Quadratic and cubic invariants of unipotent affine automorphisms

V. V. Bavula, T. H. Lenagan

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Abstract

Let K be an arbitrary field of characteristic zero, P-n := K|chi(1),...,chi(n)| be a polynomial algebra, and P-n,P-chi 1 := K|chi(-1)(1), chi(1,)...,chi(n)|, for n >= 2. Let sigma' is an element of Aut(K)(P-n) be given by

chi 1 bar right arrow chi(1), chi(2) bar right arrow chi(2) + chi(1,) ... chi(n) bar right arrow chi(n) + chi(n-1.)

It is proved that the algebra of invariants, F-n' := P-n(sigma)', is a polynomial algebra in n - 1 variables which is generated by |n/2| quadratic and |n-1/2| cubic (free) generators that are given explicitly. Let sigma is an element of Aut(K) (P-n) be given by

chi(1) bar right arrow chi(1), chi(2) bar right arrow chi(2) + chi(1), ... chi(n) bar right arrow chi(n) + chi(n-1).

It is well known that the algebra of invariants, F-n := P-n(sigma), is finitely generated (theorem of Weitzenbock [R. Weitzenbock, Uber die invarianten Gruppen, Acta Math. 58 (1932) 453-494]), has transcendence degree n - 1, and that one can give an explicit transcendence basis in which the elements have degrees 1, 2, 3, ... , n - 1. However, it is an old open problem to find explicit generators for F-n. We find an explicit vector space basis for the quadratic invariants, and prove that the algebra of invariants P-n,chi 1(sigma) is a polynomial algebra over K|chi(1), chi(-1)(1)| in n - 2 variables which is generated by |n-1/2| quadratic and |n-2/2| cubic (free) generators that are given explicitly.

The coefficients of these quadratic and cubic invariants throw light on the 'unpredictable combinatorics' of invariants of affine automorphisms and of SL2-invariants. (C) 2008 Elsevier Inc. All rights reserved.

Original languageEnglish
Pages (from-to)4132-4155
Number of pages24
JournalJournal of Algebra
Volume320
Issue number12
DOIs
Publication statusPublished - 15 Dec 2008

Keywords

  • Invariants
  • Unipotent automorphism

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