Invertible Polynomial Transformations

Document Type

Presentation Abstract

Presentation Date

4-22-1999

Abstract

The general affine group GAn(C) generalizes the more familiar general linear group GLn(C). In particular, GAn(C) is the set of one-to-one functions F = (F1, …, Fn) : Cn → Cn such that each Fi is a polynomial in n variables over C. Remarkably, such F are also onto, and F-1 is an element of GAn(C). So GAn(C) is indeed a group, consisting of the invertible polynomial transformations of Cn, with GLn(C) as a subgroup. The aim of this talk is to give an overview of what is known about this important and much-studied group.

The Structure Theorem for GA2(C) gives a fairly complete understanding in this case. For n >= 3, relatively little is known about the structure of GAn(C), except that it is amazingly complicated. For example, the tame subgroup Tn contained in GAn(C) is easy to define, and the Structure Theorem implies T2 =GA2(C), but it remains an open question whether Tn = GAn(C) for any n >= 3.

Naturally, one wishes to study certain kinds of group actions in which an "algebraic" group G acts "algebraically" on Cn, since these give rise to embeddings of G as a subgroup of GAn(C). Classically, the case in which G is a reductive group (like G = SLn(C)) has been studied since the Nineteenth Century, and many positive results are known. The case in which G is a unipotent group (like G = C+, the additive group of C) is not as well understood, though the importance of this case is widely recognized. Much of my own work has focused on actions of C+ on Cn.

Additional Details

Thursday, 22 April 1999
4:10 p.m. in MA 109
Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

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