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 {{harv|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, 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 $[a,b]+[b,a]=0$ and a ternary operation $\{a,b,c\}$ that satisfies the following identities:Irvin R. Hentzel, Luiz A. Peresi, "[https://www.researchgate.net/publication/251484095_Special_identities_for_Bol_algebras Special identities for Bol algebras]", Linear Algebra and its Applications 436(7) · April 2012

: $\{a, b, c\} + \{b, a, c\} = 0$

and

: $\{a, b, c\} + \{b, c, a\} + \{c, a, b\}= 0$

and

: $[\{a, b, c\}, d] - [\{a, b, d\}, c] + \{c, d, [a, b]\} - \{a, b, [c, d]\}+ a, b],[c, d = 0$

and

: $\{a, b, \{c, d, e\}\} - \{\{a, b, c\}, d, e\} - \{c, \{a, b, d\}, e\} - \{c, d, \{a, b, e\}\} = 0$ .

Note that {.,.,.} acts as a Lie triple system.

If {{math|A}} is a left or right alternative algebra then it has an associated Bol algebra {{math|A^b}}, where $[a,b]=ab-ba$ is the commutator and $\{a,b,c\}=\langle b,c,a\rangle$ is the Jordan associator.

References

{{Citation | last1=Bol | first1=G. | title=Gewebe und gruppen | doi=10.1007/BF01594185 | mr=1513147 | zbl = 0016.22603 | jfm = 63.1157.04 | year=1937 | journal=Mathematische Annalen | issn=0025-5831 | volume=114 | issue=1 | pages=414–431}}
{{cite book |first=H. |last=Kiechle |title=Theory of K-Loops |publisher=Springer |year=2002 |isbn=978-3-540-43262-3 }}
{{cite book |first=H.O. |last=Pflugfelder |title=Quasigroups and Loops: Introduction |publisher=Heldermann |year=1990 |isbn=978-3-88538-007-8 }} Chapter VI is about Bol loops.
{{cite journal |first=D.A. |last=Robinson |title=Bol loops |journal=Trans. Amer. Math. Soc. |volume=123 |issue=2 |pages=341–354 |year=1966 |jstor=1994661 |doi=10.1090/s0002-9947-1966-0194545-4|doi-access=free }}
{{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 }}

Category:Non-associative algebra

Category:Group theory