By Bent Orsted

ISBN-10: 0126254400

ISBN-13: 9780126254402

This booklet is a compilation of numerous works from well-recognized figures within the box of illustration conception. The presentation of the subject is exclusive in supplying numerous various issues of view, which should still makethe booklet very necessary to scholars and specialists alike. offers numerous diversified issues of view on key issues in illustration idea, from the world over identified specialists within the box

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**Extra info for Algebraic and Analytic Methods in Representation Theory (Perspectives in Mathematics)**

**Sample text**

U°). ¢~ - ~ Cq is defined by Hqo (V) = F ( H o m G- uo (Uq, V)), v~¢;. Here, Uq is considered a UqU°-module via left multiplication, and the Uq-structure on H ° ( V ) comes via right multiplication on Uq" u . f : x ~ f(xu), x , u 6 Uq, f C H ° ( V ) . H. H. Andersen 42 It is straightforward to check that H ° satisfies Frobenius reciprocity, etc. We leave to the reader the task of verifying that the results from Section 4 on remain true when we add index q. 4). We shall now indicate how one may derive the special case of this that we need from the classical case: Let Z be the A-algebra obtained by specializing v to 1.

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 , . . , 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.

I Suppose A is a k-algebra. Let k ~ A be the structure map. Then, for each r C Z, we define a new k-algebra A (r) by setting A (r) = A as a ring H. H. Andersen 32 and choosing the composite k~-2fki-~A as the new structure map. , F(x) - x p, x C k. If X is a functor from {k-algebras} into {Sets}, then we denote by X (r) the functor given by X(~)(A) - X(A(-r)), A e {k-algebras}. The r t h Frobenius morphism on X is the morphism x x given by F~:(A)" X ( A ) X ~ ) X ( A where "yr"A ~ (-~)) A (-r) is the k-algebra homomorphism that takes a into a p~, a C A .

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