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
- , 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.
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.
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]
![{\displaystyle [a,b]+[b,a]=0}](../I/m/303bb18f1f15733f21931c4ef09eaac2aaff0945.svg)
and
and
![{\displaystyle [\{a,b,c\},d]-[\{a,b,d\},c]+\{c,d,[a,b]\}-\{a,b,[c,d]\}+[[a,b],[c,d]]=0}](../I/m/44c31fef3905642c6f63136616560ae159c3f709.svg)
and
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.
![{\displaystyle [a,b]=ab-ba}](../I/m/23bf78c808a8f58515d59d520bdfac43cf80114f.svg)
References
- ↑ 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", Mathematische Annalen, 114 (1): 414–431, ISSN 0025-5831, JFM 63.1157.04, MR 1513147, Zbl 0016.22603, doi:10.1007/BF01594185
- 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. 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.