Comment.Math.Univ.Carolinae 41,2 2000415{427 415 Lo ops and quasigroups: Asp ects of current work and prosp ects for the future Jonathan D.H. Smith Abstract. This pap er gives a brief survey of certain recently developing asp ects of the study of lo ops and quasigroups, fo cussing on some of the areas that app ear to exhibit the b est prosp ects for subsequent research and for applications b oth inside and outside mathematics. Keywords: quasigroup, lo op, net, web, semisymmetric, Lindenbaum-Tarski duality,mul- tiplication group, lo op transversal, lo op transversal co de, o ctonions, Cayley numb ers, sedenions, vector eld, knot, link, quandle, surgery, Conway algebra, split extension, centraliser ring, multiplicity free, fusion rule, statistical dimension, Ising mo del Classi cation : 20N05 Intro duction Compared to the theory of groups, the theory of quasigroups is considerably older, dating back at least to Euler's work on orthogonal Latin squares. In the rst half of the twentieth century, b oth theories exp erienced comparable mo der- ate progress. But from the nineteen- fties to the nineteen-eighties, the theory of quasigroups was eclipsed by the phenomenal development of the theory of groups to such an extent that the former sometimes came to b e considered as a minor o sho ot of the latter e.g. as witnessed by the American Mathematical So ciety's 1991 Sub ject Classi cation 20N05 for lo ops and quasigroups under the heading \Other generalizations of groups". With the initial completion of the classi ca- tion of the nite simple groups, however, attention is once again b ecoming more evenly divided b etween the two theories. The current pap er aims to give a brief survey of some of the recently developing areas of research within the theory of lo ops and quasigroups, esp ecially those areas with connections to other parts of mathematics and to applications outside mathematics. The topics presented are: 1. Nets and homotopy; 2. Transversals; 3. Octonions, top ology, and knots; 4. Representation theory. For background details, see [Al63], [Br58], [CPS90], [Pf90], [Sm86], [SR99]. 416 J.D.H. Smith 1. Nets and homotopy Historically, homotopybetween lo ops and quasigroups has b een treated geo- metrically, isotopic quasigroups corresp onding to isomorphic nets [SR99, The- orem I.4.5]. Gvaramiya and Plotkin [GP92], [Vo99] reduced the isotopy of quasigroups to the isomorphism of heterogeneous algebras or \automata". It is now p ossible to give a purely homogeneous algebraic treatment of homotopy, us- ing the variety of semisymmetric quasigroups [Sm97]. Recall that a quasigroup is semisymmetric if it satis es the identity 1.1 yxy = x: Each quasigroup Q; ;=;n then has a semisymmetrisation Q, a semisymmetric 3 quasigroup structure on its direct cub e Q in which the pro duct of elements x ;x ;x and y ;y ;y is given by 1 2 3 1 2 3 1.2 y =x ;y n x ;x y : 3 2 1 3 1 2 A homotopyf ;f ;f :Q; ;=;n ! P; ;=;nbetween quasigroups corresp onds 1 2 3 to a homomorphism 1.3 f ;f ;f :x ;x ;x 7! x f ;x f ;x f 1 2 3 1 2 3 1 1 2 2 3 3 between their semisymmetrisations, so that two quasigroups are isotopic if and only if their semisymmetrisations are isomorphic. This observation places new imp ortance on the variety of semisymmetric quasigroups. Nets reapp ear within a duality theory for quasigroups. This duality theory is based on the so-called Lindenbaum-Tarski duality between sets and complete atomic Bo olean algebras, the duality relating a function f : X ! Y between 1 1 sets to the inverse image function f : }Y ! }X ; B 7! f B between their p ower sets [Jt82, VI.4.6a], [RS96, x 7]. On a quasigroup Q, consider the 2 binary op erations of multiplication p : Q ! Q;x; y 7! x y , left projection p : 3 2 2 2 Q ! Q;x; y 7! x, and right projection p : Q ! Q;x; y 7! y . Dualising, one 1 1 2 obtains the inverse image functions p : }Q ! }Q given bymultiplication, 3 1 1 2 2 p : }Q ! }Q given by left pro jection, and p : }Q ! }Q given by 2 1 right pro jection. These homomorphisms of complete