Celebratio Mathematica

A. Adrian Albert

A. A. Albert

by Daniel Zelinsky

Ab­ra­ham Ad­ri­an Al­bert died on June 6, 1972. The world lost a renowned math­em­atician, a vig­or­ous force for the ad­vance­ment of math­em­at­ics, and a very warm and un­der­stand­ing hu­man be­ing. From his birth to his death, he was as­so­ci­ated with Chica­go. As an in­vet­er­ate trav­el­er, he left that city of­ten, for far parts of the world, but he al­ways re­turned. He was born in Chica­go on Novem­ber 9, 1905, he went to school in Chica­go (ex­cept for two years when his fam­ily moved to Iron Moun­tain, Michigan), he did all his un­der­gradu­ate and gradu­ate work at the Uni­versity of Chica­go. After re­ceiv­ing his Ph.D., he left for three years at Prin­ceton and at Columbia Uni­versit­ies, then re­turned to the Uni­versity of Chica­go where he was a fac­ulty mem­ber un­til the end of his life. With this as his base, he worked in many math­em­at­ic­al cen­ters at vari­ous times in his ca­reer: The In­sti­tute for Ad­vanced Study in Prin­ceton (1933-34), Uni­versit­ies of Brazil and Buenos Aires (1947), Uni­versity of South­ern Cali­for­nia (1950), Yale Uni­versity (1956-1957), Uni­versity of Cali­for­nia at Los Angeles (1958). He op­er­ated in Wash­ing­ton in many ca­pa­cit­ies, and in the In­ter­na­tion­al Math­em­at­ic­al Uni­on. His most re­cent of­fi­cial trip was a vis­it to the USSR in 1971 as a guest of the So­viet Academy.

To his friends Pro­fess­or Al­bert was known as Ad­ri­an. Many math­em­aticians re­ferred to him af­fec­tion­ately as A\( ^3 \). He was the son of a Jew­ish fam­ily that came to Amer­ica from Eng­land. His fath­er in­sisted on a Jew­ish but not very re­li­gious train­ing. Al­bert dis­tin­guished him­self early in his schools (Herzl and Mar­shall) on the West Side of Chica­go, where the in­tel­lec­tu­al com­pet­i­tion from the oth­er bud­ding schol­ars was keen. He spent four years earn­ing his Bach­el­or’s de­gree at the Uni­versity of Chica­go, but one year later he had his Mas­ter’s de­gree, and a year after that, his Ph.D. In 1928, at age 22, his Ph.D. dis­ser­ta­tion already stamped him as one of the out­stand­ing al­geb­ra­ists of his day.

Those were the days when the math­em­at­ic­al lead­ers at the Uni­versity of Chica­go were L. E. Dick­son in al­gebra and E. H. Moore in gen­er­al to­po­logy. Dick­son was Al­bert’s thes­is ad­visor and is the one mainly re­spons­ible for steer­ing Al­bert in­to the sub­ject of al­geb­ras over fields, which is the sub­ject that primar­ily con­cerned him throughout his ca­reer.

He was one of the early Na­tion­al Re­search Coun­cil Fel­lows (1928–29). This fel­low­ship was the fore­run­ner of the mod­ern NSF Postdoc­tor­al Fel­low­ships (which un­for­tu­nately were dis­con­tin­ued re­cently) and has been held by some of the most fam­ous Amer­ic­an math­em­aticians.

The pre­co­city con­tin­ued. At the age 35, Al­bert was pro­moted to a full pro­fess­or­ship at the Uni­versity of Chica­go (at that time it was vir­tu­ally un­heard of to hold such a po­s­i­tion be­fore the age of 40). Two years later he was elec­ted to mem­ber­ship in the Na­tion­al Academy of Sci­ences, a 37-year old aca­dem­i­cian.

