Bol loop

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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

, 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.

These identities can be seen as weakened forms of

associativity, or a strengthened form of (left or right) alternativity
.

A loop is both left Bol and right Bol if and only if it is a Moufang loop. Alternatively, a right or left Bol loop is Moufang if and only if it satisfies the flexible identity a(ba) = (ab)a . Different authors use the term "Bol loop" to refer to either a left Bol or a right Bol loop.

Properties

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
.

Bruck loops

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

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
.

Bol algebra

A (left) Bol algebra is a vector space equipped with a binary operation and a ternary operation that satisfies the following identities:[1]

and

and

and

.

Note that {.,.,.} acts as a

Lie triple system
. If A is a left or right alternative algebra then it has an associated Bol algebra Ab, where is the commutator and is the Jordan associator.

References

  1. ^ Irvin R. Hentzel, Luiz A. Peresi, "Special identities for Bol algebras",  Linear Algebra and its Applications 436(7) · April 2012
  • Bol, G. (1937), "Gewebe und gruppen",
    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.
    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
    .
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