By Sam Kass; R. V. Moody
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Extra info for Affine Lie Algebras, Weight Multiplicities and Branching Rules I
Of affin e Cart"" matrices by adjoining th e neg .. li ve of th e highest ,oot to the b""" of simp le ' 0010. Fig. 2-4 illusl rat es th is for the affin inlion of C, to ql) and Fit;. 2-5 show. Ihe affini ... tion of G, 10 G~l). Th e Ca,lan matrix for ql) (row. 1e in order "'0, 0 1, 02) i. A(ql) = 2 -2 0) (o _\ 2 -2 _\ 2 . 5) Simi l.. rly Ihe Cartan matrix fo, affine-G, is -I 0) 2 -3 -I . nile Iyp .. f). , itt .. n in the form , (2. 7) " , LIE ALGEDRAS IN QUANTUM MEC HAN ICS • • n, ~ • • • n, • • n, Fig.
2-5. Th e sffiniut ion or G, by addition or lhe negative urlhe highest toot. 001 of t he lini le dimensional a lgebra; - 0'0 is the uniqu e . oot for whi ch Li e'''; is ,. muimum. The correspond ing m .. "thor8 810y untwisted ) .. mne CR r tan matrix . ffio e algebras. 8) ;,,0 with Co == 1. 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 • .
29) w.. , independent of the integers Vj(o) . Ion be al90 read M <>;«() == 0, i == 0, ... tes RI in ~. 14). As a. n m.. trix elements o o( h;) = (00 16;) = Ao;. 32) for i = 0, . ,d, aince Co = Co(no 1(0)/2 = I. ve M one expeels. nce. Howe,·e. y9 a crucia.! role in the representation theory of th ese algebras. , can be nOnZerO. tem of Ihe adjoint representation is Ihe set of affin e root. 6.. tions, whi ch we will beg in disclWling in S6, (always ""\8 118 scalar multiplication by a. n irred ucible representation.
Affine Lie Algebras, Weight Multiplicities and Branching Rules I by Sam Kass; R. V. Moody