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Nathan Jacobson (1910–1999)
Georgia Benkart, Irving Kaplansky, Kevin McCrimmon, David J. Saltman, and George B. Seligman
When a colleague was explaining how a mathe- Hebrew Immigrant Aid Society. After a few months matician can be recognized to have reached the in the rear of his father’s Nashville grocery, Jake summit of recognition by his peers, he used the and his family moved to Birmingham, Alabama, and metaphor, “He has become part of the furniture.” then, in 1923, to Columbus, Mississippi. Jake grad- That is, his contributions have become a part of uated from the S. D. Lee High School in Columbus the daily vocabulary and working equipment of in 1926. He entered the University of Alabama many of us. Such is certainly the status of Nathan that fall, intending to follow a maternal uncle into Jacobson. As my fellow authors will show more law. specifically, he earned his dominance by recasting While following a pre-law program, he took all whole theories of algebraic systems and by in- mathematics courses available. The notice of his sisting on the module-theoretic viewpoint in their professors was attracted to the extent that in his study. His expository and research monographs junior year he was offered a teaching assistantship and his ambitious textbooks have indebted a in mathematics. Two of these professors, Fred worldwide community to him for strong and Lewis and William P. Ott, were always remembered articulate leadership. The authors use this oppor- fondly as having inspired him to turn to a career tunity to remind us of some of the ways his ideas in mathematics. With their advice he applied for have shaped our thought. graduate study to Chicago, Harvard, and Princeton, “Jake”, the name all used, died on December 5, accepting an offer of a “research assistantship” 1999, at the age of eighty-nine. Extensive autobi- at Princeton. The stipend ($500) fell just a little ographical material is to be found in the “Personal short of the bill for tuition, room, and board, but History and Commentary” that he wrote in seven the following years saw increases to levels that he installments in his Collected Mathematical Papers described as “a substantial surplus over living [B14], published in three volumes by Birkhäuser in expenses.” 1989. I recommend these passages both for more His dissertation Non-commutative Polynomials details on his personal life and for his comments and Cyclic Algebras, with J. H. M. Wedderburn as on the development of his mathematical work. In advisor, was accepted for the Ph.D. in 1934. How this segment of the present article I provide a his time in Princeton and subsequently at the sketch of his career. Institute for Advanced Study led to what became His “official” birth date was September 8, 1910, his leadership in the algebraic theory of Lie alge- but Jake maintained that the correct one was bras is described below by Irving Kaplansky and October 5. His father emigrated to Nashville, Georgia Benkart. Tennessee, when Jake was five, leaving the family Emmy Noether had taken a position at Bryn in Poland until he was well enough established to Mawr. She gave weekly lectures, attended by Jake, bring them over. The First World War was nearing at the Institute. She took an interest in Jake’s work, its end when Jake, his brother, and his mother but all opportunities for collaboration ended with were able to board a Dutch ship with help from the her sudden death in the spring of 1935. Jake was appointed as her replacement at Bryn Mawr for the George B. Seligman is professor of mathematics emeritus following academic year. After a postdoctoral at Yale University. His e-mail address is selig@ fellowship with Adrian Albert at Chicago in math.yale.edu. 1936–37, he was appointed to a junior position at
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Nathan Jacobson...1945 ...1970 ...1997
the University of North Carolina. Jake praised the the academic year 1956–57, when Adrian Albert university’s president, Frank Graham, and the organized support, mainly from the research of- department head, Archibald Henderson, for their fices of the arms of the Department of Defense, rejection of the exclusionary practices concern- for some ten established and younger algebraists ing Jews that barred the doors to many positions. to be at Yale. The university cooperated by partial Although he had been on the faculty for five support for teaching in most cases. Some of Jake’s years, rising to the rank of associate professor, collaborations from that year are [58] and [59] in Jake was still subject to the Navy’s requirement of the list of bibliographic selections. special teacher training before being entrusted In July 1961 Jake represented the National Acad- with teaching in the U.N.C. wartime program for emy of Sciences at the Leningrad Fourth All-Union prospective flyers. Fortunately the pedagogical Congress of Mathematicians of the USSR. After preparation was offered in Chicago. There it enabled considerable resistance, he agreed to serve as chair Jake to renew and consolidate his relationship with of the Yale mathematics department for 1965–68, his inseparable helpmeet and companion through with assurance that no extension nor reappoint- fifty-four years of marriage. Florence Dorfman ment was expected. During his term he succeeded (“Florie”) gave up her doctoral research with Albert, in appointing Abraham Robinson, the founder of but continued in mathematics not only as an nonstandard analysis and an outstanding con- educator but also as Jake’s reader, supporter, critic, tributor to both pure and applied mathematics. An- and coauthor. When the children were older, she other coup was negotiating the return to Yale of returned as a highly successful and beloved teacher our former Ph.D., Robert Langlands. at Albertus Magnus College. The hospitality of As president of the American Mathematical their home is surely among the reasons why the Society in 1971–1972, Jake had to mediate between mathematics department at Yale has a reputation an “activist” faction, particularly in opposition to for warmth and friendliness. the Vietnam War, and