Representations of Semisimple Lie Algebras in the BGG Category $\mathscr {O}$American Mathematical Soc., 2008 - Broj stranica: 289 This is the first textbook treatment of work leading to the landmark 1979 Kazhdan-Lusztig Conjecture on characters of simple highest weight modules for a semisimple Lie algebra $\mathfrak{g}$ over $\mathbb {C}$. The setting is the module category $\mathscr {O}$ introduced by Bernstein-Gelfand-Gelfand, which includes all highest weight modules for $\mathfrak{g}$ such as Verma modules and finite dimensional simple modules. Analogues of this category have become influential in many areas of representation theory. Part I can be used as a text for independent study or for a mid-level one semester graduate course; it includes exercises and examples. The main prerequisite is familiarity with the structure theory of $\mathfrak{g}$. Basic techniques in category $\mathscr {O}$ such as BGG Reciprocity and Jantzen's translation functors are developed, culminating in an overview of the proof of the Kazhdan-Lusztig Conjecture (due to Beilinson-Bernstein and Brylinski-Kashiwara). The full proof however is beyond the scope of this book, requiring deep geometric methods: $D$-modules and perverse sheaves on the flag variety. Part II introduces closely related topics important in current research: parabolic category $\mathscr {O}$, projective functors, tilting modules, twisting and completion functors, and Koszul duality theorem of Beilinson-Ginzburg-Soergel. |
Sadržaj
1 | |
Part I Highest Weight Modules | 11 |
Part II Further Developments | 179 |
271 | |
Frequently Used Symbols | 283 |
287 | |
Back Cover | 293 |
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Representations of Semisimple Lie Algebras in the BGG Category O James E. Humphreys Ograničeni pregled - 2021 |
Uobičajeni izrazi i fraze
antidominant weight arbitrary BGG Reciprocity BGG resolution Bruhat ordering central character Chapter cohomology composition factor multiplicities compute construction contravariant form coset D(ws defined denote dim Homo dimension direct sum dominant duality element embedding equivalent example Exercise finite dimensional module flag variety follows formal characters Grothendieck group highest weight module implies indecomposable induction injective integral weights involving isomorphic Jantzen filtration Kazhdan–Lusztig KL Conjecture KL polynomials Lemma Lie algebra Lie group linkage class Loewy M(so MA(a maximal submodule maximal vector MI(X notation parabolic subalgebra parabolic Verma modules positive roots projective functors projective module proof Proposition regular root system scalar self-dual semisimple Lie algebra short exact sequence shows simple modules simple reflection simple roots simple submodule socle standard filtration subalgebra subcategory subgroup submodule summand tilting module translation functors U(g)-module weight spaces Weyl group