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Extra info for Affine Lie Algebras, Weight Multiplicities and Branching Rules I
Typ ically, il will be d ear from conte~t. Recall the inner product (-I') on (' and normalize it so thM (olet) = 2 for long root8. We ""tend (,1,) to He (' by requiring that (61 el) = for al l et E He \' . The" (Ol) i. It n . 1Y, it is the on ly one tbat i. useful in the subsequent theory. It will become usefu l in conjunction with yel ... fu rther exlension of t ' and of ('1')' For e&eh et + n6 = (n,o) E Lt:. , there is a 8ubalgeb,a . 7) 31 LIE ALGEBRAS IN QUANTUM MEC HANICS 3 -2 a:+ 2t1, " ""--, • '"" +5 a, 0, - , \ Cf, - 26 • + (, 0 • ""--0 ""---, <), a, I 0' 1 - fi 2 • ""-- " Fig.
Th e c;, i given in Fig. 2-10. = 0, ... re caUed the m .. rks of "mne-I'I(A}; t hey Me S ince the adjOiMd root is linearly dependent o n t he o thers in t· of the finite dimen.. trix (and I)A ) is ~ero. The affin e ClLllan matrix i8 now a (d + I) X (d + I) maMix with one row (or column) linearly dependent o n the o th er • . Thu. Ihe rank of the affin e Cartan matrix remains d. Wh en we make algch . ,. ponding 10 Ihese Cartan matri ces. hes among the d + 1 simple roo ts. The . cal"r product in Ihe correspo nd ing sp"-Ge ~ .
The redudive maximal subalgeb r.... with a u(l) factor of U are obtained by removing My two nodes wh""" mlLr ks are both equal to I. ubalgebra . ,·eral solutions to Ihe prob lem of finding the multiplicity til of a weight, of a representation with highest ,,·eight A. One solulion is that of [t'- 54). 13) • • >0 where p = (I, I, ... , I) E P is half the "urn of all positi'·e roots, and the sum on Ie is over positi,·e integers for which, + ka is a weight ("lH<> has "Ire"dy been comp"ted). Th e t'reudenthal formula ....