The list of oth­er hon­ors heaped on him, and of hon­or­if­ic du­ties he was asked to per­form would run to more pages than this art­icle. We men­tion just a sample: chair­man­ship (1958–1962) and dean­ship (1962–1971) at the Uni­versity of Chica­go, pres­id­ency of the Amer­ic­an Math­em­at­ic­al So­ci­ety (1965–66), trust­ee­ship of the In­sti­tute for Ad­vanced Study (1969–72), chair­man­ship of the In­ter­na­tion­al Math­em­at­ic­al Uni­on’s or­gan­iz­ing com­mit­tee for the 1970 Con­gress in Nice, mem­ber­ship in the Brazili­an and Ar­gen­tine Academies of Sci­ences, sev­er­al ed­it­or­ships, the Cole Prize in Al­gebra (1939), and three hon­or­ary de­grees. He seemed to col­lect these hon­ors with en­thu­si­asm, and ex­ecuted the du­ties with vig­or.

Al­though Al­beit worked on mat­rix the­ory, on quad­rat­ic forms, and oth­er as­pects of al­gebra, there is no ques­tion that his cent­ral in­terest was al­ways the study of fi­nite di­men­sion­al al­geb­ras over a field. In the old days, they were called hy­per­com­plex sys­tems. They are fi­nite di­men­sion­al vec­tor spaces with a mul­ti­plic­a­tion that as­so­ci­ates to every two vec­tors in the space an­oth­er vec­tor, the product. A sug­gest­ive ex­ample is the four-di­men­sion­al al­gebra of qua­ternions over the field of real num­bers. The clas­sic­al Wed­der­burn the­or­ems es­sen­tially re­duce the study of as­so­ci­at­ive al­geb­ras over a field to the clas­si­fic­a­tion of the di­vi­sion al­geb­ras (like the al­gebra of qua­ternions, for ex­ample). Over any field \( F \), a four-di­men­sion­al di­vi­sion al­gebra with cen­ter \( F \) must be an al­gebra of “gen­er­al­ized qua­ternions” whose mul­ti­plic­a­tion rules are much like the or­din­ary qua­ternions: a basis 1, \( i, j, ij \) with \( ij = ji \) and \( i^2 = a \) and \( j^2 = \beta \) ele­ments of \( F \), which have no square roots in \( F \) (but are not ne­ces­sar­ily \( - 1 \)). If one wants to gen­er­al­ize to di­men­sions high­er than 4 there are two can­did­ates: the cyc­lic al­geb­ras and the still more gen­er­al crossed product al­geb­ras. (A the­or­em as­serts that, in any case, the di­men­sion of any cent­ral di­vi­sion al­gebra is a per­fect square.) Wed­der­burn had already proved that cent­ral di­vi­sion al­geb­ras of di­men­sion 9 ate all cyc­lic al­geb­ras. In Al­bert’s dis­ser­ta­tion (1928) he proved that cent­ral di­vi­sion al­geb­ras of di­men­sion 16 are not ne­ces­sar­ily cyc­lic al­geb­ras, but are al­ways crossed products. Al­though Al­bert’s the­or­em raised the ob­vi­ous ques­tion about al­geb­ras of di­men­sion 25, 36, etc., his res­ult has stood without es­sen­tial im­prove­ment or em­bel­lish­ment (though not for lack of try­ing) un­til some nice, com­ple­ment­ary, but still not defin­it­ive res­ults of Amit­sur and oth­ers in 1971.

This study put the young Al­bert in the cen­ter of what was to be one of the ma­jor break­throughs in the the­ory of al­geb­ras: the de­term­in­a­tion of all cent­ral di­vi­sion al­geb­ras over the spe­cial field of ra­tion­al num­bers, or more gen­er­ally over any al­geb­ra­ic num­ber field. In this case, it turns out that they are all cyc­lic al­geb­ras — this is the fam­ous Hasse–Brauer–No­eth­er The­or­em (1931). An in­ter­est­ing art­icle by Hasse and Al­bert in the Trans­ac­tions of the Amer­ic­an Math­em­at­ic­al So­ci­ety (1932) traces the his­tory of this the­or­em and relates the story of Al­bert’s near miss. On the basis of his res­ults on al­geb­ras and some res­ults an­nounced by Hasse, Al­bert pub­lished some the­or­ems that nearly proved the big the­or­em, and he wrote Hasse about it. Some­how the com­mu­nic­a­tion was bad, and the Brauer–Hasse–No­eth­er manuscript was sub­mit­ted for pub­lic­a­tion without men­tion of Al­bert’s in­de­pend­ent con­tri­bu­tions. The 1932 Trans­ac­tions art­icle shows that in fact the big the­or­em fol­lows from Al­bert’s res­ults in just a few lines.

