Algebraic and Analytic Methods in Representation Theory by Bent Orsted

By Bent Orsted

This booklet is a compilation of numerous works from well-recognized figures within the box of illustration thought. The presentation of the subject is exclusive in delivering numerous assorted issues of view, which may still makethe ebook very priceless to scholars and specialists alike. provides numerous diversified issues of view on key issues in illustration thought, from the world over recognized specialists within the box

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N be a Cartan matrix, and choose a set of relatively prime integers d l , . . , dn such that (diaij) is symmetric. Fix also an indeterminate v, and set vi = v d~, i = 1 , . . , n . If m C N we write [m] = (v m - v - m ) / ( V - - V - I ) and [m]! = [ m ] [ m - 1]... [2][1]. For t C N, we set [t] - [m][m - 1]-.. [m - t + 1]/[t]!. , and [~]i for the corresponding expressions with v replaced by vi. 1 The quantum group attached to (aij) is the Q(r)-algebra U with generators Ei, Fi, Ki, K/--I, i = 1 , .

If m C N we write [m] = (v m - v - m ) / ( V - - V - I ) and [m]! = [ m ] [ m - 1]... [2][1]. For t C N, we set [t] - [m][m - 1]-.. [m - t + 1]/[t]!. , and [~]i for the corresponding expressions with v replaced by vi. 1 The quantum group attached to (aij) is the Q(r)-algebra U with generators Ei, Fi, Ki, K/--I, i = 1 , . . , n, and relations i) K i K ( -1 = 1 = K ~ I K i and KiKj = K j K i ii) K i E j K ( -1 - v ia i j Ej, iii) EiFj - FjEi iv/ z r+s=l--aij v) E K i F j K i -1 gi-gi - ~ij -- v-iaiJFj -1 vi - -1 v i 8 i (-1)s [1-aiJ 1 F[FjF~-O, r+s--1--aij 8 i#j i It turns out that U is a Hopf algebra with comultiplication A given by A ( K i ) - Ki®Ki, A ( E ~ ) - Ei®I+K~®Ei, A(Fi) - Fi®K~ - l + l ® F i , H.

4) Zr is an exact functor. The reason is that G~/B~ is affine, namely G~/B~ ~_ U¢, so that all higher sheaf cohomology of bundles on G~/Br vavishes. 5) dim Z~ (E) = pNr dim E, because dim k[U¢] = pN~ (cf. 6(iii)). 5 Let A E X(T). i) Z~()~) contains a unique irreducible Gr-submodule, which we denote Lr()~). ii) Any finite dimensional simple G~-module is isomorphic to some such L~()~). iii) If also # E X ( T ) , then L~()~) ~_ Lr(#) iff A - # (mod p X ( T ) ) . 6 Set Xp~(T) = {A E X(T) I 0 _< (A, av> < p" for all simple roots a}.

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