# Affine Lie Algebras, Weight Multiplicities and Branching by Sam Kass; R. V. Moody

By Sam Kass; R. V. Moody

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Extra info for Affine Lie Algebras, Weight Multiplicities and Branching Rules I

Sample text

Derivation of L 9 and extends uni quely to a derivation of 11 if we impose Lo( = O. A. an aside, note that additional ope,ators L .. 18) Th is expl ... 16) shows that t he L.. are derivations of L9 and O. For most of the following development, we choose to adjoin only an Lo to the affine algebra , and not th e full Wiu algebra. jor issue •. The Witt algebra and it. 1 extension , th e Vir"""ro algeb ra '11 , will be furth er discussed in Parts II and III. S.... lgebr ... is included from the beginning.

Weight, 1101 " rool, i8 adjoined, "" sho"o"n in F'ig. 2-8. This is an example o f .. o lle parameter ramily of affine matrices whi ch i8 the only family nol formed by adjoining a rool. bove , bUllhe notation i. commOn and more suggelilive. ltix for Fig. 2-8 is A(Bq'» = ( -~ -~ -~ ) . ) 2 Fig. 2·11 show8 Ihe classifi cation of aJt twisted affine diagrams. 7 Let us t"ke Oil . first glance al the root lattice. I8Ociated with the direct affin e Car tan mat,ices. Take the finite dimensional ,oot sys tem ~ of the finite dimen8ionlll a1gebrll from ....

Subalg"b . of g. s \0 <"v (X ) where X E {I' only. V(A) does llot usually remai" irreducible in {I' . a the resu lt may be ext remely complex 1lIld there is no reason to expect that V( A) will decomp06e into .. direct 8Urn of irr educi ble IJ'- roodules. If, however, g' is reduclive, t th e siluation is well beha,-ed. A subillgebra. rn;\$imple. T here Me cMeS where a or " are empty. misimp le. If V(II, A) is an irreducible mooule of 0 with weight 8ystem OrA) E P(p), Ih en V(II, A) Cl\n be written as a.