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- 1. Nathan Jacobson (1910–1999) Georgia Benkart, Irving Kaplansky, Kevin McCrimmon, David J. Saltman, and George B. SeligmanWhen a colleague was explaining how a mathe- Hebrew Immigrant Aid Society. After a few monthsmatician can be recognized to have reached the in the rear of his father’s Nashville grocery, Jakesummit of recognition by his peers, he used the and his family moved to Birmingham, Alabama, andmetaphor, “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 Columbusthe daily vocabulary and working equipment of in 1926. He entered the University of Alabamamany of us. Such is certainly the status of Nathan that fall, intending to follow a maternal uncle intoJacobson. As my fellow authors will show more law.specifically, he earned his dominance by recasting While following a pre-law program, he took allwhole theories of algebraic systems and by in- mathematics courses available. The notice of hissisting on the module-theoretic viewpoint in their professors was attracted to the extent that in hisstudy. His expository and research monographs junior year he was offered a teaching assistantshipand his ambitious textbooks have indebted a in mathematics. Two of these professors, Fredworldwide community to him for strong and Lewis and William P. Ott, were always rememberedarticulate leadership. The authors use this oppor- fondly as having inspired him to turn to a careertunity to remind us of some of the ways his ideas in mathematics. With their advice he applied forhave 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 littleographical material is to be found in the “Personal short of the bill for tuition, room, and board, butHistory and Commentary” that he wrote in seven the following years saw increases to levels that heinstallments 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 Polynomialsdetails on his personal life and for his comments and Cyclic Algebras, with J. H. M. Wedderburn ason the development of his mathematical work. In advisor, was accepted for the Ph.D. in 1934. Howthis segment of the present article I provide a his time in Princeton and subsequently at thesketch 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 andOctober 5. His father emigrated to Nashville, Georgia Benkart.Tennessee, when Jake was five, leaving the family Emmy Noether had taken a position at Brynin 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 withwere 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 theGeorge B. Seligman is professor of mathematics emeritus following academic year. After a postdoctoralat Yale University. His e-mail address is selig@ fellowship with Adrian Albert at Chicago inmath.yale.edu. 1936–37, he was appointed to a junior position atOCTOBER 2000 NOTICES OF THE AMS 1061
- 2. 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 19801062 NOTICES OF THE AMS VOLUME 47, NUMBER 9
- 3. issue of the Notices in response to a vicious personal for all subsequent work in the field, work thatattack 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 onlyafter he had earned the honor of carrying the four years beyond the doctorate. His thesis advisoruniversity’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 piecefrom around the world. Kevin McCrimmon and will also read the autobiography and (toDavid 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 Princetonbut his right hand was nearly useless for writing, was the presence of Hermann Weyl at theand he could not walk unaided. With Florie taking newly founded Institute for Advancedon much of the mechanics, he finished the book Study. Weyl gave a course on Lie groups andon division algebras [B16] for publication in 1996, Lie algebras for which notes were writtencompleting 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 firstment took her from Jake’s side in 1996. No visitor paper to use the term “Lie algebra”; thethereafter could fail to be reminded how much she change from “infinitesimal group” washad 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 lemmato make the trip to Baltimore in January of 1998 (Lemma 2 on page 877): If A and B areto be honored with the Society’s Leroy P. Steele matrices over a field of characteristic 0 andPrize for Lifetime Achievement. A photo accom- A commutes with AB − BA , then AB − BApanying this article shows his radiance at that is nilpotent. I fell in love with this lemmaevent. His only lament was the absence of Florie. With wife Florie, and came back to it repeatedly. Just say “Ja-May they now have found reunion. around 1960. 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 beenthe world of mathematics has lost a giant of one novel example of a simple algebra: the Witttwentieth-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 toJake. 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 thatwas attracted by the special program in algebra that Jake was overoptimistic in wondering whether allsummer. 