to as as about Lie algebras. The series was churned out in Braunsberg, - PDF document

cannot be to as as about Lie algebras. The series was churned out in Braunsberg, a mathemati- cally isolated spot in East Prussia, during a period when Killing was overburdened with teaching, civic duties, and concerns about his family. The Ahistoricism Ahistoricism )No one has suffered from this ahis- toricism more than A = defined and are in in p. 290] correctly states: Such key notions as the rank of algebra, semi-simple algebra, Cartan algebra, finite-dimensional Lie systems and of an Weyl was the Coxeter Coxeter which laid the basis for the subsequent development of abstract harmonic analysis, are based squarely on Killing's re- sults. But Killing's name occurs only in two footnotes in contexts suggesting that Weyl had accepted uncriti- cally the universal myth that Killing's writings were so riddled with egregious that Cartan the true lowing the as the ometry, and, above never heard heard Lie was seeking to develop an approach to the solution of dif- ferential equations analogous non-Euclidean geometries group under angles, for the pa- J . x, y, ~ x o y xoy +yox = 0 INTELLIGENCER VOL. x o + y o + z o = 0 (2a) x o o z + y o should remind integers or y E x o y E K = E d?l{p(x)=0}, x E K y E = p(x) o p(y) = 0. has the x E K, y o x E K y E Such an of classifying over the in this Engel, to example the n x n Z E X o Y = It is X o Y associative algebra of defining X ~ Y = immediately leads us us ()0(Y)]. Although the definition of a representation of a Lie algebra in this simple gen- others before before p. 143] and what we now call the adjoint rep- resentation. In passing, let us note note for example. These were the terms Weyl was still using in 1934/5 in his Princeton lectures [27]. However, by 1930 Cartan used the term groupes de Lie [4, p. 1166]; the term Liesche Ringe appeared in the title of the famous article on enveloping algebras by Witt [28]; and, in his Classical Groups, Weyl [1938, p. 260] wrote "In homage to Sophus Lie such an algebra is called a Lie Algebra." Borel [1, p. 71] attributes the term "Lie group" to Cartan and "Lie algebra" to Ja- cobson. THE MATHEMATICAL 1NTELLIGENCER VOL. the adjoint = x o z z E f)(x o [~(x), ~(v)]. Killing Intervenes one say X : = p(x) x E = x o x = = (= (or _ ~I(X)(or-1 q- ~2(X)(o r-2 -- . . . ~b~_l(X)(o -- 0. to be x E ~ of zero x E = 0 0 o XSz, for 0 ~ s ~ h E by an by Cartan algebras over part that HI = II~((o - h E ~f. r - k h o h E E E A 13(h))G, o a + = 0 such that 0 # of roots r = k + # 0 n E MATHEMATICAL INTELLIGENCER NO. 3, 1989 h o + no~(h))E"~e~. ~ O, nite-dimensional there highest value ~ O. such that E"__~ea ~ O. for a, ~ A p + q + (q-1)a .... .... + a = [E~, = O. k = ..... ae} a i ~ that each E A a i { B a i - o 9 Coxeter transformations. the two i + e 9 + i + from the system of i, 1 i # Wilhelm Killing his later years. which an ajk ~ have been a 0 way one obtains the f ( 4, 6, 7, various results in fully later to am not that have as it � 3 f = 1 f = 2 three. Replacing a i i # As far far p. 793] on the geometry of simple groups. The one minor error in Killing's classification was the exhibition of two exceptional groups of rank four. Cartan noticed that Killing's two root systems are easily seen to be equivalent. It is peculiar that Killing overlooked this since his mastery of calculation modified by A n B n C n D n D n n pp. 146-150]. The exceptional algebra of rank two [ 15, 15, p. 130]. The exceptional al- gebras F4, E a, E7, E 8 4, 6, respectively. The largest E 8 8 7]). Coxeter employed a graph to classify this type of group. During that the and which which contain an Appendix for charac- of Table Table p. 22] that aijaji {0, 1, 2, 3}. There is a one-to-one correspondence be- tween the Cartan matrices of graph correspond as in G 2 = 3. use the = 2 i # j, it difficult to A = = p. 21] imply that A is symmetrisable--that is, there exist non-zero numbers d i zero or non-zero non-zero in the USSR and Robert Moody [22] in Canada noticed that if Killing's conditions on (aij) were relaxed, it was still possible to associate to the Cartan matrix A a proving the the This paper was also basic to to and LeMire [19], who discussed infinite dimensional