Read e-book online Affine Lie Algebras, Weight Multiplicities and Branching PDF

By Sam Kass; R. V. Moody

ISBN-10: 0520067681

ISBN-13: 9780520067684

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

Example text

Tiolls of affine algebra>! at i. l'ticu l .. rl y uscful in physi co, i. t o cOllside r what h appen. to V(A) when it is vie wed .... II. ( _module whe", {I' is,. 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.

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 • . Thu. Ihe rank of the affin e Cartan matrix remains d.

Where some generators in th e adjoiD t rep resentation are antihermitian lUId olhers hermitian. are all inlinite dimens;omll .. nd ran inlo several different series. 4 Consider a finite di"",nsional module V eAl of a finite dimensional Lie al_ gebra {I(A). 9. ub/lpac ... "r E P, of the act ion of Ihe Cartan subalg ... bra On VeAl. T E P. This is merely II r... llltemeni of our beginning point in §I. where we emph""ized the physical significance of d-dimensiona[ latti ce. of addili,·e quantum numbers.

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


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