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, thetopological 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 hisby Albert I heard Jake give a talk on locally com- infinite simple pseudo-groups. In my own study ofpact 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 heday. Pontrjagin had done the pioneering work by launched his general structure theory for rings inshowing that the only connected locally compact [31] and [32]. Let A be a ring with unit element. Letdivision rings are the reals, complexes, and quater- J be the intersection of the maximal left ideals innions. The paper [3], joint with Olga Taussky, took A . There is no apparent reason why J should be aa big step forward by studying a general two-sided ideal, but it is. There is no apparentlocally 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 itIrving Kaplansky is director emeritus of the Mathemati- vanishes, A is called semisimple. (Warning: Otherscal Sciences Research Institute, Berkeley. His e-mail address say “semiprimitive”, reserving “semisimple” foris kap@msri.org. the Artinian case.) Now the famous WedderburnOCTOBER 2000 NOTICES OF THE AMS 1063
- 4. 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 toSteele 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 toWith 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 benkart@math.wisc.edu.1064 NOTICES OF THE AMS VOLUME 47, NUMBER 9
- 5. A . Weakly closed means that for each ordered pair p pof elements a, b ∈ S , there is a scalar γ(a, b) so D p (ab) = D p−k (a)D k (b), kthat ab + γ(a, b)ba ∈ S . If every element a of S is k=0nilpotent (ak = 0 for some k ), then the associativesubalgebra S ∗ of A generated by S is nilpotent that D p (ab) = D p (a)b + aD p (b) . In other words, D p((S ∗ )m = 0 for some m). Jacobson perfectly phrased is a derivation. It was Jacobson’s great insightthis lovely little gem so that it can be invoked for that the property of being closed under p-powersLie and Jordan algebras and Lie superalgebras. conveys important structural information. ThisIt is noteworthy as one of the few general results idea led him to introduce the notion of a restrictedthat apply over any field, even fields of prime Lie algebra [11].characteristic. Rather than present the general abstract defin- One crowning achievement of nineteenth- ition, let us assume for simplicity that the centercentury mathematics was the classification by Z(L) = {z ∈ L | [z, x] = 0 for all x ∈ L}Cartan and Killing of the finite-dimensionalsimple Lie algebras over an algebraically closed of the Lie algebra L is zero. In that case, L is restrictedfield F of characteristic zero. These Lie algebras if for each x ∈ L , the mapping (ad x)p , which is aare (up to isomorphism): derivation of L , in fact equals ad y for some y in L . a) sln (F) , the special linear Lie algebra of n × n Usually the element y is written x[p] to indicate its matrices over F of trace 0 for n ≥ 2 ; dependence on both x and the p-power. b) son (F) , the orthogonal Lie algebra of n × n The Lie algebras associated to algebraic groups (the analogues of Lie groups over arbitrary fields) matrices x over F such that xt = −x for n ≥ 5 , are always restricted, so the characteristic p ver- t denoting transpose; sions of the Lie algebras in (a) through (d) are re- c) spn (F) , the symplectic Lie algebra of n × n stricted. They are simple too, except when p | n for matrices x over F such that xt J + Jx = 0 for sln (F) , where it is necessary to factor out scalar mul- n ≥ 4 . Here n must be even, and J is the n × n tiples of the identity matrix. However, they are matrix of a nondegenerate skew-symmetric not the only finite-dimensional simple Lie alge- bilinear form. bras over algebraically closed fields of character- d) one of 5 exceptional Lie algebras e6 , e7 , e8 , f4 , g2. istic p > 0 . The Witt algebra, which is the deriva- When the underlying field F is not algebraically tion algebra of the truncated polynomial algebraclosed, it is possible to describe the simple Lie F[x | xp = 0] , provides an example, as do the Ja-algebras L over F that upon extension to the cobson-Witt algebras, which are the derivations ¯ ¯algebraic closure F are isomorphic to sln (F) , son (F) ¯ p of F[x1 , . . . , xm | xi = 0] . The latter algebras werefor n = 8, or spn (F ¯) . Jacobson’s ground-breaking discovered and investigated by Jacobson in thepapers of 1937–38 [9], [14], [15] showed that L is early 1940s as part of his efforts to develop a Ga-the Lie algebra (A, [ · , · ])/F1 constructed from a lois theory for purely inseparable field extensionssimple associative algebra A whose center is F1 using derivations rather than automorphisms [28].