Algebraic Theory of Locally Nilpotent Derivations (Encyclopaedia of Mathematical Sciences Book 136)

This book explores the theory and application of locally nilpotent derivations, a subject motivated by questions in affine algebraic geometry and having fundamental connections to areas such as commutative algebra, representation theory, Lie algebras and differential equations. The author provides a unified treatment of the subject, beginning with 16 First Principles on which the theory is based. These are used to establish classical results, such as Rentschler's Theorem for the plane and the Cancellation Theorem for Curves. More recent results, such as Makar-Limanov's theorem for locally nilpotent derivations of polynomial rings, are also discussed. Topics of special interest include progress in classifying additive actions on three-dimensional affine space, finiteness questions (Hilbert's 14 th Problem), algorithms, the Makar-Limanov invariant, and connections to the Cancellation Problem and the Embedding Problem. A lot of new material is included in this expanded second edition, such as canonical factorization of quotient morphisms, and a more extended treatment of linear actions. The reader will also find a wealth of examples and open problems and an updated resource for future investigations. This Book Explores The Theory And Application Of Locally Nilpotent Derivations, A Subject Motivated By Questions In Affine Algebraic Geometry And Having Fundamental Connections To Areas Such As Commutative Algebra, Representation Theory, Lie Algebras And Differential Equations. The Author Provides A Unified Treatment Of The Subject, Beginning With 16 First Principles On Which The Theory Is Based. These Are Used To Establish Classical Results, Such As Rentschler's Theorem For The Plane And The Cancellation Theorem For Curves. More Recent Results, Such As Makar-limanov's Theorem For Locally Nilpotent Derivations Of Polynomial Rings, Are Also Discussed. Topics Of Special Interest Include Progress In Classifying Additive Actions On Three-dimensional Affine Space, Finiteness Questions (hilbert's 14th problem), Algorithms, The Makar-limanov Invariant, And Connections To The Cancellation Problem And The Embedding Problem. A Lot Of New Material Is Included In This Expanded Second Edition, Such As Canonical Factorization Of Quotient Morphisms, And A More Extended Treatment Of Linear Actions. The Reader Will Also Find A Wealth Of Examples And Open Problems And An Updated Resource For Future Investigations. Introduction -- 1 first Principles -- 2 Further Properties Of Lnds -- 3 Polynomial Rings -- 4 Dimension Two -- 5 Dimension Three -- 6 Linear Actions Of Unipotent Groups -- 7 Non-finitely Generated Kernels -- 8 Algorithms -- 9 Makar-limanov And Derksen Invariants -- 10 Slices, Embeddings And Cancellation -- 11 Epilogue -- References -- Index. Gene Freudenburg. Previous Edition: 2006. Includes Bibliographical References (pages 299-313) And Index. Front Matter ....Pages i-xxii First Principles (Gene Freudenburg)....Pages 1-39 Further Properties of LNDs (Gene Freudenburg)....Pages 41-72 Polynomial Rings (Gene Freudenburg)....Pages 73-112 Dimension Two (Gene Freudenburg)....Pages 113-136 Dimension Three (Gene Freudenburg)....Pages 137-165 Linear Actions of Unipotent Groups (Gene Freudenburg)....Pages 167-191 Non-Finitely Generated Kernels (Gene Freudenburg)....Pages 193-216 Algorithms (Gene Freudenburg)....Pages 217-243 Makar-Limanov and Derksen Invariants (Gene Freudenburg)....Pages 245-264 Slices, Embeddings and Cancellation (Gene Freudenburg)....Pages 265-285 Epilogue (Gene Freudenburg)....Pages 287-298 Back Matter ....Pages 299-319