representations of finite Lie al- gebras. Chevalley's paper also initiated the current widespread exploitation of the universal associative enveloping algebras of Lie algebras--a concept first rigorously defined by Witt [28]. Among the Kac-Moody algebras the most tractable are the symmetrisable. The most extensively studied and applied are the affine Lie algebras which satisfy all Killing's conditions except that the the is 0. The Cartan matrices for the affine Lie algebras are in one-to-one correspondence with the graphs in the right-hand column of Table 1, which first appeared in [27]. Wilhelm Killing the Man Killing was born in Burbach in Westphalia, Germany, on 10 May 1847 and died in M~inster on 11 February 1923. Killing began university study in M~inster in 1865 but quickly moved to Berlin and came under the influence of Kummer and Weierstrass. His thesis, completed in March 1872, was supervised by At one THE MATHEMATICAL INTELLIGENCER VOL. 3, 1989 o... O~O 2 3 O n .., O~O 1 2 3 2 3 5 6 1 2 3 4 5 6 E 8 1 2 3 4 5 6 7 1 1 1 1 1 1 1 2 2 2 2 1 1 2 3 2 1 1 2 3 4 3 1 2 3 4 5 2 1 6 4 2 A 1 \f9 \f9 A 1 \f9 1 1 A I I 2 F 4 2 3 G 2 \f9 \f9 0 \f9 \f9 \f9 F 4 o \f9 1 2 3 4 2 \f9 \f9 \f9 F 4 \f9 1 2 3 2 1 B 1 0 O~O~O... \f9 n 1 2 2 2 2 1 1 1 1 O:=~=OmO~O.-. O~O=~=O 2 2 2 Cn:O~O~O... 0~0~0 O=~OmO~O... 0~0~---0, 2 2 2 2 1 2 i Cn: I~O~I,.- 1 2 2 2 Although he in his his active active from Braunsberg. These dealt with (i) Non-Euclidean geometries in n-dimensions (1883); (ii) "The Extension of the Concept of Space" (1884); and (iii) his first tentative thoughts about Lie's trans- formation groups (1886). Killing's original treatment of Lie algebras first appeared in (ii). It was only after this librarian to similar to Halley with to his period and gether with with p. 399]. His students loved and ad- 36 THE MATHEMATICAL INTELLIGENCER VOL. 3, 1989 Nor was Killing satisfied for them to become narrow spe- cialists, so he spread his lectures over many topics beyond geometry and groups. Killing was conservative in his the War old age for his Killing's Work achievement. His Once Z.v.G.IV the regular Fine Hall has ever dislike for for we find the following less than generous comment about . . to Engel Engel p. 221/2] there was no love lost between Lie and Killing. This comes through in the nine references to Killing's work in volume III of Wilhelm Killing, probably about 1889-1891. [20]. With one exception they are negative and seem to have the purpose of proving that anything of value about transformation groups was first discovered by Lie. Even if this were true, it does not do justice to the fact that there was no possibility of Killing in Brauns- berg knowing Lie's results published in Christiana. So if Lie's results are wonderful, Killing's independent discovery of them is equally wonderful! It seems to me that even Hawkins, more than else to the trouble example which will to of semi- basis for of abstract abstract (4) The Weyl group and the Coxeter transformation are in Z.v.G.II. There they are realized not as orthog- onal motions of Euclidean space but as permutations of the roots. In my view, this is the proper way to think of them for general Kac-Moody algebras. Fur- ther, the conditions for symmetrisability which play a key role in Kac's book [17] are given on p. 21 of Z.v.G.II. (5) It was Killing who discovered the exceptional Lie algebra Ea, which apparently is the main hope for saving Super-String Theory--not that I expect it to be saved! (6) Roughly one third of A. Borel 1885-1985," ed. 2. I. Lie Algebras, (1968), 344-361. W. Burnside, la Classification Comptes Rendus, (1948), 1136-1138. M. Coxeter, Annals of Math. (1934), 588-621. Bd. V F. Engel, Deut. Math. W. Feit (1963), 775-1029. Principia Mathematica for Hist. Historia Mathematica Archive for Hist. G. Kac, Lie algebras Nauk, USSR 1923-1967; English translation: Math. USSR Iz- 2 (1968), 1271-1311. G. Kac, "Infinite W. Killing, stetigen, endli- W. LeMire, "Weight spaces Leipzig (1888-1893). 21. R. V. P. Oellers, "Wilhelm Killing: Ein Religi6se Quellenschriften, Stimmen der Weyl "Mathematische H. Weyl, uous groups," Richard Brauer; Coxeter (1934-35). Darstellung Liescher (1937), 152-160. Department of Mathematics Queen's University Kingston, Ontario