(or what is now called a central simple associative His work set the stage for Albert and Frank [AF],algebra), or it is the set of skew elements of a [F], who constructed simple Lie algebras from thecentral simple associative algebra with involution. Jacobson-Witt algebras. Kostrikin and Safarevic ˇ ˇThis work began Jacobson’s general program on [KS] extended the ideas in [AF] and [F] by identi-“forms of algebras” [89] that ultimately led to his fying four unifying families of simple Lie algebrasclassification of the forms of the Lie algebra that live in the Jacobson-Witt algebra. These fourg2 using composition algebras [19] and to the are called the Cartan-type Lie algebras becauseclassification of forms of simple Jordan algebras they correspond to Cartan’s four infinite families[40], [58]. (Witt, special, Hamiltonian, contact) of infinite-di- For any sort of algebra A (associative, Lie, mensional complex Lie algebras.Jordan, etc.), the linear transformations D : A → A ˇ Kostrikin and Safarevic conjectured that over an ˇthat satisfy the “derivative property” D(ab) algebraically closed field of characteristic p > 5 a= D(a)b + aD(b) are said to be derivations. Deriva- finite-dimensional restricted simple Lie algebra istions are very natural objects to study [11], [28] es- classical (as in (a) through (d) above) or of Cartanpecially in Lie theory, because the adjoint transfor- type. Almost one hundred years after the classifi-mation ad x : L → L , given by ad x(y) = [x, y] , of a cation of the simple Lie algebras of characteristicLie algebra L is always a derivation. This statement zero, Block and Wilson [BW] in 1988 succeeded inis equivalent to the Jacobi identity. In general, the proving this conjecture. If the notion of Cartan-typecomposition D1 D2 of two derivations need not be Lie algebras is expanded to include the simple al-a derivation; however, the set of all derivations is a gebras arising from Cartan-type algebras that areLie algebra under the commutator product twisted by an automorphism, then one can for-[D1 , D2 ] = D1 D2 − D2 D1 . If the underlying field ˇ mulate the Generalized Kostrikin-Safarevic Con- ˇhas characteristic p > 0 , then it is a consequence jecture by erasing the restrictedness assumptionof Leibniz’s formula, in the statement above. In the absence ofOCTOBER 2000 NOTICES OF THE AMS 1065
- 6. The nonassociative algebra of octonions (or Books of Nathan Jacobson Cayley numbers) is responsible for most of the ex- [B1] The Theory of Rings, Mathematical Surveys, No. II, Amer. Math. ceptional phenomena in Lie and Jordan theory. Soc., Providence, RI, 1943. Just as the complex numbers C = R ⊕ Ri are the [B2] Lectures in Abstract Algebra, Vol. 1, Basic Concepts, Van Nostrand, Princeton, NJ, 1951; Springer-Verlag reprint, 1975. double of the real numbers R, and the quaternions [B3] Lectures in Abstract Algebra, Vol. 2, Linear Algebra, Van Nostrand, H = C ⊕ Cj are the double of the complex numbers, Princeton, NJ, 1953; Springer-Verlag reprint, 1975. so the octonions O = H ⊕ H can be regarded as [B4] Lectures in Abstract Algebra, Vol. 3, Theory of Fields and Galois The- the double of the quaternions. The derivation al- ory, Van Nostrand, Princeton, NJ, 1964; Springer-Verlag reprint, 1975. gebra Der(O) of the octonions is the simple ex- [B5] Structure of Rings, Colloquium Publications, vol. 37, Amer. Math. ceptional Lie algebra g2 ([19] or [B6]). The space of Soc., Providence, RI, 1956 and 1964. 3 × 3 Hermitian matrices H3 (O) with entries in O [B6] Lie Algebras, Interscience-Wiley, New York-London, 1962; Dover is the exceptional 27-dimensional simple Jordan reprint, 1979. algebra, now called the Albert algebra, and its de- [B7] Structure and Representations of Jordan Algebras, Colloquium Pub- lications, vol. 39, Amer. Math. Soc., Providence, RI, 1968. rivation algebra Der(H3 (O)) is the simple excep- [B8] Lectures on Quadratic Jordan Algebras, Tata Institute of Fundamental tional Lie algebra f4 . The Lie algebras of types e6, Research, Bombay, 1969. e7, and e8 can be constructed using the octonions [B9] Exceptional Lie Algebras, Lecture Notes in Pure and Appl. Math., as well. This is the tale told in Exceptional Lie Al- Dekker, New York, 1971. gebras by an author whose own contributions to [B10] Basic Algebra I, Freeman, New York, 1974; second edition, 1985. that story are immense. [B11] PI-Algebras. An Introduction, Springer-Verlag, Berlin-New York, My only class with Jacobson was an exceptional 1975. Lie algebras course—it was truly an exceptional Lie [B12] Basic Algebra II, Freeman, New York, 1980; second edition, 1989. [B13] Structure Theory of Jordan Algebras, Lecture Notes in Math., Uni- algebras course. About twelve years after I took this versity of Arkansas, 1981. class, a colleague at Wisconsin, who knew what the [B14] Collected Mathematical Papers, volumes 1–3, Birkhäuser, Boston, book meant to me, brought me nine copies of Ex- 1989. ceptional Lie Algebras that he had found on sale [B15] A. ADRIAN ALBERT, Collected Mathematical Papers, 2 vols. (R. Block, in New York. They are long gone to good homes, N. Jacobson, M. Osborn, D. Saltman, and D. Zelinsky, eds.), Amer. as inquiring minds wanted to know, and there is Math. Soc., Providence, RI, 1993. no better place to start. [B16] Finite Dimensional Division Algebras over Fields, Springer-Verlag, As president of the American Mathematical So- Berlin-New York, 1996. ciety and vice president of the International Math- ematical Union, Jacobson was extraordinarily busy during my graduate years at Yale. Yet he was a restrictedness, Strade’s p-envelopes, which are re- calm, reassuring mentor who never seemed rushed stricted Lie algebras, save the day and enable