Algebraic Theory of Locally Nilpotent Derivations / by Gene Freudenburg.

Format:

• Book

Author/Creator:

• Freudenburg, Gene, Author.

Series:

• Encyclopaedia of Mathematical Sciences ; 136

Language:

English

Subjects (All):

• Commutative algebra.
• Commutative rings.
• Algebraic geometry.
• Topological groups.
• Lie groups.
• Commutative Rings and Algebras.
• Algebraic Geometry.
• Topological Groups and Lie Groups.

Local Subjects:

• Commutative Rings and Algebras.
• Algebraic Geometry.
• Topological Groups and Lie Groups.

Physical Description:

1 online resource (XXII, 319 p.)

Edition:

2nd ed. 2017.

Place of Publication:

Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2017.

Summary:

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 includedin 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.

Contents:

• 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.

Notes:

Includes bibliographical references and index.

ISBN:

3-662-55350-3