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Preface; Introduction; References; Contents; List of Contributors; 1 A Short Survey on Lie Theory and Finsler Geometry; 1.1 Generalities; 1.1.1 Representations, Derivations, Killing Form; 1.1.2 The Weyl Algebra of Differential Operators; 1.1.3 Lie Groups, Exponential Map; 1.1.4 Compact Lie Algebras; 1.2 Non-associative Structures; 1.2.1 Loops and Geometry; 1.2.2 Gerstenhaber Algebras; 1.3 Finsler Geometry; 1.3.1 Finsler Metric and Minkowski Norm; 1.3.2 Geodesic and Curvature; 1.3.3 Local Homogeneity and Global Homogeneity; 1.3.4 Some Recent Progress; References
2.5 ConclusionsReferences; 3 Character, Multiplicity, and Decomposition Problems in the Representation Theory of Complex Lie Algebras; 3.1 Introduction; 3.2 The Character Problem and the Kazhdan-Lusztig Conjecture; 3.2.1 Highest Weights; 3.2.2 Modules with Highest Weight; 3.2.3 The Character; 3.2.4 The Character Formula of Weyl; 3.2.5 The Hecke Algebra and Kazhdan-Lusztig Polynomials; 3.2.6 A Generalization of Weyl's Character Formula: The Conjecture of Kazhdan and Lusztig; 3.2.7 The Multiplicity Version of the Character Problem; 3.3 Category O; 3.3.1 The Block Decomposition of O
3.3.2 Objects Admitting a Verma Flag3.3.3 Projectives in O; 3.3.4 Translation Functors; 3.3.5 The Decomposition Problem I; 3.4 Deforming the Category O; 3.4.1 Deformed Verma Modules and Verma Flags; 3.4.2 Simple Objects in OA; 3.4.3 Projectives in OA; 3.4.4 The BGG-Reciprocity; 3.4.5 Another Result of Bernstein-Gelfand-Gelfand; 3.4.6 The Block Decomposition of OA; 3.4.7 On the Structure of Some Projectives; 3.4.8 Endomorphism Rings in the Subgeneric Situations; 3.5 Soergel's Theory; 3.5.1 Endomorphisms of Multiplicity Free Projectives; 3.5.2 Big Projectives
3.5.3 The Strukturfunktor and the Struktursatz3.5.4 The Combinatorics of Translation Functors; 3.6 Soergel Bimodules, Parity Sheaves, and Moment Graph Sheaves; 3.6.1 Moment Graph Sheaves; 3.6.2 Soergel Bimodules; 3.6.3 Parity Sheaves; 3.6.4 Another Proof of the Kazhdan-Lusztig Conjecture; 3.7 An Epilogue: The Modular Case; References; 4 The BCH-Formula and Order Conditions for Splitting Methods; 4.1 Introduction; 4.2 Formal Power Series; 4.3 Reformulation Using Formal Differentiation; 4.4 The Baker-Campbell-Hausdorff Formula; 4.5 Splitting Schemes; 4.6 Computing Order Conditions in Examples
2.5 ConclusionsReferences; 3 Character, Multiplicity, and Decomposition Problems in the Representation Theory of Complex Lie Algebras; 3.1 Introduction; 3.2 The Character Problem and the Kazhdan-Lusztig Conjecture; 3.2.1 Highest Weights; 3.2.2 Modules with Highest Weight; 3.2.3 The Character; 3.2.4 The Character Formula of Weyl; 3.2.5 The Hecke Algebra and Kazhdan-Lusztig Polynomials; 3.2.6 A Generalization of Weyl's Character Formula: The Conjecture of Kazhdan and Lusztig; 3.2.7 The Multiplicity Version of the Character Problem; 3.3 Category O; 3.3.1 The Block Decomposition of O
3.3.2 Objects Admitting a Verma Flag3.3.3 Projectives in O; 3.3.4 Translation Functors; 3.3.5 The Decomposition Problem I; 3.4 Deforming the Category O; 3.4.1 Deformed Verma Modules and Verma Flags; 3.4.2 Simple Objects in OA; 3.4.3 Projectives in OA; 3.4.4 The BGG-Reciprocity; 3.4.5 Another Result of Bernstein-Gelfand-Gelfand; 3.4.6 The Block Decomposition of OA; 3.4.7 On the Structure of Some Projectives; 3.4.8 Endomorphism Rings in the Subgeneric Situations; 3.5 Soergel's Theory; 3.5.1 Endomorphisms of Multiplicity Free Projectives; 3.5.2 Big Projectives
3.5.3 The Strukturfunktor and the Struktursatz3.5.4 The Combinatorics of Translation Functors; 3.6 Soergel Bimodules, Parity Sheaves, and Moment Graph Sheaves; 3.6.1 Moment Graph Sheaves; 3.6.2 Soergel Bimodules; 3.6.3 Parity Sheaves; 3.6.4 Another Proof of the Kazhdan-Lusztig Conjecture; 3.7 An Epilogue: The Modular Case; References; 4 The BCH-Formula and Order Conditions for Splitting Methods; 4.1 Introduction; 4.2 Formal Power Series; 4.3 Reformulation Using Formal Differentiation; 4.4 The Baker-Campbell-Hausdorff Formula; 4.5 Splitting Schemes; 4.6 Computing Order Conditions in Examples