the and who always had time to talk. We, his thirty- classification to be carried out (see [SW], [St]). It is three Ph.D. students who felt his gentle guidance impossible to imagine how the classification might and experienced his gracious kindness, owe him have been achieved without Jacobson’s notion of a special debt that perhaps can be repaid only in a restricted Lie algebra and his guiding light, for kind by emulating his behavior with our own grad- Jacobson kindled in his students and grandstu- uate students. dents a great interest in the classification problem. I last saw “Jake” about a year ago, when I briefly As a result, he and his descendants—Curtis, Selig- stopped in New Haven en route to a colloquium in man, Wilson, Gregory—and I all have been involved Boston. Knowing that I had arrived from Princeton, in this enterprise. he was eager to reminisce about the exciting early Although after the mid-1950s Jacobson devoted days of the Institute there. He also had just received much of his research to associative and Jordan al- a copy of Kevin McCrimmon’s new book, A Taste gebras, he wrote two books, Lie Algebras [B6] and of Jordan Algebras [McC2]. What delighted him Exceptional Lie Algebras [B9], and supervised a most about the book, dedicated to “Jake” and his number of graduate students in Lie theory. Lie Al- wife Florie, was that the contributions of each of gebras transformed the beautiful classification them had been acknowledged. That is exactly how picture of Cartan and Killing into highly under- he wanted it to be. The mathematical community, standable text. Its status as a “classic” having been their family, and their friends will miss them both confirmed by its 1979 republication in the Dover very much. series, Lie Algebras still remains the best basic reference for restricted Lie algebras and for an ex- Kevin McCrimmon position of the famous embedding result known I would like to say a few words about Jake’s as the Jacobson-Morosov theorem. This book, like legacy for Jordan algebras. Jordan algebras were Jacobson’s papers, has a timeless quality, and one must marvel at just how readable his works are Kevin McCrimmon is professor of mathematics at the even now, over sixty years after many of them University of Virginia. His e-mail address is kmm4m@ were written. virginia.edu.1066 NOTICES OF THE AMS VOLUME 47, NUMBER 9
- 7. Papers of Jacobson Referred to in the Articleintroduced by P. Jordan as an attempt to provide The numbers are taken from the bibliography in Collected Math-an algebraic setting for quantum mechanics that ematical Papers, [B14].enjoyed all the properties of the usual model yetdid not presuppose an underlying associative al- [1] Non-commutative polynomials and cyclic algebras, Annals of Math.gebra. In the usual interpretation, quantum-me- 35 (1934), 197–208 Princeton University dissertation.chanical observables are represented by operators [3] (with O. Taussky), Locally compact rings, Proc. Nat. Acad. Sci. USAon Hilbert space, but only hermitian operators are 21 (1935), 106–108.physically observable. A linear algebra is called a [4] Rational methods in Lie algebas, Annals of Math. 36 (1935), 875–881. [9] A class of normal simple Lie algebras of characteristic zero, AnnalsJordan algebra if it satisfies the identities of Math. 38 (1937), 508–517.(J1) xy = yx, [11] Abstract derivation and Lie algebras, Trans. Amer. Math. Soc. 42 (1937), 206–224.(J2) (x y)x = x2 (yx). 2 [12] p-Algebras of exponent p, Bull. Amer. Math. Soc. 43 (1937), 667–670. [14] Simple Lie algebras of type A , Annals of Math. 39 (1938), 181–188.If A is an associative algebra, then the vector space [15] Simple Lie algebras over a field of characteristic zero, Duke Math.A , together with the “anti-commutator” multipli- 1 Jour. 4 (1938), 534–551.cation x · y := 2 (xy + yx) , forms a Jordan algebra, [19] Cayley numbers and normal simple Lie algebras of type G , Dukedenoted A+ . A Jordan algebra J is called special if Math. Jour. 5 (1939), 775–783.it arises as a Jordan subalgebra of some associa- [21] A note on hermitian forms, Bull. Amer. Math. Soc. 46 (1940),tive algebra, J ⊆ A+ . The most important example 264–268.is the subspace H(A, ∗) of ∗ -hermitian elements [23] Classes of restricted Lie algebras of characteristic p, I, Amer. Jour.x∗ = x with respect to an involution ∗ on A (i.e., Math. 63 (1941), 15–25.an involutive antiautomorphism). Any symmetric [24] Classes of restricted Lie algebras of characteristic p, II, Duke Math. Jour. 10 (1943), 107–121.bilinear form · , · on a vector space V over a field [28] Galois theory of purely inseparable fields of exponent one, Amer.F somewhat accidentally gives rise to a special Jour. of Math. 66 (1944), 645–648.Jordan algebra (a “spin factor”) on the space [31] Structure theory of simple rings without finiteness assumptions,F · 1 ⊕ V by having 1 act as identity element and Trans. Amer. Math. Soc. 57 (1945), 228–245.having vectors multiply by v · w := v, w 1 ; this is [32] The radical and semi-simplicity for arbitrary rings, Amer. Jour.a Jordan subalgebra of the Clifford algebra of the Math. 67 (1945), 