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Historical Notes on Loop Theory 361 Comment.Math.Univ.Carolinae 41,2 2000359{370 359 Historical notes on lo op theory Hala Orlik Pflugfelder Abstract. This pap er deals with the origins and early history of lo op theory, summarizing the p erio d from the 1920s through the 1960s. Keywords: quasigroup theory, lo op theory, history Classi cation : Primary 01A60; Secondary 20N05 This pap er is an attempt to map, to t together not only in a geographical and achronological sense but also conceptually, the various areas where lo op theory originated and through which it moved during the early part of its 70 years of history. 70 years is not very much compared to, say,over 300 years of di erential calculus. But it is precisely b ecause lo op theory is a relatively young sub ject that it is often misinterpreted. Therefore, it is extremely imp ortant for us to acknowledge its distinctive origins. To give an example, when someb o dy asks, \What is a lo op?", the simplest way to explain is to say, \it is a group without asso ciativity". This is true, but it is not the whole truth. It is essential to emphasize that lo op theory is not just a generalization of group theory but a discipline of its own, originating from and still moving within four basic research areas | algebra, geometry, top ology, and combinatorics. Lo oking back on the rst 50 years of lo op history, one can see that every decade initiated a new and imp ortant phase in its development. These distinct p erio ds can b e summarized as follows: I. 1920s the rst glimmerings of non-asso ciativity II. 1930s the de ning p erio d Germany III. 1940s-60s building the basic algebraic frame and new approaches to pro jective geometry United States IV. 1950s-60s the return of lo ops to Western Europ e England, France, Germany, Holland, and Italy V. 1960s the ascendance of lo ops in the Soviet Union Each of the p erio ds I I-V is represented in our literature bya classic b o ok of that era, which to this day remains a main reference source in its resp ective area: II. Geometry of Webs,by Blaschke and Bol, 1938 German This pap er is in nal form and no version of it will b e submitted for publication elsewhere. 360 H.O. P ugfelder III. A Survey of Binary Systems,by Bruck, 1958 English IV. Projective Planes,by Pickert, 1955 German V. Foundations of the Theory of Quasigroups and Loops,by Belousov, 1967 Russian One aim of this pap er is to shed light on the original motivations for the rst publications on quasigroups by Moufang and Bol. The events of those years are to o far in the past for many p eople to know rst-hand or to have heard ab out from witnesses. But they are also to o recent to b e found in math-history b o oks or even on any of the new math-historical web-sites. Let us b egin with p erio d I, or rather with the prehistory of non-asso ciativity. I. 1920s | the rst glimmerings of non-asso ciativity In the history of science we know many cases where at certain times certain revolutionary ideas were, so to sp eak, \in the air" until they manifested themselves in di erent places, sometimes indep endently and in di erent forms. During the last two centuries, two such prominent cases have o ccurred in mathematics and physics. Hyp erb olic geometry was discovered almost simul- taneously by Lobatschevski and Bolyai in the 1820s, and would subsequently combine with Riemannian geometry announced in 1866 to form the eld of Non-Euclidean Geometry. Similarly, around the turn of the century,anentirely new conception of Space-Time emerged out of the Lorentz transformation of 1895, which replaced earlier Galilean notions, and Einstein's Sp ecial Relativity of 1905. Wenow know that these two ideas, Non-Euclidean Geometry and Curved Space- Time, are not utterly unrelated and b oth help ed to prepare the ground for the notion of non-asso ciativity. The oldest non-asso ciative op eration used by mankind was plain subtraction of natural numb ers. But the rst example of an abstract non-asso ciative system was Cayley numb ers, constructed by Arthur Cayley in 1845. Later they were generalized by Dickson to what we know as Caley-Dickson algebras. They b ecame the sub ject of vigorous study in the 1920s b ecause of their prominent role in the structure theory of alternative rings. Another class of non-asso ciative structures was systems with one binary op er- ation. One of the earliest publications dealing with binary systems that explicitly mentioned non-asso ciativitywas the pap er On a Generalization of the Associative Law 1929 byAnton K. Suschkewitsch, who was a Russian professor