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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.

Here and from now on, it will be assumed that g is semisimple. Identify g* with g through the Killing form. Call x E g* nilpotent if ad~ x is a nilpotent derivation. This property is preserved under the action of G, and so it is natural to define a nilpotent orbit in g* to be one consisting of just nilpotent elements. 1 ([BBr2, J5, Vo3]) For each J e PrimU(g), the associated variety V(gr J) of J is the closure of a nilpotent orbit. Thus, for each nilpotent orbit 49 in g* it makes sense to consider the subset X(O) of X - Prim U(g) of primitive ideals whose associated variety is 49.

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. Choose also q E k (where k is an algebraically closed field of characteristic p > 0 as in previous sections) such that q is a primitive lth root of unity (so p ~ 1).