atomic Bo olean algebras are sp eci ed by their e ects on the atoms of their domains, namely on the singleton subsets of Q: 8 1 2 1 p : }Q ! }Q ; fbg7!fx; b j x 2 Qg; > 1 < 1 2 1.4 2 p : }Q ! }Q ; fag7!fa; y j y 2 Qg; 2 > : 1 2 3 p : }Q ! }Q ; fcg7!fx; y j x y = cg: 3 1 In net terminology, p fbg in 1.41 is just the 1-line lab elled by b. Similarly, 1 1 1 p fag in 1.42 is the 2-line lab elled by a, while p fcg in 1.43 is the 3- 2 3 line lab elled by c. Thus the dual of a quasigroup is a net. The dual ob ject is a Lo ops and quasigroups: Asp ects of currentwork and prosp ects for the future 417 set that decomp oses as the direct pro duct of anytwoofaset of three isomor- phic images [SR99, p. 88]. This de nition of a net is purely categorical, and may thus be interpreted in the category of top ological spaces or other categories of geometric interest. It o ers new approaches to the problem of co ordinatising con- gurations in web geometry that has b een studied by Akivis, Goldb erg, Shelekov et al. [Ak92], [Go88]. In geometric contexts, one should replace the \combina- torial" Lindenbaum-Tarski dualityby a duality appropriate to the context, e.g. Gelfand dualitybetween compact Hausdor spaces and C*-algebras [Jt82, IV.4], or the dualitybetween lo cally de ned smo oth maps and their pullbacks acting on di erential forms [Ol95, p. 26]. 2. Transversals Lo ops, or more generally right lo ops, app ear naturally as algebraic structures on transversals or sections of a subgroup of a group. This observation, going back to R. Baer [Ba39], lies at the heart of much current research on lo ops, e.g. work of Karzel, Kreuzer, Strambach, Wefelscheid et al. in geometry [Ka93], [Kr98], [KW94], [Na94] or work of Kikkawa, Sabinin, Ungar et al. in di erential geometry and analysis [CPS, Chapter XI I], [Fr94], [Ki98], [Sa72-98], [Un91-94]. It is also closely related to the question of when or how a given group action arises as the multiplication group action of a quasigroup or lo op, cf. work of Dr apal, Kepka, Niemenmaa, Phillips et al. [CPS90, x I I I.6], [Dr93-94], [Ke93-97], [Ni95d-97], [NK90-94], [NR92], [NV94], [PS99]. Let H b e a subgroup of a group G; ;=;n; 1, and let T b e a right transversal S to H in G such that 1 represents H . Thus the group is partitioned as G = Ht. t2T " De ne a map " : G ! T ; g 7! g by " 2.1 g 2 Hg ; " so that g or g" is the unique representativein T for the right coset of H that contains g . It is also convenient to de ne a map : G ! H ; g 7! g by " 2.2 g = g g : " " Note that 1 =1 =1. Moreover h = h and h = 1 for h in H , while t = 1 and " t = t for t in T . Now de ne a binary multiplication and a binary right division k on T by 2.3 t u =tu"; tku =t=u" 1 for t; u in T , i.e. by tu 2 H t u and t=u = tu 2 H tku. Then T; ; k; 1 is a right lo op. If this right lo op happ ens to b e two-sided, then the transversal T is called a loop transversal . To within right lo op isomorphism, every rightloopQ may b e obtained via 2.3 from the transversal T = R Q to the stabiliser H of 1 in the rightmultiplication group G of Q. 418 J.D.H. Smith The lo op theoretical concept of a lo op transversal even has fruitful applications within ab elian groups. In co ding theory, the lo op transversal T; to a linear co de C; + within an ab elian group channel G; + is the set of errors corrected by the co de [Sm92l], [SR99, I x 4.4]. Using the notation of 2.2: If a word g is received, it is deco ded to the co deword g under the assumption that the transmitted co deword was sub jected to the error g". The most ecientwayto de ne a co de in a go o d channel is to sp ecify the lo op structure an ab elian group m P t inductively by on the set of errors. For elements t ;t ;::: of T , de ne 1 2 i i=1 0 m m1 m 0 Q Q P P P t = 0 and t inductively by t . De ne t = t + t = 0 and m i i i i i i=1 i=1 i=1 i=1 i=1 m1 m Q Q t = t t . m i i i=1 i=1 Principle of Lo cal Duality. Let T b e a lo op transversal to a linear co de C in achannel G. Supp ose that T is a set of generators for G.