a “purist” faction, who felt In 1943 Jake left the Navy and North Carolina the Society should adhere strictly to scientific aims. for the Army training program and an associate Although his personal sentiments were with the professorship at Johns Hopkins, where he had ear- activists, he preserved the respect of all parties by lier spent a year as a visitor. It was during his time offering all a hearing and by following an open at Hopkins that he developed much of the general and democratic process in discussion and deci- theory of rings that is his most famous achieve- sions. His term as vice president of the International ment. The offer of a tenured associate professor- Mathematical Union (IMU) (1972–74) was more ship from Yale that he received and accepted in stormy. The issue at the center of contention was 1947 represented more than an appreciation of his the refusal of the Soviet authorities, as represented outstanding research and teaching. The anti-Semitic by L. S. Pontrjagin, the other vice president of the barrier to senior appointments in the faculty of Yale IMU, to permit many outstanding Soviet mathe- College had fallen only in 1946, and there were still maticians to participate in International Congresses. misgivings about that step in too many quarters; Beyond that, anti-Semitic and antidissident prac- but the time had come when merit could prevail. tices kept promising students from being admit- The events of his early years at Yale and his ted to universities and senior scholars who had visits to Paris and elsewhere are covered in the fallen out of favor from being allowed to emigrate. Collected Papers, to which we owe lists of his pub- The determination with which Jake protested may lications and of his Ph.D. students. Outstanding was be gathered from his comments in the June 1980
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issue of the Notices in response to a vicious personal for all subsequent work in the field, work that attack by Pontrjagin. eventually answered all the major questions. His retirement from Yale in 1981 came only At the time of the summer of 1938, Jake was only after he had earned the honor of carrying the four years beyond the doctorate. His thesis advisor university’s mace as senior professor at the com- at Princeton was Wedderburn. The thesis [1] mencement ceremonies. Students, colleagues, and concerned finite-dimensional associative al- fellow scholars gathered to honor him and to pre- gebras. Thus there is a remarkable continu- sent him with their contributions in a volume, ity in the passing of the mantle from Wed- Algebraists’ Homage [AH]. Retirement made it derburn to Jacobson. possible for him to accept numerous invitations I hope that many readers of this piece from around the world. Kevin McCrimmon and will also read the autobiography and (to David Saltman write of his activity and influence borrow a word from Halmos) the au- on research in the retirement years. tomathography contained in the three vol- In February of 1992 he suffered a crippling umes of [B14]. From this we learn that a sec- stroke. The effect on his speech gradually wore off, ond major influence on Jake at Princeton but his right hand was nearly useless for writing, was the presence of Hermann Weyl at the and he could not walk unaided. With Florie taking newly founded Institute for Advanced on much of the mechanics, he finished the book Study. Weyl gave a course on Lie groups and on division algebras [B16] for publication in 1996, Lie algebras for which notes were written completing the journey he had started with Wed- by Jake and by Richard Brauer. A second Yale, 1981. derburn. Meanwhile, Florie was receiving powerful lifelong interest was planted in Jake at that time. It promptly bore fruit in the influen- medication. The combination of illness and treat- tial paper [4]. (I believe that this is the first ment took her from Jake’s side in 1996. No visitor paper to use the term “Lie algebra”; the thereafter could fail to be reminded how much she change from “infinitesimal group” was had meant to him. made in Weyl’s lectures.) This elegant paper There was still one happy occasion. He was able is probably best known for a lemma to make the trip to Baltimore in January of 1998 (Lemma 2 on page 877): If A and B are to be honored with the Society’s Leroy P. Steele matrices over a field of characteristic 0 and Prize for Lifetime Achievement. A photo accom- A commutes with AB BA, then AB BA panying this article shows his radiance at that is nilpotent. I fell in love with this lemma With wife Florie, event. His only lament was the absence of Florie. and came back to it repeatedly. Just say “Ja- around 1960. May they now have found reunion. cobson’s lemma” to just about anyone, and —George B. Seligman, organizer he or she is likely to light up in recognition. Irving Kaplansky His early papers on Lie algebras were also note- worthy for launching the theory of Lie algebras of With the death of Nathan Jacobson (“Jake”) characteristic p>0. Thus far there had only been the world of mathematics has lost a giant of one novel example of a simple algebra: the Witt twentieth-century algebra. algebra. In [24] he broadened this to a family of I shall begin by recalling my first contact with algebras. Once again we find his name attached to Jake. It was in the summer of 1938 at the Univer- an object, for they came to be called the Witt- sity of Chicago. With a fresh bachelor’s degree, I Jacobson algebras. At first blush it might seem that was attracted by the special program in algebra that Jake was overoptimistic in wondering whether all summer. I attended Jake’s course on continuous the simple ones were now at hand [23, page 481]. groups. This carried me from the definition of a But when the classification finally came, the topological space (new to me) to exciting topics at answer was that one had only to modify the Witt- the frontier. Also, in a seminar course conducted Jacobson algebras in the way that Cartan did in his by Albert I heard Jake give a talk on locally com- infinite simple pseudo-groups. In my own study of pact division rings. This kindled in me an interest Lie algebras