300–320.bilinear form. [33] Structure theory for algebraic algebras of bounded degree, Annals A Jordan algebra is called exceptional if it is not of Math. 46 (1945), 695–707.special; Jordan was seeking an exceptional Jordan [37] Isomorphisms of Jordan rings, Amer. Jour. of Math. 70 (1948), 317–326.model for quantum mechanics. In 1934 Jordan, J. [39] Lie and Jordan triple systems, Amer. Jour. Math. 71 (1949), 149–170.von Neumann, and E. Wigner made a complete [40] (with F. D. Jacobson), Classification and representation of semi-sim-classification of finite-dimensional formally real ple Jordan algebras, Trans. Amer. Math. Soc. 65 (1949), 141–169.Jordan algebras and showed they were direct sums [58] (with A. A. Albert), On reduced exceptional simple Jordan algebras,of five types of simple algebras: spin factors and Annals of Math. 66 (1957), 400–417.hermitian n × n matrices Hn (R), Hn (C), Hn (H) over [59] (with L. J. Paige), On Jordan algebras with two generators, J. Ratio-the reals R, the complexes C, or Hamilton’s quater- nal Mech. and Anal. 6 (1957), 895–906.nions H , together with a totally unexpected Hn (O) [69] Generic norm of an algebra, Osaka Math. J. 15 (1963), 25–50. [70] Clifford algebras for algebras with involution of type D , J. Alge-over Cayley’s octonions O (but only for n = 3). This bra 1 (1964), 288–300.latter 27-dimensional Jordan algebra has since be- [73] Structure theory for a class of Jordan algebras, Proc. Nat. Acad. Sci.come the celebrated exceptional Jordan algebra USA 55 (1966), 243–251.(often referred to as the Albert algebra). [76] Structure groups and Lie algebras of Jordan algebras of symmet- The exceptional Jordan algebra proved to be ric elements of associative algebras with involutions, Advances inan invaluable ingredient in explicit constructions Math. 20 (1976), 106–150.of the exceptional Lie algebras, especially those [79] Some applications of Jordan norms to associative algebras, Advancesover arbitrary fields. These Lie constructions and in Math. 48 (1983), 149–165. [89] Forms of algebras, Some Recent Advances in Basic Sciences, Acad-related foundational work Jake did with Albert emic Press, New York, 1966.and others on classifying finite-dimensional nonas-sociative algebras have already been discussed byGeorgia Benkart. I will concentrate on the new in- Ji ⊆ Ai any Jordan isomorphism J1 → J2 extendedsights, new concepts, and new tools he brought to to an isomorphism or anti-isomorphism of theJordan algebras. given associative envelopes A1 → A2 . In his 1949 papers [40] (with his wife Florie) and [39], he in-Universal Gadgets troduced the universal specialization µ in the uni-In his very first paper on Jordan algebras [31] in versal special envelope U of J (analogous to the well-1948, describing the isomorphisms between the known universal associative enveloping algebraspecial simple Jordan rings classified by Albert in of Lie theory). This was introduced to reduce Jor-1946, Jake used results of Ancochea and Kalisch dan homomorphisms (called specializations)showing that for certain special Jordan algebras J → A+ to associative homomorphisms U → A ofOCTOBER 2000 NOTICES OF THE AMS 1067
- 8. the universal gadget. It was characterized by The U-Operator and the Fundamental its universal property that all specializations Formula ϕ J −→ A (not just the isomorphisms) were reduced One particular case of the Jordan triple product oc- ˜ ϕ to associative homomorphisms of U −→ A by fac- curs when the two outside variables coincide, lead- toring through the universal specialization, ing to an important quadratic product, 1 Ux y = {x, y, x}. 2 This is equal to xyx in special algebras. Jake in- troduced these operators and the U -notation (its origins are obscure) and conjectured the Funda- mental Formula in operator terms, the right vertical arrow in the second diagram in- UUx y = Ux Uy Ux . dicating the forgetful map. This is easy to verify in associative algebras, since This was especially effective, since U is finite di- on an element z it becomes (xyx)z(xyx) = mensional when J is (unlike in the Lie case), so that xyxzxyx = x(y(xzx)y)x . After hearing Jake lec- the homomorphisms of U were well understood ture on this, I. G. Macdonald went home and proved from the associative theory. This strategy made the the conjecture. Moreover, using a deep theorem of extension of Jordan specializations to the asso- Shirshov on two-generated Jordan algebras, he ciative envelope automatic. There was of course went on to establish a general principle that any work to be done in describing the universal gad- Jordan polynomial identity in three variables that get for any particular Jordan algebra, but then the is linear in one of them will hold in all Jordan al- entire question of Jordan specializations was re- gebras as soon as it holds in all associative alge- duced to the study of this one associative algebra. bras. The study of specialization is essentially the The U -operator and its Fundamental Formula study of all Jordan “modules”. Jake also intro- have completely recast our view of the Jordan duced a universal gadget for multiplication