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Extra resources for Affine Lie Algebras, Weight Multiplicities and Branching Rules I
Product for the affine ~ . will h modified in j5, since VA is not invertible. 5. E XT ENDE D W E IG HT LATTICES 5 .
Sllion. wback to the gene ratoR and relations approach to constructing Lie algebr .... i8 Ih .. jlll to m&ke it .. pp .... ent what \I(A) really looks like. For in stance, when i. g(A) finite dimensional? With the extension c .. 22), we add to (3. ;] = ~o,(Lo)J;. where o;(Lo) is constr .. 25) IhIlt 6(Lo) = ~ l. 10 The loop .. Igebra ronstruelion of Ihe dired alIine ILlgebrM yields .. y of the root system in ter"", of the root. 1 Lie .. Igebra g from whi ch the alIine algebra is const ructed: one adjoint representation of g occurS ..
S_ (e) T he weight sp""" wi t h highest weighl ll has dimension I. In olhe r wo. s, 11 th;), i "" l, ... ,d. These numbers lie in Z ~". epre8C ntation: if V' i. an_ ol he. nd V' areequi,·"lent ifll-n d only if II "" II '. ThIlS, II uniquely II-nd comp letely labels an irrM ucib[e representll-lion, 00 "'e ofi en denote II-n irreducihle mo