Twisted forms of the determinant
type: Item
instance of: scholarly article
title: Twisted forms of the determinant
zbMATH Open document ID: 0531.10025
DOI: 10.1016/0021-8693(84)90055-3
author: William C. Waterhouse
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: Let f be a polynomial in n variables over the field k homogeneous of degree m. Think of f as a function on a k-vector space V of dimension n. Another homogeneous polynomial \(f_ 1\) on a k-space \(V_ 1\) is called a twisted form of f split by the extension L of k if there is an L-linear isomorphism \(V_ 1\otimes L\quad \to \quad V\otimes L\) transforming \(f_ 1\otimes id_ L\) to \(f\otimes id_ L\). Two twisted forms \(f_ 1,f_ 2\) are equivalent over k (up to scalar) if there is a k- isomorphism \(V_ 1\to V_ 2\) transforming \(f_ 1\) into \(f_ 2\) (respectively into a scalar multiple of \(f_ 2).\) In the paper under review the special case \(f=\det(x_{ij})\) is considered. It can be shown: Each twisted form of \(\det(x_{ij})\) can be split by a separable extension field and the equivalence classes of forms up to scalar correspond to the elements of \(H^ 1(Gal(k_ s/k),G(k_ s))\) where \(k_ s\) denotes the separable closure of k and G is the group of automorphisms leaving \(\det(x_{ij})\) semi-invariant, i.e. those automorphisms sending \(\det(x_{ij})\) to a scalar multiple of itself. For special fields k all equivalence classes of twisted forms of the determinant can be determined by the standard machinery of Galois cohomology. For instance over the reals the only real forms obtained from \(\det(x_{ij})\) by a complex linear change of variables are up to real equivalence the form \(\det(x_{ij})\), plus or minus the determinant on \(n\times n\) Hermitian matrices or plus or minus the reduced norm on quaternion matrices of suitable dimension.
Mathematics Subject Classification ID: 11E76
Mathematics Subject Classification ID: 12G05
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 20G10
zbMATH DE Number: 3841964
zbMATH Keywords: higher degree form
zbMATH Keywords: semisimple algebras
zbMATH Keywords: equivalence classes of twisted forms of determinant
zbMATH Keywords: Galois cohomology
reviewed by: Hans-Jochen Bartels
MaRDI profile type: Publication
full work available at URL: 0021-8693(84)90055-3
OpenAlex ID: W2025254879
cites work: A structure theory for Jordan algebras
cites work: Zur Arithmetik von Konjugationsklassen in algebraischen Gruppen
cites work: Q5596912
cites work: Q5622399
cites work: Structure groups and Lie algebras of Jordan algebras of symmetric elements of associative algebras with involution
cites work: Lectures on Galois cohomology of classical groups. With an appendix by T. Springer
cites work: Q3293461
cites work: The corestriction of algebraic structures
cites work: Q5509718
cites work: Galois Cohomology
cites work: Automorphism schemes and forms of Witt Lie algebras
cites work: Linear maps preserving reduced norms
cites work: Q3845400
description: scientific article; zbMATH DE number 3841964