spe- landscape: we have slowly come to realize that cializations (corresponding to Jordan “bimodules”) the fundamental product in a Jordan system is and showed how it related to a certain “meson al- the quadratic product Ux y , not the bilinear prod- gebra” introduced by physicists. uct {x, y} or the trilinear product {x, y, z} , which results by polarizing x → (x, z) in the quadratic Triple Products expression Ux y . This basic product is as associa- 1 To avoid messy factors 2 in Jordan products, we tive as such a product can be: unital Jordan alge- can introduce the brace product (or 2 -tad) bras are described axiomatically by {x, y} := 2xy . In 1949 [39] Jake recognized the im- • U1 = 1J , portance of the so-called 3 -tad {x, y, z} defined by • Ux {y, x, z} = {x, y, Ux z} , • UUx y = Ux Uy Ux . 2{x, y, z} := {{x, y}, z} + {{z, y}, x} − {y, {x, z}}. Jake used the U -operator to obtain the basic facts about inverses. He showed that the proper In associative algebras these products take the definition of x invertible is that Ux be an invert- simple form {x, y} = xy + yx and {x, y, z} = xyz ible operator, with Ux−1 = (Ux )−1 . There is no cor- + zyx. Jordan triple systems are algebraic struc- responding result for the bilinear multiplication. tures closed under a triple product behaving like Once more, while the U -operator and Funda- {x, y, z} . Jordan algebras are of course closed mental Formula were proving their worth alge- under their triple products, but certain subspaces braically, they popped their heads up again in dif- might be closed under the triple but not the bilinear ferential geometry in work of Koecher: Ux arises product, so Jordan triple systems were a wider naturally out of the inversion map j(x) = −x−1 by class of algebraic structures (as Jake had shown for Lie triple systems). Once more an unexpected con- Ux = (∂j |x )−1 nection appeared between Jordan and Lie theory: for ∂j |x the usual differential (best linear ap- 3 -graded Lie algebras L = L1 ⊕ L0 ⊕ L−1 lead nat- proximation) of the nonlinear map j at the point urally to Jordan pairs (V+ , V− ) = (L1 , L−1 ) —a pair of x . This allowed T. A. Springer [Sp] to base an en- spaces acting on each other, but not on them- tire theory and classification of Jordan algebras on selves, as Jordan triple systems—via the Jordan the operation of inversion. Another illustration of triple product {x+ , y − , z + } = [[x+ , y − ], z + ] . Jordan how Ux arises from inversion is the Hua identity, triples and pairs are now seen as important fea- which can be written as tures of the mathematical landscape, with Jordan algebras as especially exemplary members of this family. Ux (y) = x − (x−1 − (x − y −1 )−1 )−11068 NOTICES OF THE AMS VOLUME 47, NUMBER 9
- 9. or as recognized im- x −1 + (y −1 −1 − x) −1 = (x − Ux y) mediately the utility of thesewhenever x , y , and y −1 − x are invertible. This as an analogueidentity is relatively easy to derive by an argument of one-sidedappealing to Zariski density and change of unit el- ideals for gen-ement. It was Jake who showed how to change eral Jordanunits in Jordan algebras (passing from J with unit algebras and1 to an “isotope” with unit u , for any invertible u ) developed aand demonstrated the power of this method in Jor- structure the-dan theory. ory for Jordan rings with de-Generic Norms scending chainJake made frequent use of the concept of generic conditionnorm. This is a generalization of the determinant (d.c.c.) on qua-for matrices, more generally of the “reduced norm” dratic ideals Yale, about 1980. Left to right: Ying Cheng,on finite-dimensional associative algebras. He that was com- Jacobson, and Walter Feit.showed that the generic norm could be defined for pletely analo-any finite-dimensional power-associative algebra, gous to thei.e., an algebra in which each element generates an Artin-Wedder-associative subalgebra, so that the usual rules of burn theory forpowers apply, though the algebra as a whole need associativenot be associative [69]. The key idea was that the rings with“generic element” satisfied a generic minimum d.c.c. on leftpolynomial ideals [73]. Like Athena spring- xn − σ1 (x)xn−1 + . . . + (−1)n σn (x)1 = 0 ing full grownin which each σi is a homogeneous polynomial from the mindfunction of degree i. Here σ1 is the generic trace of Zeus, thisand σn the generic norm. For ordinary associative theory sprangmatrix algebras these are just the usual trace and full grown from the minddeterminant, and the generic minimum polyno- of Jake.mial is the characteristic polynomial. The key This was theplayer here was the generic norm N( · ) , since the first truly ring- Jacobson at his 75th birthday celebration,minimum polynomial could be recovered as theoretic ap- 1985, with Efim Zelmanov. N(λ1 − x) = λn − σ1 (x)λn−1 + . . . + (−1)n σn (x)1 proach to Jor- dan algebras: the requisite idempotents (for nofor an indeterminate λ. theory seemed possible without a rich supply of Any quadratic form Q is the generic norm of a idempotents) arose from the minimal quadraticJordan algebra of “degree 2 ”, x2 − T (x)x + ideals instead of elements