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loop (algebra)

Contents

This entry is about loops in algebra. For loops in topology see loop (topology).


Contents

Idea

In algebra a loop is a quasigroup with (two-sided) identity element.

Definition

With multiplication, divisions and identities

A left loop is a unital magma (G,()():G×GG,1:G)(G, (-)\cdot(-):G\times G\to G,1:G) equipped with a left division ()\():G×GG(-)\backslash(-):G \times G \to G such that x(x\y)=yx \cdot (x \backslash y) = y and x\(xy)=yx \backslash (x \cdot y) = y. A right loop is a unital magma (G,()():G×GG,1:G)(G, (-)\cdot(-):G\times G\to G,1:G) equipped with a right division ()/():G×GG(-)/(-):G \times G \to G such that (x/y)y=x(x / y) \cdot y = x and (xy)/y=x(x \cdot y) / y = x. A two-sided loop or just a loop is a unital magma that is both a left loop and a right loop.

With multiplication, inverses, and identity

Equivalently, one could speak of left and right inverse elements instead of left and right division:

A left loop is a unital magma (G,()():G×GG,1:G)(G, (-)\cdot(-):G\times G\to G,1:G) equipped with a left inverse 1():GG^{-1}(-):G \to G such that x( 1xy)=yx \cdot (^{-1}x \cdot y) = y and 1x(xy)=y^{-1}x \cdot (x \cdot y) = y. A right loop is a unital magma (G,()():G×GG,1:G)(G, (-)\cdot(-):G\times G\to G,1:G) equipped with a right inverse () 1:GG(-)^{-1}:G \to G such that (xy 1)y=x(x \cdot y^{-1}) \cdot y = x and (xy)y 1=x(x \cdot y) \cdot y^{-1} = x. A two-sided loop or just a loop is a unital magma that is both a left loop and a right loop. One then defines left division to be x\y=1xyx \backslash y = {^{-1}}x \cdot y and right division to be x/y=xy 1x / y = x \cdot y^{-1}.

With divisions and identity

There is another definition of a loop using only division and identity:

A left loop is a pointed magma (G,\,1)(G,\backslash,1) such that:

  • For all aa and bb in GG, a\a=1a\backslash a=1
  • For all aa in GG, (a\1)\1=a(a\backslash 1)\backslash 1=a

A right loop is a pointed magma (G,/)(G,/) such that:

  • For all aa and bb in GG, a/a=1a/a=1
  • For all aa in GG, 1/(1/a)=a1/(1/a)=a

A loop is a left and right loop as defined above (G,\,/,1)(G,\backslash,/,1) such that a/(1/b)=(a\1)\ba/(1/b) = (a\backslash 1)\backslash b for all aa and bb in GG.

Properties

Loops are described by a Lawvere theory.

Note that, even in a loop, left and right inverses need not agree. See the discussion on the English Wikipedia for convenient inverse properties. A loop with a two-sided inverse is a nonassociative group.

Examples

  • Any group is a loop.

  • Any nonassociative group is a loop.

  • The nonzero elements of a (not necessarily associative) unital division algebra (such as the octonions) form a quasigroup; this fact is basically the definition of ‘division algebra’.

  • code loops are loops which are central extensions of abelian groups (actually vector spaces over the finite field 𝔽 2\mathbb{F}_2) by 2\mathbb{Z}_2.

  • A Moufang loop is a loop.

Applications

Local analytic loops have interesting induced structure on the tangent space at the identity, generalizing the Lie algebra of a group, see Sabinin algebra. Sabinin algebras are closely related to the local study of affine connections on manifolds. They include some known important classes of nonassociative algebras, namely Lie algebras, Mal’cev algebras, Lie triple systems (related to the study of symmetric spaces), Bol algebras as simplest cases.

algebraic structureoidification
truth valuetransitive relation
magmamagmoid
unital magmaunital magmoid
quasigroupquasigroupoid
looploopoid
semigroupsemicategory
monoidcategory
associative quasigroupassociative quasigroupoid
groupgroupoid
flexible magmaflexible magmoid
alternative magmaalternative magmoid
absorption monoidabsorption category
(left,right) cancellative monoid(left,right) cancellative category
rigCMon-enriched category
nonunital ringAb-enriched semicategory
nonassociative ringAb-enriched unital magmoid
ringringoid
differential ring?differential ringoid?
nonassociative algebralinear magmoid
nonassociative unital algebraunital linear magmoid
nonunital algebralinear semicategory
associative unital algebralinear category
C-star algebraC-star category
differential algebradifferential algebroid
flexible algebraflexible linear magmoid
alternative algebraalternative linear magmoid
Lie algebraLie algebroid
strict monoidal categorystrict 2-category
strict 2-groupstrict 2-groupoid
monoidal poset?2-poset
monoidal groupoid?(2,1)-category
monoidal category2-category/bicategory
2-group2-groupoid/bigroupoid

Literature

  • Kenneth Kunen, Quasigroups, loops, and associative laws, J. Algebra 185 (1) (1996), pp. 194–204

  • Péter T. Nagy, Karl Strambach, Loops as invariant sections in groups, and their geometry, Canad. J. Math. 46(1994), 1027-1056 doi

  • Lev Vasilʹevich Sabinin, Smooth quasigroups and loops: forty-five years of incredible growth, Commentationes Mathematicae Universitatis Carolinae 41 (2000), No. 2, 377–400 cdml pdf

See also:

Last revised on May 24, 2021 at 11:20:39. See the history of this page for a list of all contributions to it.