Celebratio Mathematica

Irving Kaplansky

Irving Kaplansky and supersymmetry

by Peter G. O. Freund

I ar­rived in Chica­go some two dec­ades after Irving Ka­plansky, and I met Kap, as we all called him, shortly after my ar­rival here. We be­came friends later, in 1975, while col­lab­or­at­ing on a pa­per on su­per­sym­metry. Lie su­per­al­geb­ras, graded coun­ter­parts of or­din­ary Lie al­geb­ras, play a cent­ral role in string the­ory and oth­er uni­fied the­or­ies. A clas­si­fic­a­tion of the simple ones was of es­sence. I took some ini­tial steps, but the real work star­ted when Yitz Her­stein put me in touch with Kap. At first, com­mu­nic­a­tion was not easy. We couldn’t quite make out each oth­er’s reas­on­ing, much as we agreed on res­ults. It didn’t take long however, to get used to the oth­er’s way of look­ing at things. Math­em­aticians and phys­i­cists think in sim­il­ar ways after all, all that was needed was a dic­tion­ary. This was dur­ing the early phase of the rap­proche­ment between math­em­at­ics and the­or­et­ic­al phys­ics. After the glor­i­ous first half of the twen­ti­eth cen­tury — when the likes of Poin­caré, Hil­bert, Weyl, von Neu­mann, Élie Cartan, Emmy No­eth­er, and oth­ers made ma­jor con­tri­bu­tions to the then-new phys­ic­al the­or­ies of gen­er­al re­lativ­ity and quantum mech­an­ics, while phys­i­cists like Jordan, Dir­ac, Casimir, and Feyn­man made ma­jor con­tri­bu­tions to math­em­at­ics — phys­ics entered a peri­od best de­scribed as phe­nomen­o­lo­gic­al. Dur­ing this peri­od, some ad­vanced com­plex func­tion the­ory aside, very little mod­ern math­em­at­ics was drawn on. To give you an idea, when in his cel­eb­rated “Eight­fold Way” pa­per, Mur­ray Gell-Mann wrote down a basis of the three-di­men­sion­al rep­res­ent­a­tion of the \( \mathfrak{su}(3) \) Lie al­gebra, this was her­al­ded by phys­i­cists as a great math­em­at­ic­al feat. “Ima­gine, he found a \( 3{\times}3 \) gen­er­al­iz­a­tion of the fam­ous \( 2{\times}2 \) Pauli matrices,” is what most people said. To get there, Mur­ray had con­sul­ted with Block and Serre! It was in the fields of su­per­sym­metry and gauge the­ory that the ini­tial steps in mod­ern math­em­at­ic­al phys­ics were taken. This con­ver­gence of the paths of math­em­at­ics and of the­or­et­ic­al phys­ics is typ­ic­al of times when ma­jor new phys­ic­al the­or­ies — gauge the­ory and string the­ory in this case — are be­ing born. The earli­est ex­ample of such a con­ver­gence is the cre­ation of cal­cu­lus at the birth of New­ton’s mech­an­ics and of his the­ory of grav­it­a­tion. Weyl’s spec­tac­u­lar work on group the­ory un­der the im­pact of the new­born quantum mech­an­ics is an­oth­er such ex­ample.

A few words about our joint pa­per [1] are in or­der here. In it we found all the in­fin­ite fam­il­ies of simple Lie su­per­al­geb­ras, as well as 17-, 31- and 40-di­men­sion­al ex­cep­tion­al ones. We also dis­cussed real forms and ex­plained why su­per­sym­metry can act on 4-di­men­sion­al anti-de Sit­ter but not on de Sit­ter space, a res­ult es­sen­tial for un­der­stand­ing why the re­mark­able du­al­ity dis­covered by Mal­da­cena [e4] in the 1990s, is of the AdS/CFT and not of the dS/CFT type. We were con­vinced that we had found all simple Lie su­per­al­geb­ras (as we ac­tu­ally had), but we lacked a proof of this fact. The proof came from the power­ful in­de­pend­ent work of Vic­tor Kac [e1], [e2], [e3]. Amus­ingly, in his beau­ti­ful proof, Kac some­how over­looked one of the ex­cep­tion­al su­per­al­geb­ras, namely the 31-di­men­sion­al su­per-al­gebra \( G(3) \), whose Bose (even) sec­tor con­sists of the or­din­ary Lie al­gebra \( \mathfrak{g}_2 + \mathfrak{sl}(2) \), the only simple Lie su­per­al­gebra to have an ex­cep­tion­al or­din­ary Lie al­gebra as one of the two con­stitu­ents of its Bose sec­tor. I said “amus­ingly” above be­cause, as I learned from Kap, in the clas­si­fic­a­tion of or­din­ary simple Lie al­geb­ras, in his ex­tremely im­port­ant early work, Killing had found al­most all of them, but he “some­how over­looked one,” namely the ex­cep­tion­al 52-di­men­sion­al simple Lie al­gebra \( F_4 \), which re­mained to be dis­covered later by Élie Cartan. Ap­par­ently, \( G(3) \) is the ex­cep­tion­al Lie su­per­al­gebra which car­ries on that curse of the or­din­ary ex­cep­tion­al Lie al­gebra \( F_4 \).