algebraic over a field.Q(x)1 = 0 . Only certain very special cubic forms, After Jake’s paper, one could paraphrasesuch as with the Albert algebra, arise as generic Archimedes and say, “Give me the Fundamentalnorms of “degree 3 ” Jordan algebras, x3 − T (x)x2 + Formula and I can move the world.” I mimickedS(x)x − N(x)1 = 0 . We no longer say, with Cartan, Jake’s entire paper [McC1] to get a theory of qua-that the exceptional Lie group E6 arises as a group dratic Jordan rings based entirely on the productof transformations preserving a certain cubic form Ux y ≈ xyx , which had no need of a scalar 1 and2on 27-dimensional space; we say that E6 arises as hence was applicable not only to fields of charac-the group of linear transformations on the 27-di- teristic 2 but also to arithmetic situations, such asmensional Albert algebra that preserve the generic algebras over the integers.norm, or equivalently, the surface N(x) = 1. Simi- Later Jake rechristened these B ’s inner ideals.larly, F4 arises not as the isotropy group of a point The Jordan product xyx does not have a left oron the cubic surface, but as the isotropy group of right like the associative product xy ; it has an in-the identity element, or better, as precisely the side and an outside. An inner ideal is a subspace Bautomorphism group of the Albert algebra. closed under inner multiplication by J, UB J ⊆ B, while an outer ideal is closed under outer multi-Inner Ideals 1 plication, UJ B ⊆ B . If there is a scalar 2 available,David Topping [T] introduced quadratic ideals these outer ideals are the same thing as ideals(subspaces B ⊆ J closed under the quadratic prod- (two-sided, both inner and outer).uct UB J ⊆ B ) in his study of Jordan algebras of self- The final achievement of the classical age inadjoint operators on Hilbert space. Jake Jordan algebra was Jake’s structure theory forOCTOBER 2000 NOTICES OF THE AMS 1069
- 10. algebras with capacity [B13]. The d.c.c. leads to min- imal inner ideals, which generate division idem- Ph.D. Students of Nathan Jacobson potents, so Jake just started from a decomposition of the unit 1 into a finite sum of division idem- Charles L. Carroll Jr. (1945) potents. Serendipitously, this was just what was Eugene Schenkman (1950) needed when Efim Zelmanov ushered in the new Charles W. Curtis (1951) age in Jordan theory with his classification of sim- William G. Lister (1951) ple Jordan algebras of arbitrary dimension. Henry G. Jacob (1953) George B. Seligman (1954) And so the torch was passed to a new genera- Morris Weisfeld (1954) tion. Bruno Harris (1956) Earl J. Taft (1956) Dallas W. Sasser (1957) David J. Saltman Maria J. Woneburger (1957) Tae-il Suh (1961) Nathan Jacobson has had an important and Herbert F. Kreimer Jr. (1962) deep influence on the theory of central simple al- Charles M. Glennie (1963) gebras despite the fact that he wrote on the sub- David A. Smith (1963) ject only at isolated points of his career. His con- Dominic C. Soda (1964) tributions can be divided into distinct time periods, Harry P. Allen (1965) separated by periods when he published in other Eugene A. Klotz (1965) areas. His influence was not only exercised by Kevin M. McCrimmon (1965) means of research papers but also through his ex- Joseph C. Ferrar (1966) position of known results in monographs. Daya-Nand Verma (1966) To begin with, Jake’s Ph.D. thesis was in the area Lynn Barnes Small (1967) of central simple algebras [1]. In that work he John R. Faulkner (1969) traced the connection between cyclic algebras and Samuel R. Gordon (1969) twisted polynomial rings. The deepest part con- Michel Racine (1971) cerned the so-called Schur index, which is the de- Jerome M. Katz (1972) gree of the division algebra associated to a central Ronald Infante (1973) simple algebra. In his thesis Jake shows that the Louis H. Rowen (1973) Schur index of a cyclic algebra can be computed Georgia M. Benkart (1974) by knowing the factorization of a polynomial in the David J. Saltman (1976) twisted polynomial ring. Though this work dates Robert A. Bix (1977) to 1934, the point of view is still being exploited Leslie Hogben (1978) and generalized, for example, in the work of Louis Craig L. Huneke (1978) Rowen. The next phase of Jake’s work in central simple algebras came in two papers, [11] and [12], in 1937. (but not the crossed-product approach) extends In the then-current theory of central simple alge- beautifully to Azumaya algebras over rings. Fur- bras, an important place was occupied by the so- thermore, behind Jacobson’s derivation extension called Noether-Skolem Theorem, which concerned result lay a useful fact that has become key in the automorphisms of these algebras. Jacobson began theory, namely, that the Brauer group map is sur- the investigation of derivations and their place in jective over purely inseparable extensions. the theory. His first important result was analogous The next paper on central simple algebras, of to the automorphism case. He proved that every particular note, is on another topic still. In [15] Ja- derivation of a central simple algebra, trivial on the cobson began