Al­bert was hurt and dis­ap­poin­ted by this in­cid­ent. But the depth of that hurt could not com­pare with his feel­ings about the sub­sequent Nazi scourge which caused some im­port­ant Ger­man math­em­aticians to be­gin dis­tin­guish­ing between “Ary­an” and “Semit­ic” math­em­at­ics, and which res­ul­ted in the ex­odus of so many Ger­man sci­ent­ists, Jews and non-Jews alike, in­clud­ing both Richard Brauer and Emmy No­eth­er. Al­bert was in­vited to be a mem­ber of the In­sti­tute for Ad­vanced Study in Prin­ceton dur­ing its open­ing year in 1933–34. (An­oth­er dis­tin­guished mem­ber, who ar­rived that year and re­mained on a per­man­ent basis was Al­bert Ein­stein.) This con­tact with Prin­ceton was prof­it­able for Al­bert. His as­so­ci­ations with Lef­schetz in par­tic­u­lar res­ul­ted in one of Al­bert’s math­em­at­ic­al ac­com­plish­ments that he al­ways re­garded with greatest pleas­ure, and for which he later won the Amer­ic­an Math­em­at­ic­al So­ci­ety’s Cole Prize in al­gebra. Already in 1929, Lef­schetz had in­ter­ested Al­bert in a ma­jor un­solved prob­lem in the the­ory of al­geb­ra­ic func­tions, Riemann sur­faces and Abeli­an vari­et­ies. In a series of pa­pers (1929–1934) Al­bert pro­duced a defin­it­ive solu­tion. What was re­quired was a clas­si­fic­a­tion of the al­geb­ra­ic cor­res­pond­ences of a Riemann sur­face (auto­morph­isms of a com­plex curve). This had been re­duced to the prob­lem of find­ing the matrices that com­mute with a cer­tain “Riemann mat­rix” of peri­ods of ba­sic Abeli­an in­teg­rals on the Riemann sur­face. These com­mut­ing matrices form an al­gebra, and in the ba­sic cases, a cent­ral simple al­gebra over the ra­tion­al num­ber field. This ver­sion of the prob­lem was right in the cen­ter of Al­bert’s spe­cial ex­pert­ise, and he de­mol­ished it.

Later, he at­tacked the prob­lem of gen­er­al nonas­so­ci­at­ive al­geb­ras that are fi­nite di­men­sion­al over a field. Al­most single-handed he in­flu­enced a large num­ber of young math­em­aticians to break this seem­ingly un­prom­ising ground. Spe­cial al­geb­ras had been stud­ied that were not as­so­ci­at­ive but which obeyed ax­ioms sub­sti­tut­ing for the as­so­ci­at­ive law (Lie al­geb­ras, Jordan al­geb­ras, al­tern­at­ive al­geb­ras). Res­ults like the Wed­der­burn the­or­ems had been proved for some of them; in fact, the res­ults for Lie al­geb­ras over the com­plex num­ber field were proved by E. Cartan be­fore Wed­der­burn ob­tained his cor­res­pond­ing the­or­ems for the as­so­ci­at­ive al­geb­ras. But Al­bert had the idea of us­ing as­so­ci­at­ive al­gebra the­ory to prove ana­logs of the Wed­der­burn the­or­ems for quite ar­bit­rary nonas­so­ci­at­ive al­geb­ras (even a nonas­so­ci­at­ive al­gebra has an as­so­ci­at­ive “reg­u­lar rep­res­ent­a­tion” al­gebra). It is a sign of his geni­us that he was ac­tu­ally able to de­vel­op a reas­on­able the­ory and also sig­ni­fic­antly in­flu­ence the the­ory and ap­plic­a­tions of the spe­cial al­geb­ras we men­tioned.

