64 IX. The Exceptional Lie Algebras
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1 64 IX. Te Exceptional Lie Algebras IX. Te Exceptional Lie Algebras We ave displayed te four series of classical Lie algebras and teir Dynkin diagrams. How many more simple Lie algebras are tere? Surprisingly, tere are only ve. We may prove tis by considering a set of vectors (candidates for simple roots) i H 0 and dening a matrix (analogous to te Cartan matrix) 1 : M ij =2 i j i j j i (IX:1) and an associated diagram (analogous to te Dynkin diagram), were te i t and j t points are joined by M ij M ji lines. Te set i is called allowable, (in Jacobson's usage) if i: Te i are linearly independent, tat is, if det M 6= 0. ii: M ij 0 for i 6= j. iii: M ij M ji =0 1 2 or 3:
2 IX. Te Exceptional Lie Algebras 65 Wit tese denitions, we can prove a series of lemmas: 1. Any subset of an allowable set is allowable. Proof: Since a subset of a linearly independent set is linearly independent, (i) is easy. Equally obvious are (ii) and (iii). 2. An allowable set as more points tan joined pairs. Proof: Let = Pi i i i i ; 1 2. Since te set is linearly independent, 6= 0so i > 0. Tus X i j i 0 < i = 2 1 i<j i i i 2 j j i 0 < ;X i<j no. of points [M ij M ji ] no. of points : (IX:2) For eac pair of joined points, M ij M ji is at least unity, so no. of joined pairs < no. of points. 3. An allowable set's diagram as no loops. Proof: If it did, tere would be a subset wit at least as many joined pairs as points. 4. If an allowable set as a diagram wit a cain of points joined only to successive points by single lines, tere is an allowable set wose diagram is te same except tat te cain P is srunk to a point. Proof: Let te cain be 1 2 ::: m and let = i i.now i = X i i i i +2 X i<j i j i = m 1 1 i;(m ; 1) 1 1 i = 1 1 i (IX:3) so is te same size as te individual points in te cain. Moreover, if is joined to te cain at te end,say to 1, ten 1 i = i, since j i = 0 for all j 6= 1.
3 66 IX. Te Exceptional Lie Algebras 5. No more tan tree lines emanate from a vertexofanallowable diagram. Proof: Suppose 1 2 ::: m are connected to 0. Ten i j i =0 for i j 6= 0 since tere are no loops. Since 0 is linearly independent of te i,its magnitude squared is greater tan te sum of te squares of its components along te ortogonal directions i i i i ; 1 2 : 0 0 i >X 0 i i 2 i i i ;1 : (IX:4) i Tus 4 > P i M 0iM i0. But M 0i M i0 is te number of segments joining 0 and i. 6. Te only allowable conguration wit a triple line is 7. An allowable diagram mayave onevertex wit tree segments meeting at a point, but not more. It may ave one double line segment, but not more. It may not ave bot. Proof: In eac of tese instances, it would be possible to take a subset of te diagram and srink a cain into a point so tat te resulting diagram would ave apoint wit more tan tree line emanating from it. Note tat tis means tat a connected diagram can ave roots of at most two sizes, and we encefort darken te dots for te smaller roots. 8. Te diagrams x x and x x x
4 IX. Te Exceptional Lie Algebras 67 are not allowable. Proof: Consider te determinant ofm for te rst diagram: 2 ; ;1 2 ; ;2 2 ; ;1 2 ;175 : ;1 2 We see tat if we add te rst and last columns, plus twice te second and fourt, plus tree times te tird, we get all zeros. Tus te determinant vanises. Te matrix for te second diagram is just te transponse of te rst. 9. Te only diagrams wit a double line segment wicmay be allowable are of te form: x x... x... x x x 10. By (7) above, te only diagrams wit a branc in tem are of te form:
5 68 IX. Te Exceptional Lie Algebras Te diagram below is not allowable Proof: 2 Te matrix for te diagram is: 64 2 ; ;1 2 ; ;1 2 ;1 0 ; ;1 2 ; ; ; ; ; Straigtforward manipulation like tat above sows tat te determinant vanises.
6 IX. Te Exceptional Lie Algebras Te only allowable diagrams wit a branc in tem are of te form:... /// \ \\\ / / \ Te diagram below is not allowable. Tis is proved simply by evaluating te associated determinant and sowing it vanises.
7 70 IX. Te Exceptional Lie Algebras 14. Te complete list of allowable congurations is A n... B n... x C n x x... x D n... /// \ \\\ / / \ G 2 x
8 IX. Te Exceptional Lie Algebras 71 F 4 x x E 6 E 7 E 8 Above are given te names use to designate te ve exceptional Lie algebras. So far we ave only excluded all oter possibilities. In fact, tese ve diagrams do correspond to simple Lie algebras.
9 72 IX. Te Exceptional Lie Algebras Footnote 1. Trougout te capter we follow te approac of JACOBSON, pp. 128{ 135. Exercise 1. Prove #11 and #13 above.
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1. Consider the trigonometric function f(t) whose graph is shown below. Write down a possible formula for f(t).
. Consider te trigonometric function f(t) wose grap is sown below. Write down a possible formula for f(t). Tis function appears to be an odd, periodic function tat as been sifted upwards, so we will use
, meant to remind us of the definition of f (x) as the limit of difference quotients: = lim
Mat 132 Differentiation Formulas Stewart 2.3 So far, we ave seen ow various real-world problems rate of cange and geometric problems tangent lines lead to derivatives. In tis section, we will see ow to
Solution for the Homework 4
Solution for te Homework 4 Problem 6.5: In tis section we computed te single-particle translational partition function, tr, by summing over all definite-energy wavefunctions. An alternative approac, owever,