of mathe- matics in Voronezh. In his pap er, Suschkewitsch observes that, in the pro of of the Lagrange theorem for groups, one do es not make any use of the asso ciative law. So he rightly conjectures that it could b e p ossible to have non-asso ciative binary systems which satisfy the Lagrange prop erty. He constructs twotyp es of such so-called \general groups", satisfying his Postulate A or Postulate B. In Suschkewitsch's approach, one can detect some early attempts in the direction of mo dern lo op theory as a generalization of group-theoretical notions. His \general groups" seem to b e the Historical notes on loop theory 361 1 predecessors of mo dern quasigroups as isotop es of groups. Unfortunately, Suschkevitsch's ideas did not takerootin his own country at the time. In the 1930s, a ruthless p erio d of p olitical repression b egan in the Soviet Union and authorities did not consider Suschkewitsch to b e p olitically trustwor- thy. He had descended from minor Russian nobility. What was even worse, he had studied mathematics in Berlin b efore the revolution and published his 1929 pap er in an American journal. During his later teaching career in Khar'kov, Suschke- witschwas not p ermitted to sup ervise any dissertations | a circumstance that precluded any further development or dissemination of his ideas. It would not b e until the late 1930s, with Murdo ch in America, and in the 1960s, with Belousov in Russia, that Suschkevitsch's work would once again attract attention. I I. 1930s | the de ning p erio d For the next stage, we must lo ok to Germany in the 1930s, where interest was arising simultaneously from algebra, geometry, and top ology. Geometry was traditionally strong in Germany from the age of Gauss, whereas the algebra of the XIX century was predominantly British. The state of German algebra, however, b egan to change considerably from the time of Hilb ert and his axiomatic approach to algebra and geometry. By the 1920s, along with Go ettingen University, the relatively young Univer- sity of Hamburg was b ecoming a new center of research, a circumstance that contributed imp ortantly to the emergence of quasigroups. This researchmoved along several paths, which so on b ecame interwoven. On the algebraic scene, brilliant algebraists happ ened to b e in Hamburg at the time, such as Erich Hecke, a student of Hilb ert; Emil Artin; and Artin's students, Max Zorn and Hans Zassenhaus. Algebraic interest in non-asso ciativity rst came not from binary systems, as was the case with Suschkevitsch, but from alternative algebras. Many hop ed that they could b e useful in the mathematics of quantum mechanics. That hop e was never realized, but in the pro cess the signi cance of non-asso ciativity b egan to emerge. It was around this time that Artin proved a theorem that Moufang would later use in her famous pap er on quasigroups. Artin's theorem: In an alternative algebra, if any three elements multiply asso ciatively, they generate a subalgebra. Simultaneously, along the geometric path, researchwas following Hilb ert's prin- ciple that geometric axioms of planes corresp ond to algebraic prop erties of their co ordinatizing systems. Very exciting developments were also under way in di erential geometry. There is no question that the most prominent gure at the Hamburg Mathematical Sem- inar at the time was Wilhelm Blaschke. His magnetic p ersonality and innovative 1 It was broughttomy attention after the original presentation of this pap er that binary systems with left and right division, whichwenow call quasigroups, were mentioned by Ernst Schro eder in his b o ok Lehrbuch der Arithmethik und Algebra 1873, and in his Vorlesungen ueber die Algebra der Logik 1890. 362 H.O. P ugfelder work in di erential geometry attracted to him and to his ideas manyyoung and talented mathematicians. The new branch of mathematics that he was creating was Web Geometry. Blaschke's b o ok on the sub ject, Web Geometry, co-authored with Bol, came out only in 1938, but was preceded by many separate publica- tions on this topic by himself and by his followers, including the 66 pap ers of the series \Webs and Groups". Bol alone contributed 14 pap ers to this series. Among the earliest were pap ers by Thomsen and Reidemeister, whose names we now know from corresp onding web-con gurations. The title of the series, \Webs and Groups", subtitled \Top ological Questions of Di erential Geometry", was also signi cant for having neatly combined the three main areas from whichloop theory emerged: geometry, algebra, and top ology.
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