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

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The 8tarting point for this theory is the importanl resu lt Ih "t every such rep rese ntati on is completely reducih[e. esenlalions. 2 le t V be a finil e di mensional irreducible II-module. The follow ing wsu[to Me due to \Vey[: ( a) T here is a basis {v~} of V consistingof slales partially [ELhe[ed by addi_ ti ve quantum numbers 0 . weighl" in P. (b) There is a ,,·eight II lh"t is th e highest in Ih e sense of t he orderi ng defin ed in §L7, an d its coo. s_ (e) T he weight sp""" wi t h highest weighl ll has dimension I.

As do ne by Weyl (W- 39J. The 8tarting point for this theory is the importanl resu lt Ih "t every such rep rese ntati on is completely reducih[e. esenlalions. 2 le t V be a finil e di mensional irreducible II-module. The follow ing wsu[to Me due to \Vey[: ( a) T here is a basis {v~} of V consistingof slales partially [ELhe[ed by addi_ ti ve quantum numbers 0 . weighl" in P. (b) There is a ,,·eight II lh"t is th e highest in Ih e sense of t he orderi ng defin ed in §L7, an d its coo. s_ (e) T he weight sp""" wi t h highest weighl ll has dimension I.

Pac"s are isomorphic 10 ~ontinue8 . I,(el. G. 4 Consider the root system A of affine-Iua' imag ine an infinite number of sheets of pape r in II . tack exle nding in both d irect ions with an In3 root system d rawn o n each p"ge, "" shown ill F ig. 3-1. The n th page in Ihe . U3) + Z6. 8) The """lar product of two root. d oes not depend on the page Ilumb er. ero wi th the 0 1 on page 70

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