Lectures on Noncommutative Geometry

Description

These Lectures are based on a course on noncommutative geom- etry given by the author in 2003. The lectures contain some standard material, such as Poisson and Gerstenhaber algebras, deformations, Hochschild cohomology, Serre functors, etc. We also discuss many less known as well as some new results such as noncommutative Chern-Weil theory, noncommutative symplectic geometry, noncommutative differential forms and double-tangent bundles 1. Introduction Basic notation. Acknowledgments. 2. Morita Equivalence 2.1. Categories and functors. 2.2. Algebras and spaces. 2.3. Morita theorem. 3. Derivations and Atiyah algebras 3.1. 3.2. Square-zero construction. 3.3. Super-derivations. 3.4. The tensor algebra of a bimodule. 3.5. Picard group of a category. 3.6. Atiyah algebra of a vector bundle. 4. The Bar Complex 4.1. Free product of algebras 4.2. 4.3. Second construction of the bar complex (after Drinfeld). 4.4. Third construction of the bar complex. 4.5. Reduced Bar complex. 5. Hochschild homology and cohomology 5.1. 5.2. Hochschild cohomology 5.3. Interpretation of HH0. 5.4. Interpretation of HH1. 5.5. Interpretation of HH2. 5.6. Interpretation of HH3. 5.7. Reduced cochain complex 6. Poisson brackets and Gerstenhaber algebras 6.1. Polyvector fields 6.2. Poisson brackets. 6.3. Gerstenhaber algebras. 6.4. -extension of a Gerstenhaber algebra. 6.5. Lie algebroids. 6.6. Gerstenhaber structure on Hochchild cochains. 6.7. Noncommutative Poisson algebras. 7. Deformation quantization 7.1. Star products. 7.2. 7.3. A Lie algebra associated to a deformation. 7.4. Example: deformations of the algebra `39`42`"613A``45`47`"603AEnd(E). 7.5. 7.6. Moyal-Weyl quantization. 7.7. Weyl Algebra 8. Kähler differentials 8.1. 9. The Hochschild-Kostant-Rosenberg Theorem 9.1. Smoothness. 9.2. From Hochschild complex to Chevalley-Eilenberg complex. 9.3. Proof of Theorem ??. 9.4. Digression: Applications to formality. 10. Noncommutative differential forms 10.1. 10.2. The differential envelope. 10.3. The universal square-zero extension. 10.4. Hochschild differential on non-commutative forms. 10.5. Relative differential forms 10.6. The Commutative Case 10.7. The Noncommutative Case 11. Noncommutative Calculus 11.1. 11.2. Operations on Hochschild complexes. 11.3. The functor of `functions' 11.4. Karoubi-de Rham complex. 11.5. The Quillen sequence. 11.6. The Karoubi Operator. 11.7. Harmonic decomposition. 11.8. Noncommutative polyvector fields. 12. The Representation Functor 12.1. 12.2. Traces. 12.3. Noncommutative Rep-scheme (following LBW). 12.4. The Rep-functor on vector fields. 12.5. Rep-functor and the de Rham complex. 12.6. Equivariant Cohomology 13. Double-derivations and the double-tangent bundle. 13.1. 13.2. 13.3. Double-derivations for a free algebra. 13.4. The Crawley-Boevey construction. 13.5. Sketch of proof of Theorem ??. 14. Noncommutative Symplectic Geometry 14.1. 14.2. Noncommutative `flat' space 15. Kirillov-Kostant Bracket 15.1. Coordinate formula. 15.2. Geometric approach. 15.3. Symplectic structure on coadjoint orbits. 15.4. The algebra OA 15.5. Drinfeld's bracket. 16. Review of (commutative) Chern-Weil Theory 16.1. 16.2. Connections on G-bundles. 16.3. Transgression map. 16.4. Chern-Simons formalism. 16.5. Special case: g=gln. 16.6. Quantized Weil algebra. 17. Noncommutative Chern-Weil theory 17.1. 17.2. Example: case A=. 17.3. Gelfand-Smirnov bracket. 18. Chern Character on K-theory 18.1. Infinite matrices. 18.2. The group K0(A). 18.3. Chern class on K0 and K1. 18.4. Chern classes via connections. 19. Formally Smooth Algebras 19.1. 19.2. Examples of formally smooth algebras. 19.3. Coherent modules and algebras. 19.4. Smoothness via torsion-free connection 20. Serre functors and Duality 20.1. 20.2. Serre duality. 20.3. Calabi-Yau categories. 20.4. Homological duality. 20.5. Auslander-Reiten functor. 20.6. Homologically smooth algebras. 21. Geometry over an Operad 21.1. References