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2017 | Book

First Principles Read chapter Read first chapter

Algebraic Theory of Locally Nilpotent Derivations

Author: Prof. Dr. Gene Freudenburg

Publisher: Springer Berlin Heidelberg

Book Series : Encyclopaedia of Mathematical Sciences

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

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.

Table of Contents

Frontmatter
Chapter 1. First Principles
Abstract
Throughout this chapter, assume thatB is an integral domain containing a fieldk of characteristic zero.B is referred to as ak-domain.B ∗ denotes the group of units ofB and frac(B) denotes the field of fractions ofB. Further, Aut(B) denotes the group of ring automorphisms ofB, and Aut k (B) denotes the group of automorphisms ofB as ak-algebra. IfA ⊂B is a subring, then tr.deg A B denotes the transcendence degree of frac(B) over frac(A). The ideal generated byx 1, …, x n ∈B is denoted by either (x 1, …, x n ) orx 1 B + ⋯ +x n B. The ring ofm ×n matrices with entries inB is indicated by \(\mathcal{M}_{m\times n}(B)\) and the ring ofn ×n matrices with entries inB is indicated by \(\mathcal{M}_{n}(B)\). The transpose of a matrixM isM T .
Gene Freudenburg
Chapter 2. Further Properties of LNDs
Abstract
The first three sections of this chapter investigate derivations in the case B has one or more nice divisorial properties, in addition to the ongoing assumption that B is a commutative k-domain, where k is a field of characteristic zero. Subsequent sections discuss quasi-extensions, G-critical elements, the degree of a derivation, trees and cables, exponential automorphisms, construction of kernel elements by transvectants and Wronskians, and recognition of polynomial rings.
Gene Freudenburg
Chapter 3. Polynomial Rings
Abstract
This chapter investigates locally nilpotent derivations in the case B is a polynomial ring in a finite number of variables over a field k of characteristic zero. Equivalently, we are interested in the algebraic actions of \(\mathbb{G}_{a}\) on \(\mathbb{A}_{k}^{n}\).
Gene Freudenburg
Chapter 4. Dimension Two
Abstract
This chapter examines locally nilpotent R-derivations of R[x, y] = R [2] for certain rings R containing \(\mathbb{Q}\). This set is denoted LND R (R[x, y]).
Gene Freudenburg
Chapter 5. Dimension Three
Abstract
In the study of \(\mathbb{G}_{a}\)-actions on \(\mathbb{A}^{n}\), the dimension-three case stands between the fully developed theory in dimension two, and the wide open possibilities in dimension four. The fundamental theorems of this chapter show that many important features of planar \(\mathbb{G}_{a}\)-actions carry over to dimension three: Every invariant ring is a polynomial ring; the quotient map is always surjective; and free \(\mathbb{G}_{a}\)-actions are translations. It turns out that none of these properties remains generally true in dimension four. In addition, there exist locally nilpotent derivations in dimension three of maximal rank 3, and these have no counterpart in dimension two. Such examples will be explored in this chapter.
Gene Freudenburg
Chapter 6. Linear Actions of Unipotent Groups
Abstract
Invariant theory originally concerned itself with groups of vector space transformations, so the linear algebraic \(\mathbb{G}_{a}\)-actions were the first \(\mathbb{G}_{a}\)-actions to be studied. The action of \(SL_{2}(\mathbb{C})\) on the vector space V n of binary forms of degree n has an especially rich history, dating back to the mid-Nineteenth Century. The ring of SL 2-invariants, together with the ring of \(\mathbb{G}_{a}\)-invariants for the subgroup \(\mathbb{G}_{a} \subset SL_{2}(\mathbb{C})\), were the focus of much research at the time. A fundamental result due to Gordan (1868) is that both \(k[V _{n}]^{SL_{2}(\mathbb{C})}\) and \(k[V _{n}]^{\mathbb{G}_{a}}\) are finitely generated [182]. Gordan calculated generators for these rings up to n = 6. These historical developments are discussed in Sect.  6.3.1 .
Gene Freudenburg
Chapter 7. Non-Finitely Generated Kernels
Abstract
The Maurer-Weitzenbr̈ock Theorem for linear \(\mathbb{G}_{a}\)-actions on affine space does not generalize to non-linear \(\mathbb{G}_{a}\)-actions. In 1990, Paul Roberts [359] gave the first examples of non-affine invariant rings for \(\mathbb{G}_{a}\)-actions on an affine space. These examples involved actions of \(\mathbb{G}_{a}\) on \(\mathbb{A}^{7}\) over a field k of characteristic zero, and are counterexamples to Hilbert’s Fourteenth Problem. Subsequent examples of \(\mathbb{G}_{a}\)-actions of non-finite type were constructed in Freudenburg [163] and in Daigle and Freudenburg [79] for \(\mathbb{A}^{6}\) and \(\mathbb{A}^{5}\), respectively.
Gene Freudenburg
Chapter 8. Algorithms
Abstract
We have seen that the invariant ring of a \(\mathbb{G}_{a}\)-action on an affine variety need not be finitely generated as a k-algebra. But in many cases, most notably in the linear case, the invariant ring is known to be finitely generated, and in these cases it is desirable to have effective means of calculating invariants. In this chapter, we consider constructive invariant theory for \(\mathbb{G}_{a}\)-actions.
Gene Freudenburg
Chapter 9. Makar-Limanov and Derksen Invariants
Abstract
In 1994, a meeting entitled “Workshop on Open Algebraic Varieties” was held at McGill University. This meeting was organized by Peter Russell, who at the time was working with Mariusz Koras to solve the Linearization Problem for \(\mathbb{C}^{{\ast}}\)-actions on \(\mathbb{C}^{3}\). A key remaining piece of their work was to decide whether certain hypersurfaces in \(\mathbb{C}^{4}\) were algebraically isomorphic to \(\mathbb{C}^{3}\).
Gene Freudenburg
Chapter 10. Slices, Embeddings and Cancellation
Abstract
The Zariski Cancellation Problem can be viewed as a descendant of Zariski’s cancellation question for fields; see Sect.  1.1.2 . It can be stated as follows.
Gene Freudenburg
Chapter 11. Epilogue
Abstract
Many open questions, ranging from specific cases to broader themes, have already been posed and discussed in the foregoing chapters. A solution to the Embedding Problem or Cancellation Problem for complex affine spaces would reverberate across the whole of algebra, and we have seen how locally nilpotent derivations might play a role in their solution. Following are several additional directions for future inquiry.
Gene Freudenburg
Backmatter
Metadata
Title
Algebraic Theory of Locally Nilpotent Derivations
Author
Prof. Dr. Gene Freudenburg
Copyright Year
2017
Publisher
Springer Berlin Heidelberg
Electronic ISBN
978-3-662-55350-3
Print ISBN
978-3-662-55348-0
DOI
https://doi.org/10.1007/978-3-662-55350-3

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