his study of central simple algebras center, was inner. His next result had no auto- with involution, a subject he concerned himself morphism version: Jake proved that every deriva- with until the end of his career. An involution of tion on the center extended to the whole algebra, A is an anti-automorphism of order 2 , and for the a result clearly false for automorphisms. These ob- moment an involution will always be the identity servations turned out to be crucial in the theory, on the center F of A . If A = EndF (V ) , then involu- then just begun, of so-called p-algebras, which are tions correspond to similarity classes of symmet- central simple algebras of prime characteristic p ric or antisymmetric nonsingular bilinear forms on and degree a power of p. Jake could rewrite cyclic V . In the symmetric case (that of a quadratic form) algebras of degree p in terms of derivations instead the involution is called orthogonal, a concept that of automorphisms. It turns out that this approach generalizes to an arbitrary central simple A . Given a quadratic form, one can define the use- David J. Saltman is professor of mathematics at the ful and important Clifford algebra and even Clif- University of Texas, Austin. His e-mail address is ford subalgebra. In [70] Jake showed that one could saltman@fireant.ma.utexas.edu. define an even Clifford algebra for an orthogonal1070 NOTICES OF THE AMS VOLUME 47, NUMBER 9
- 11. involution on an arbitrary central simple A . This [G] C. M. GLENNIE, Some identities valid in special Jordanwas the important first result in a continuing long algebras but not in all Jordan algebras, Pacific J.program of many people (e.g., J. P. Tignol) that has Math. 16 (1966), 47–59. [KS] A. I. KOSTRIKIN and I. R. ˇAFAREVIC, Graded Lie algebras S ˇextended many parts of the theory of quadratic of finite characteristic, Izv. Akad. Nauk SSSR Ser. Mat.forms to involutions. 33 (1969), 251–322. While on the subject of involutions, let me jump [McC1] K. MCCRIMMON, A general theory of Jordan ringsahead to one of Jake’s last published results. An Proc. Nat. Acad. Sci. USA 56 (1966), 1455–1459.involution on A , still trivial on the center, which [McC2] — —, A Taste of Jordan Algebras, submitted for —is not orthogonal is called symplectic. If S ⊂ A is publication.the space of elements fixed by a symplectic invo- [Sp] T. A. SPRINGER, Jordan Algebras and Algebraic Groups, Ergebnisse der Mathematik, vol. 75, Springer-Verlag,lution, then there is a form p on S , called the Berlin-New York, 1973.Pfaffian, whose square is a form of the determi- [St] H. STRADE, The classification of the simple modularnant restricted to S . In [79] Jake showed that the Lie algebras, VI. Solving the final case, Trans. Amer.function field L of the zero set of the Pfaffian was Math. Soc. 350 (1978), 2553–2628. 1a so-called “generic 2 splitting field”. That is, A ⊗F L [SW] H. STRADE and R. L. WILSON, Classification of simplehas Schur index 2 in the nontrivial cases, and any Lie algebras over algebraically closed fields of primeother field with this property is a specialization of characteristic, Bull. Amer. Math. Soc. (N.S.) 24 (1991), 357–362.L . This was the first (and best understood) exam- [T] D. M. TOPPING, Jordan algebras of self-adjoint oper-ple of what one calls generalized Brauer-Severi va- ators, Mem. Amer. Math. Soc. 53 (1965).rieties. The last subject of Jake’s interest we highlightis reduced norms. If A is a central simple algebra ¯with center F and F is the algebraic closure of F , ¯then A ⊗F F is isomorphic to a ring of matrices ¯Mn (F ) . Such an example has the well-known de- ¯ ¯terminant map d : Mn (F ) → F . However, this mapcan be defined on A itself and is then called thereduced norm n : A → F . This is a polynomial mapand can be thought of as a polynomial on A . Jakeshowed that this polynomial carries a surprisingamount of information about the algebra-structureof A . Namely, he showed that a linear mapf : A → B that preserves 1 and preserves the norm,must be an isomorphism or anti-isomorphism.This result is a special case of the Jordan theoryin [76], where it arises, in the above form, only aftercombining other results. The book mentioned above, [B16], was Jake’s lastpublication and will very likely influence the fieldsignificantly. In [B11] Jake wrote a monograph giv-ing a complete exposition of Amitsur’s noncrossedproduct result. His clear, careful, and streamlinedapproach was a huge influence on mathematiciansfollowing up on Amitsur’s result.References References to books and articles by NathanJacobson appear in sidebars.[AF] A. A. ALBERT and M. S. FRANK, Simple Lie algebras of characteristic p, Rendiconti del Sem. Math. Torino 14 (1954/55), 117–139.[AH] S. A. AMITSUR, D. J. SALTMAN, and G. B. SELIGMAN, eds., Algebraists’ Homage, Contemp. Math., vol. 13, Amer. Math. Soc., Providence, RI, 1982.[BW]R. E. BLOCK and R. L. WILSON, Classification of the re- stricted simple Lie algebras, J. Algebra 114 (1988), 115–259.[F] M. S. Frank, Two new classes of simple Lie algebras, Trans. Amer. Math. Soc. 112 (1964), 456–482.OCTOBER 2000 NOTICES OF THE AMS 1071