I cut my teeth reading these papers. in locally compact rings that has lasted to this I have now reached the time period when he day. Pontrjagin had done the pioneering work by launched his general structure theory for rings in showing that the only connected locally compact [31] and [32]. Let A be a ring with unit element. Let division rings are the reals, complexes, and quater- J be the intersection of the maximal left ideals in nions. The paper [3], joint with Olga Taussky, took A. There is no apparent reason why J should be a a big step forward by studying a general two-sided ideal, but it is. There is no apparent locally compact ring. This laid the foundation reason why J should be a left-right symmetric, but it is. J is of course the Jacobson radical. When it Irving Kaplansky is director emeritus of the Mathemati- vanishes, A is called semisimple. (Warning: Others cal Sciences Research Institute, Berkeley. His e-mail address say “semiprimitive”, reserving “semisimple” for is [email protected]. the Artinian case.) Now the famous Wedderburn
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structure theorems survive, in a 1947, during which his course on rings was a preview somewhat weakened form. A semi- of the forthcoming book. Polynomial identities and simple ring is a subdirect sum of central polynomials surfaced at that time. This tale primitive ones, and a primitive has been told twice—as he remembered it and as I ring resembles matrices over a di- did. I shall not repeat it here. But let me record how vision ring, with the matrices indebted I am to him for this inspiration. And I would allowed to be infinite. like also to thank him again for the overly generous This splendid theory works. footnote [33, page 702] in which he gave me credit Over the years there have been re- for extending his commutativity theorem from peated uses of it to settle prob- xn = x to xn(x) = x. lems not stated in terms of the Let me pay tribute to his wife Florence (“Florie”). theory. Not only did she offer him support through a long The Colloquium volume [B5] in- and happy marriage, she was a joint author [40]. cludes his account of his structure Jake’s final three years were saddened by the loss theory. It was definitive when it ap- of Florie. Friends, students, and colleagues are peared. It remains indispensable mourning the loss of both. We will always re- Receiving the AMS Leroy P. today; I think it will continue to be member the hospitality they were always ready to Steele Prize, Baltimore, 1998. indispensable for a long time. Late offer and their outgoing, charming personalities. in life [B16] he returned to the ba- In closing I would like to mention three more sic classical topic of finite-dimen- gems: (1) His inauguration of the fertile concept of sional division algebras and pre- triple systems [39]. (2) His reduction of Hermitian sented a remarkable new view of forms to quadratic forms [21]. Every linear alge- this venerable subject. braist should put this into his or her armory. (3) This There are three great classes of last is due to his student Glennie [G]: the amazing algebras: associative, Lie, and Jor- identity satisfied by special Jordan algebras. dan. The date of his associative book is 1956. Just six years later came his Lie algebra book [B6]. It set a high standard for the fairly Georgia Benkart numerous books that have fol- It was spring 1934, and Nathan Jacobson was just lowed. Among other things, I find finishing his doctoral dissertation on division al- the abundance of challenging ex- gebras at Princeton under J. H. M. Wedderburn. ercises to be a big plus. After six Richard Brauer, who had been designated Hermann more years came his book [B7] on Weyl’s research assistant at the newly established Jordan algebras, completing his Institute for Advanced Study, was delayed in ar- trio on the three classes of alge- riving until the fall, so Jacobson was asked to bras. He did the hat trick! Again, bridge the gap and write up Weyl’s lecture notes this book was polished, eminently on continuous groups. This proved to be a mo- readable, and definitive at the mentous event for Lie theory as well as the start time. But subsequent dramatic de- of young Jacobson’s distinguished writing career. velopments, above all at the hands Weyl felt that it would be of interest to study Lie of McCrimmon and Zelmanov, algebras over arbitrary fields without recourse to With Dick and Alice Shafer at have transformed the subject. the group or to the algebraic closure of the field. the Steele Prize ceremony. It is amazing but true that in ad- Jacobson, who was well versed in Wedderburn’s dition to writing these three books similar investigations on associative algebras, read- Jake found the time to write an algebra textbook not ily took to the task. His first paper on the subject, once, but twice. I am referring to [B2], [B3], [B4] and “Rational methods in Lie algebras” [4], which ap- [B10], [B12]. The citation for the Steele Prize for Life- peared in 1935, acknowledged Weyl’s profound time Achievement (Notices 45 (1998), 508) said that influence. It rederives the well-known theorems the first is superseded by the second. I disagree. I am of Lie and Engel on solvable and nilpotent Lie al- glad that we have both; they will both be studied gebras by using methods from elementary linear and enjoyed for a long time. algebra that set the stage for “rationalizing” other Let me return to the debts I owe him. After his parts of the theory. structure theory of rings appeared, I ventured to A beautiful example of the rationalizing process begin a steady stream of correspondence with him involves Jacobson’s notion of a weakly closed sub- about this and about locally compact rings. He was set S in a finite-dimensional associative algebra always prompt in replying, and his replies were always helpful. He gently tolerated my often naive Georgia Benkart is professor of mathematics at the Uni- stabs. It was like doing a second Ph.D. thesis. This versity of Wisconsin–Madison. Her e-mail address is climaxed in his visit to Chicago in the summer of [email protected].
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