Bol loop

Short description: Algebraic structure

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

[math]\displaystyle{ a(b(ac))=(a(ba))c }[/math], for every a,b,c in L,

while L is said to be a right Bol loop if it satisfies

[math]\displaystyle{ ((ca)b)a=c((ab)a) }[/math], 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 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.

Bol algebra

A (left) Bol algebra is a vector space equipped with a binary operation [math]\displaystyle{ [a,b]+[b,a]=0 }[/math] and a ternary operation [math]\displaystyle{ \{a,b,c\} }[/math] that satisfies the following identities:

[math]\displaystyle{ \{a, b, c\} + \{b, a, c\} = 0 }[/math]

and

[math]\displaystyle{ \{a, b, c\} + \{b, c, a\} + \{c, a, b\}= 0 }[/math]

and

[math]\displaystyle{ [\{a, b, c\}, d] - [\{a, b, d\}, c] + \{c, d, [a, b]\} - \{a, b, [c, d]\}+ a, b],[c, d = 0 }[/math]

and

[math]\displaystyle{ \{a, b, \{c, d, e\}\} - \{\{a, b, c\}, d, e\} - \{c, \{a, b, d\}, e\} - \{c, d, \{a, b, e\}\} = 0 }[/math].

Note that {.,.,.} acts as a Lie triple system. If A is a left or right alternative algebra then it has an associated Bol algebra A^b, where [math]\displaystyle{ [a,b]=ab-ba }[/math] is the commutator and [math]\displaystyle{ \{a,b,c\}=\langle b,c,a\rangle }[/math] is the Jordan associator.

References

• Bol, G. (1937), "Gewebe und gruppen", Mathematische Annalen 114 (1): 414–431, doi:10.1007/BF01594185, ISSN 0025-5831 
• 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. doi:10.1090/s0002-9947-1966-0194545-4. 
• 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.