I men­tioned the al­most total lack of con­tact between the­or­et­ic­al phys­i­cists and math­em­aticians, when this work got go­ing. It went so deep that in 1975 most phys­i­cists, if asked to name a great mod­ern math­em­atician, would come up with Her­mann Weyl, or John von Neu­mann, both long dead. Math­em­aticians had it a bit easi­er, for if they read the news­pa­pers, they could at least keep track of the No­bel Prizes, where­as news­pa­per ed­it­ors rarely treated Fields Medal awards as “news fit to print.”

I re­call that while stand­ing by the state-of-the-art Xer­ox ma­chine to pro­duce some ten cop­ies of our pa­per in about… half an hour’s time, I asked Kap, “Who would you say, is the greatest math­em­atician alive?” He im­me­di­ately took me to task: my ques­tion was ill-defined, did I mean al­geb­ra­ist, or to­po­lo­gist, or num­ber-the­or­ist, or geo­met­er, or dif­fer­en­tial geo­met­er, or al­geb­ra­ic geo­met­er, etc. … I replied that I did not ask for a rig­or­ous an­swer, but just a “gut-feel­ing” kind of an­swer. “Oh, in that case the an­swer is simple: An­dré Weil,” he replied, without the slight­est hes­it­a­tion, a reply that should not sur­prise any­one, who has heard today’s talks. “You see,” Kap went on, “We all taught courses on Lie al­geb­ras or Jordan al­geb­ras, or whatever we were work­ing on at the time. By con­trast, Weil called all the courses he ever taught simply ‘math­em­at­ics’ and he lived up to this title.”

Kap went on to tell me about Weil’s le­gendary first col­loqui­um talk in Chica­go. This was the first time I heard that very funny story. Weil had been re­cruited for the Chica­go math­em­at­ics de­part­ment by its chair­man, Mar­shall Stone. With Stone sit­ting in the first row, Weil began his first Chica­go col­loqui­um talk with the ob­ser­va­tion, “There are three types of de­part­ment chair­men. A bad chair­man will only re­cruit fac­ulty worse than him­self, thus lead­ing to the gradu­al de­gen­er­a­tion of his de­part­ment. A bet­ter chair­man will settle for fac­ulty roughly of the same caliber as him­self, lead­ing to a pre­ser­va­tion of the qual­ity of the de­part­ment. Fi­nally, a good chair­man will only hire people bet­ter than him­self, lead­ing to a con­stant im­prove­ment of his de­part­ment. I am very pleased to be at Chica­go, which has a very good chair­man.” Stone laughed it off; he did not take of­fense.

The lack of com­mu­nic­a­tion between math­em­aticians and phys­i­cists was to end soon. By 1977, we all knew about Atiyah and Sing­er, and then the floodgates came down fast, to the point that an ex­tremely close col­lab­or­a­tion between math­em­aticians and phys­i­cists got star­ted and, un­der the lead­er­ship of Ed Wit­ten and oth­ers, is on­go­ing and bear­ing beau­ti­ful fruit to this day. By the way, on Kap’s desk I no­ticed some work of his on Hopf al­geb­ras. I asked him about Hopf al­geb­ras, and got the reply, “They are of no rel­ev­ance what­so­ever for phys­ics.” I took his word on this, was I ever gull­ible. In the wake of our joint work, Kap and I be­came good friends. This friend­ship was fueled also by our shared love of mu­sic; he was a fine pi­an­ist, and I used to sing. For me, the most mar­velous part of my col­lab­or­a­tion and friend­ship with Kap was that for the first time I got to see up-close how a great math­em­atician thinks.


[1]P. G. O. Fre­und and I. Ka­plansky: “Simple su­per­sym­met­ries,” J. Math­em­at­ic­al Phys. 17 : 2 (1976), pp. 228–​231. MR 0403438 Zbl 0399.​17005

[2]I. Ka­plansky: “Su­per­al­geb­ras,” Pa­cific J. Math. 86 : 1 (1980), pp. 93–​98. MR 586871 Zbl 0438.​17003