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23 (1990), 625-644. [Pa] J. Paradowski, Filtrations of modules over quantum algebras, Proc. Symp. Pure Math. 56 (1994), Part 2, 93-108. [Hi] C. M. Ringel, The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences, Math. Z. 208 (1991), 209-223. [RT] N. Reshetikhin and V. Turaev, Invariants of 3-manifolds via link polynomials and quantum groups, Invent. Math. 103, 547-597. Chapter 1 Modular Representations [Sw] 51 M. Sweedler, ttopf Algebras, Benjamin, 1969.

We prove next t h a t 9 ~ is surjective. 3). For arbitrary A E X ( T ) +, we choose r E N such t h a t (p'r-1)p+wo,k E X ( T ) + Note t h a t wo((p ~ - l ) p + w o t ) =-(S-1)p+I is the minimal weight in H°((p " - 1)p + woA), so t h a t we have an exact B-sequence 0 ~ A ~ H°((p r - 1)p + woA)® (p~ - 1)p -+ Q -~ 0. This gives us the diagram 0 1 , ; 2p) * + H°((p " - 1)p + w0A) ®H°((p ~ - 1)p) ' H°(Q) , H°((p r - 1)p + w0A) ®Hg((--p r- H x (Q* ® - 2 p ) * 1)p)* H. H. Andersen 28 As we have observed, the middle horizontal map 1 ® ~ is an isomorphism.

7 to a similar problem for affine Lie algebras. 7 is also solved (although there still seems to be some problem if (aij) is not symmetric) for all 1. 7. We conclude this section with a few remarks on injective modules for Uq. Set Stz = L q ( ( l - 1)p) (We are again in the situation where q E C is a n / t h root of 1. ) Then Stl is injective in ~q (this is in contrast with the situation for representations of G in characteristic p > 0, where no finite dimensional representation is ever injective).