Algebraic and Analytic Methods in Representation Theory by Bent Ørsted and Henrik Schlichtkrull (Eds.)

By Bent Ørsted and Henrik Schlichtkrull (Eds.)

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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.

2), 103 (1976), 557-591. [Ke2] ~ , Representations of algebraic groups in prime characteristics, Ann. Sci. Ecole Norm. Sup. 14 (1981), 61-74. [Lull G. Lusztig, Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, J. Amer. Math. Soc. 3 (1990). 257-296. [Lu2] ~ , Introduction to quantum groups, Progress in Mathematics, BirkhSuser, 1993. [Ma] O. Mathieu, Filtrations of G-modules, Ann. Sci. Ecole Norm. Sup. 23 (1990), 625-644. [Pa] J. Paradowski, Filtrations of modules over quantum algebras, Proc.

On a finer level, modules may differ appreciably in size and yet still have the same associated variety because the latter only sees the radical of grAn n U(0) m. Recall V(w) of Lecture 1. The involutivity of V(w) has an easy proof based on the observation that the orthogonal of n + N w(n +) in n - is a subalgebra. 1), which asserts that dim(O M n +) - 1 dim (_9 for any nilpotent orbit (9. A deeper analysis of orbital varieties results [St] from considering the Steinberg variety 8. This is defined as follows.

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