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group theory

Contents

Idea

Quasigroup is a binary algebraic structure in which every equation of the form $a\cdot x = b$ or of the form $y\cdot a = b$ has a unique solution for $x$, resp. $y$. The notion of quasigroup is a generalization of the notion of group without the associativity law or identity element. A quasigroup with identity is called a loop (French la boucle, Russian лупа).

Note that, in the absence of associativity, it is not enough (even for a loop) to say that every element has an inverse element (on either side); instead, you must say that division is always possible. This is because the definition $x/y = x y^{-1}$ won't work right without associativity.

Some consider the concept of quasigroup to be an example of centipede mathematics, see more at historical notes on quasigroups.

Definitions

The usual definition is this:

Definition

A quasigroup is a set $G$ equipped with a binary operation $G \times G \to G$ (which we will write with concatenation) such that:

• for all $x$ and $y$, there exist unique $l$ and $r$ such that $l x = y$ and $x r = y$.

Then $l$ is called the left quotient $y / x$ ($y$ divided by $x$, $y$ over $x$) and $r$ is called the right quotient $x \backslash y$ ($x$ dividing $y$, $x$ under $y$).

Note that we must specify, in the definition, that $l$ and $r$ are unique; without associativity, we cannot prove this.

As with the inverse elements of a group, we can make the quotients into operations so that all axioms are equations:

Definition

A quasigroup is a set $G$ equipped with three binary operations (product, left quotient, and right quotient) such that these equations always hold:

• $(x / y) y = x$,
• $x (x \backslash y) = y$,
• $(x y) / y = x$,
• $x \backslash (x y) = y$.

Also, without the right quotient we have left quasigroups, and without the left quotient the right quasigroups. Thus quasigroups are described by a Lawvere theory and can therefore be internalized into any cartesian monoidal category. There are weaker structures, say left and right quasigroups in which either $\backslash$ or $/$ is well defined.

In any case:

Definition

A loop is a quasigroup with an identity element.

Loops are also 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.

For good measure, here is another special kind of quasigroup:

Definition

A group is an associative loop.

Examples

• Any group is a loop, of course.
• Any abelian group is a quasigroup in two other ways: the product switches places with one of the quotients. (The other quotient remains a quotient.)
• H-spaces are (homotopy-) loops — this is because the shearing map?s $(x,y)\mapsto (x,x y)$ and $(x,y)\mapsto (x y, y)$ are equivalences. This generalizes the octonion examples. Note that a loop space is always equivalent to a group, hence not all homotopy loops are loop spaces. In particular …
• The nonzero elements of a (not necessarily associative) 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 $\mathbb{F}_2$) by $\mathbb{Z}_2$.

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.

There are interesting subvarieties of quasigroups and loops (which are still not associative). Also, left racks (and quandles in particular) are precisely left distributive left quasigroups, with abundance of recent applications in the study of knots and links. Finite racks have been studied in the connection to classification of finite dimensional pointed Hopf algebras. Local augmented Lie racks appeared as integration objects in the local integration theory of Leibniz algebras.

TS-quasigroups are related to Steiner triple systems.

Cayley multiplication tables of finite quasigroups are Latin squares (basically the ‘sudoku squares’ from the quotation here).

As a sample of centipede mathematics, we have the following result on smooth quasigroups, i.e., quasigroups internal to the category of smooth manifolds:

Proposition

The tangent bundle of a smooth quasigroup $Q$ is trivial.

Proof

Suppose WLOG that $Q$ is inhabited by an element $x$, and let $V = T_x(Q)$ be the tangent space at $x$. Define a map

$\phi \colon Q \times V \to T Q: (y, v) \mapsto (d (K \circ L_y))(v)$

where $L_y: Q \to Q$ is the smooth map $z \mapsto y z$ and $K: Q \to Q$ is the map $z \mapsto z/x$. The map $\phi$ commutes with the bundle projections $\pi_Q: Q \times V \to Q$, $\pi: T Q \to Q$. The map $K$ has an inverse $J: z \mapsto z x$ and each map $L_y$ has an inverse $M_y: z \mapsto y \backslash z$. We may therefore write down an inverse to $\phi$:

$T Q \to Q \times V: w \mapsto (\pi(w), d(M_{\pi(w)} \circ J))(w)).$

This shows $T Q$ is isomorphic to the product bundle $Q \times V$.

Literature

• eom: quasi-group, Webs, geometry of, Net (in differential geometry)
• R.H. Bruck, A survey of binary systems, Springer-Verlag 1958
• 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
• Momo Bangoura, Bigèbres quasi-Lie et boucles de Lie, Bull. Belg. Math. Soc. Simon Stevin 16:4 (2009), 593-616 euclid arXiv:math.SG/0607662; Quasi-bigèbres de Lie et cohomologie d’algèbre de Lie, arxiv/1006.0677
• 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

Last revised on December 10, 2014 at 01:27:37. See the history of this page for a list of all contributions to it.