract algebra, a Bol loop is an algebraic structure generalizing the notion of group. Bol loops are named for the Dutch mathematician Gerrit Bol who introduced them in (Bol 1937).

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.

These identities can be seen as weakened forms of associativity.

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.

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 (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.

References

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: 341-354.
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.

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