Al­bert’s style of al­gebra was al­most in­im­it­able. He had a diabol­ic­al fa­cil­ity with ma­nip­u­la­tion of iden­tit­ies — an en­ter­prise in which most math­em­aticians founder, nev­er be­ing able to see the forest for the trees. Some­how, Al­bert could see through mazes of sym­bols to the in­ner work­ings of all those poly­no­mi­als in sev­er­al vari­ables or mul­ti­plic­a­tion tables of com­plic­ated al­geb­ras.

Math­em­at­ics was Al­bert’s great en­thu­si­asm. It was im­possible to as­so­ci­ate with him for any length of time without feel­ing the vig­or with which he pur­sued his the­or­ems. He was al­ways will­ing to talk about his latest math­em­at­ic­al ex­ploits. When his son, Alan, was still very young, Al­bert in­sisted on ex­plain­ing even to him his latest the­or­ems, pa­tiently de­scrib­ing the ne­ces­sary in­gredi­ents to the in­trigued school­boy who had not yet form­ally seen any real math­em­at­ics.

Al­bert was a pro­lif­ic au­thor of text­books, re­search treat­ises, and more than a hun­dred and thirty re­search pa­pers, the last of which is due to ap­pear soon.

A minor theme run­ning through Al­bert’s life was his fas­cin­a­tion with cam­er­as, ra­di­os and oth­er gad­gets. I have al­ways thought that this streak was re­spons­ible for his activ­ity in the Ap­plied Math­em­at­ics Group at North­west­ern Uni­versity dur­ing World War II and his as­so­ci­ation with the Rand Cor­por­a­tion (1951 and 1952), South­ern Cali­for­nia Ap­plied Math­em­at­ics Pro­ject (1953–55; he was its chair­man 1959–60) and the In­sti­tute for De­fense Ana­lys­is (dir­ect­or, Com­mu­nic­a­tion Re­search Di­vi­sion 1961–62, trust­ee 1969–72). His prin­cip­al con­tri­bu­tion in these activ­it­ies was to crypt­ana­lys­is and cod­ing. He was also a lifelong afi­cion­ado of de­tect­ive stor­ies, which he de­voured at an enorm­ous rate.

To his friends, Pro­fess­or Al­bert is re­membered as a de­voted fam­ily man. He mar­ried Frieda Dav­is in his second year of gradu­ate study, and they shared a close re­la­tion­ship for all the sub­sequent forty-four years. They had two sons and a daugh­ter and five grand­chil­dren. (Tra­gic­ally one son died of an ill­ness at the age of 25.) Per­haps one should also count his 29 Ph.D. stu­dents whom he treated al­most as mem­bers of his fam­ily.

He was al­ways pleased to use his in­flu­ence in Wash­ing­ton to im­prove the status of math­em­aticians in gen­er­al, and he was will­ing to do the same for in­di­vidu­al math­em­aticians whom he con­sidered worthy. One of the more homey causes to which he lent the weight of his repu­ta­tion was re­tain­ing an apart­ment build­ing at the Uni­versity of Chica­go for vis­it­ing math­em­at­ics fac­ulty and their fam­il­ies. There are fam­il­ies throughout the world that re­mem­ber this little math­em­at­ic­al mi­cro­cosm with pleas­ure.

Every­one who knew him will re­mem­ber his vig­or­ous but round, me­di­um build, curly hair, and of­ten boy­ish de­mean­or; but es­pe­cially one must re­mem­ber his great, pleased grin that he flashed to wel­come news of new suc­cesses for any of his ex­ten­ded fam­ily any­where in the world of math­em­at­ics.