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 .

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, 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 B''² ''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 ,b ,a0 and a ternary operation $\$ that satisfies the following identities:Irvin R. Hentzel, Luiz A. Peresi,
Special identities for Bol algebras
,  ''Linear Algebra and its Applications'' 436(7) · April 2012

:$\ + \ = 0$
and
:$\ + \ + \= 0$
and
:$$, d- , c+ \ - \+ a, b, d = 0
and
:$\ - \ - \ - \ = 0$.
Note that acts as a Lie triple system.
If is a left or right alternative algebra then it has an associated Bol algebra , where ,bab-ba is the commutator and $\=\langle b,c,a\rangle$ is the Jordan associator.

References

*
*
* Chapter VI is about Bol loops.
*
* {{cite book , first=A.A. , last=Ungar , title=Beyond the Einstein Addition Law and Its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces , publisher=Kluwer , year=2002 , isbn=978-0-7923-6909-7

Non-associative algebra
Group theory